Numerical Investigation of
Convection-Diffusion-Reaction Systems
By
Waqas Ashraf
CIIT/FA09-PMT-004/ISB
PhD Thesis
In
Mathematics
COMSATS Institute of Information Technology
Islamabad-Pakistan
Fall, 2013
ii
COMSATS Institute of Information Technology
Numerical Investigation of
Convection-Diffusion-Reaction Systems
A Thesis Presented to
COMSATS Institute of Information Technology, Islamabad
In partial fulfillment
of the requirement for the degree of
PhD Mathematics
By
Waqas Ashraf
CIIT/FA09-PMT-004/ISB
Fall, 2013
iii
Numerical Investigation of
Convection-Diffusion-Reaction Systems
A Post Graduate Thesis submitted to the Department of Mathematics as partial
fulfillment of the requirement for the award of Degree of PhD.
Name Registration No.
Waqas Ashraf CIIT/FA09-PMT-004/ISB
Supervisor
Dr. habil. Shamsul Qamar
Associate Professor Department of Mathematics
Islamabad Campus.
COMSATS Institute of Information Technology (CIIT)
Islamabad.
November, 2013
iv
Final Approval
This thesis titled
Numerical Investigation of
Convection-Diffusion- Reaction Systems
By
Waqas Ashraf
CIIT/FA09-PMT-004/ISB
Has been approved
For the COMSATS Institute of Information Technology, Islamabad.
External Examinaer1:_______________________________________________
External Examinaer2:_______________________________________________
Supervisor:_______________________________________________
Dr. habil. Shamsul Qamar
Associate Professor CIIT, Islamabad.
Head of Department:_______________________________________________
Dr. Moiz-ud-Din Khan
Professor CIIT, Islamabad
Dean, Faculty of Sciences:_______________________________________________
Professor Dr. Arshad Saleem Bhatti
v
Declaration
I, Waqas Ashraf registration# FA09-PMT-004/ISB, hereby declare that I have produced
the work presented in this thesis, during the scheduled period of study. I also declare that
I have not taken any material from any source except referred to wherever due that
amount of plagiarism is within acceptable range. If a violation of HEC rules on research
has occurred in this thesis, I shall be liable to punishable action under the plagiarism rules
of the HEC.
Date: ____________ Signature of student:
Waqas Ashraf
CIIT/FA09-PMT-004/ISB
vi
Certificate
It is certified that Waqas Ashraf registration# FA09-PMT-004/ISB has carried out all the
work related to this thesis under my supervision at the department of Mathematics,
COMSATS Institute of Information Technology, Islamabad and the work fulfills
the requirement for award of PhD degree.
Date: _______________
Supervisor:
Dr. habil. Shamsul Qamar
Associate Professor CIIT, Islamabad
Head of Department:
Prof. Dr. Moiz-ud-Din Khan
Professor CIIT, Islamabad
vii
DEDICATION
I dedicate this thesis to my loving family
.
ABSTRACT
Numerical Investigation of Convection Diffusion Reaction Systems
This work is concerned with the numerical solution of selected convection-diffusion-reaction
(CDR) type mathematical models with dominating convective and reactive terms, coupled
with some algebraic equations. Five established CDR-type models are analyzed namely, the
gas-solid reaction, chemotaxis, liquid chromatography, radiation hydrodynamical, and hy-
perbolic heat condition models. These models are encountered in various scientific and engi-
neering fields, such as chemical engineering, biological systems, astrophysics, heat transfer,
and fluid dynamics. The Laplace transformation is applied as a basic tool to find the ana-
lytical solutions of linear CDR models for different types of boundary conditions. However,
for the nonlinear models, numerical techniques are the only tools to get physical solutions.
The nonlinear transport and stiff source (reaction) terms of the governing differential equa-
tions produce discontinuities and narrow peaks in the solution. It is difficult to capture
steep variations in the solution through a less accurate numerical scheme. Therefore, ef-
ficient and accurate numerical methods are needed to obtain physically reliable solutions
in acceptable computational time. The objective of this thesis project is to develop and
implement simpler, robust, and accurate numerical frameworks for the solution of one and
two-dimensional CDR type systems. The space-time CE/SE-method, the discontinuous
Galerkin (DG) finite element method , and different high resolution finite volume schemes
(FVSs) are proposed to numerically approximate the solution of these models. Several
case studies are carried out. The validity and performance of the suggested numerical
techniques are revealed through test problems and by comparing their results with each
other, analytical solutions, and the results of some available finite volume schemes in the
literature.
ix
Contents
1 Introduction 1
1.1 Problem and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Investigated CDR-Type Models . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Gas-solid reaction and chemotaxis models . . . . . . . . . . . . . . 3
1.2.2 Chromatographic models . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Radiation hydrodynamical models . . . . . . . . . . . . . . . . . . . 5
1.2.4 Hyperbolic heat conduction model . . . . . . . . . . . . . . . . . . 5
1.3 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Desired Performance of a Numerical Method . . . . . . . . . . . . . . . . . 8
1.5 Proposed Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5.1 The space time CE/SE-method . . . . . . . . . . . . . . . . . . . . 10
1.5.2 The discontinuous Galerkin finite element method . . . . . . . . . . 10
1.5.3 The central schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5.4 The flux-limiting finite volume schemes (FVSs) . . . . . . . . . . . 12
1.6 The Laplace Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 Achievements of the Project . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.8 Thesis Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.8.1 Published and accepted articles: . . . . . . . . . . . . . . . . . . . . 15
2 Fundamentals of Convection-Diffusion-Reaction Systems 17
2.1 Derivation of Linear Scalar CDR Equation . . . . . . . . . . . . . . . . . . 18
2.2 Analytical Solutions of Linear CDR Models . . . . . . . . . . . . . . . . . 19
x
2.2.1 A linear single CDR-equation . . . . . . . . . . . . . . . . . . . . . 20
2.2.2 A linear CDR-system of two equations . . . . . . . . . . . . . . . . 21
2.3 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Hyperbolic Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Models of Gas-Solid Reaction and Kelar-Segal for Chemotaxis 28
3.1 The Gas-Solid Reactions (GSRs) Model . . . . . . . . . . . . . . . . . . . . 29
3.1.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1.2 The one-dimensional CE/SE-method . . . . . . . . . . . . . . . . . 32
3.1.3 The one-dimensional central schemes . . . . . . . . . . . . . . . . . 36
3.2 The Keller-Segel (KS) model . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.2 The two-dimensional CE/SE-method . . . . . . . . . . . . . . . . . 39
3.2.3 The two-dimensional central scheme . . . . . . . . . . . . . . . . . . 43
3.3 Numerical Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.1 Test problems for gas-sold reaction model . . . . . . . . . . . . . . 46
3.3.2 Test problems for KS-model . . . . . . . . . . . . . . . . . . . . . . 52
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 Theoretical Investigation of Liquid Chromatographic Models 60
4.1 Motivation and Background . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Chromatographic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.1 The equilibrium dispersive model (EDM) . . . . . . . . . . . . . . . 63
4.2.2 The lumped kinetic model (LKM) . . . . . . . . . . . . . . . . . . . 65
4.3 Analytical Solutions of EDM for Linear Isotherms . . . . . . . . . . . . . . 66
4.4 Analytical solutions of LKM for Linear Isotherms . . . . . . . . . . . . . . 70
4.5 The Discontinuous Galerkin Scheme for Solving LKM . . . . . . . . . . . . 72
xi
4.6 The Flux-Limiting Finite Volume Schemes . . . . . . . . . . . . . . . . . . 77
4.7 Numerical Test Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.7.1 One-component elution with linear isotherm . . . . . . . . . . . . . 81
4.7.2 One-component elution with nonlinear isotherm . . . . . . . . . . . 88
4.7.3 Two-component elutions . . . . . . . . . . . . . . . . . . . . . . . . 90
4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5 Numerical Approximations of Radiation Hydrodynamics Model 96
5.1 One-dimensional RHD model . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.1.1 One-dimensional central upwind scheme . . . . . . . . . . . . . . . 99
5.1.2 One-dimensional central schemes . . . . . . . . . . . . . . . . . . . 101
5.2 Two-dimensional RHD model . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2.1 Central upwind scheme . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2.2 Two-dimensional central scheme . . . . . . . . . . . . . . . . . . . . 104
5.3 Numerical Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6 Application of CE/SE Method to Hyperbolic Heat Conduction Model 126
6.1 The Hyperbolic Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . 127
6.2 One-Dimensional CE/SE Method . . . . . . . . . . . . . . . . . . . . . . . 130
6.3 Two-Dimensional CE/SE Method . . . . . . . . . . . . . . . . . . . . . . . 134
6.4 Application of Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 136
6.5 Numerical Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.5.1 One-dimensional test problems . . . . . . . . . . . . . . . . . . . . 137
6.5.2 Two-dimensional test problems . . . . . . . . . . . . . . . . . . . . 139
6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7 Conclusions and Future Recommendations 148
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.2 Future Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
xii
8 References 153
xiii
List of Figures
3.1 Staggered grid near SE(j, n), the CE−(j, n) and CE+(j, n), as well as the
CE(j, n). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Space-time geometry of the modified CE/SE-method: (a) representative grid
points in x-y plane, (b) the definitions of CE and SE. . . . . . . . . . . . . 40
3.3 Case 1: Results of single specie test problem at td=50. . . . . . . . . . . . 45
3.4 Case 2: Results of two species test problem at t=50. . . . . . . . . . . . . . 47
3.5 Problem 1: errors at different numbers of grid points. . . . . . . . . . . . . 48
3.6 Results of problem 1 at td = 0.6. . . . . . . . . . . . . . . . . . . . . . . . . 49
3.7 Results of problem 2 at td = 2.5. . . . . . . . . . . . . . . . . . . . . . . . . 50
3.8 Results of problem 3 at td = 1.0. . . . . . . . . . . . . . . . . . . . . . . . . 51
3.9 Results of problem 4 at t = 1× 10−4. . . . . . . . . . . . . . . . . . . . . . 55
3.10 Results of problem 4 at t = 4.4 × 10−6, t = 4.4 × 10−5 and t = 4.4 × 10−4
(from top to bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.11 Results of problem 5 at t = 5× 10−6, t = 1× 10−4 and t = 4.5× 10−4 (from
top to bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.12 Results of problem 6 at t = 3× 10−4. . . . . . . . . . . . . . . . . . . . . . 58
3.13 Results of problem 6 at t = 1.5×10−5, t = 4.5×10−5 and t = 3.0×10−4(from
top to bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1 Cell centered finite volume grid . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 Grids near the boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3 Case 1 (EDM): Comparison of results for Dirichlet boundary conditions. . 83
4.4 Case 1 (EDM): Comparison of results for Danckwert’s boundary conditions. 83
xiv
4.5 Case 1 (EDM): Effect of boundary conditions for different values of Peclet
numbers (or diffusion coefficients). . . . . . . . . . . . . . . . . . . . . . . . 84
4.6 Case 3 (LKM): Comparison of results for Dirichlet boundary conditions. . 87
4.7 Case 3 (LKM): Comparison of results for Danckwert’s boundary conditions. 87
4.8 Results comparison in the case of one-component nonlinear EDM. . . . . . 91
4.9 Linear LKM solution for one-component elution. . . . . . . . . . . . . . . . 91
4.10 Left: Two component nonlinear elutions profile at the column outlet. Right:
different injection volumes are used, first injection: c1,0 = 4 mol/l and
c2,0 = 2 mol/l, second injection: c1,0 = 2 mol/l and c2,0 = 1 mol/l, third
injection: c1,0 = 1 mol/l, c2,0 = 0.5 mol/l. . . . . . . . . . . . . . . . . . . 94
4.11 Error analysis for two component nonlinear elutions: top: L1-Error for com-
ponent 1, middle: L1-Error for component 2, bottom: CPU time of different
schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1 Problem 1: The central upwind scheme results for different values of K on
50 grid points at t = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.2 Problem 1: L1-error plots of density for different values of K. . . . . . . . . 111
5.3 Problem 1: L1-error plots of pressure for different values of K. . . . . . . . 112
5.4 Problem 1: L1-error plots of temperature for different values of K. . . . . . 113
5.5 Problem 1: Results on 400 mesh cells at t = 0.2 for κ(T ) = 0. . . . . . . . 114
5.6 Problem 2: Results on 400 mesh cells at t = 0.04. . . . . . . . . . . . . . . 115
5.7 Problem 2: L1-Error plots at κ(T ) = 0. . . . . . . . . . . . . . . . . . . . . 117
5.8 Problem 3: Results on 400 mesh cells at t = 0.02. . . . . . . . . . . . . . . 118
5.9 Problem 4: Results on 400 mesh cells at t = 0.18. . . . . . . . . . . . . . . 119
5.10 Problem 5: 2D results on 256× 128 mesh cells at t = 0.6, κ(T ) = 0. . . . . 120
5.11 Problem 5: 2D results on 256× 128 mesh cells at t = 0.6, κ(T ) = 10−3(1 +
10T 3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.12 Problem 5: 1D results along x = 0.25 at t = 0.6, κ(T ) = 0. . . . . . . . . . 121
5.13 Problem 5: 1D results along y = 0.5 at t = 0.6, κ(T ) = 0. . . . . . . . . . . 122
xv
5.14 Problem 5: Density and temperature contour at different time steps, κ(T ) = 0.123
5.15 Problem 5: 1D results along y = 0.55 at t = 0.5, κ(T ) = 0. . . . . . . . . . 123
5.16 Problem 5: 1D results along x = 0.30 at t = 0.5, κ(T ) = 0. . . . . . . . . . 124
5.17 Problem 6: 2D results on 128× 128 mesh cells at t = 0.5, κ(T ) = 0. . . . . 124
5.18 Problem 6: 1D results along y = 1.0 at t = 0.5, κ(T ) = 0. . . . . . . . . . . 125
6.1 Illustration of the one-dimensional staggered grid. . . . . . . . . . . . . . . 133
6.2 Problem 1: Interaction of two pulses. . . . . . . . . . . . . . . . . . . . . . 141
6.3 Problem 2: A single shock reflection. . . . . . . . . . . . . . . . . . . . . . 141
6.4 problem 3: Inflow periodic boundary conditions. . . . . . . . . . . . . . . . 142
6.5 Problem 4: Evolution of energy density and heat flux (explosion in a box). 143
6.6 Problem 4: One-dimensional plots along x-axis at y = 1. . . . . . . . . . . 144
6.7 Problem 5: Evolution of energy density and heat flux (interacting heat pulses).145
6.8 Problem 5: One-dimensional plots along x-axis at y = 1. . . . . . . . . . . 146
6.9 Problem 6: Evolution of energy density, heat flux, and 1D plots at y = 0.75
(inflow boundary conditions in 2D). . . . . . . . . . . . . . . . . . . . . . . 147
xvi
List of Tables
3.1 Case 1: Data used in the numerical simulation of single CDR model. . . . 45
3.2 Case 1: Comparison of numerical L1-errors at different grid points. . . . . 46
3.3 Case 2: Data used in the numerical simulation . . . . . . . . . . . . . . . . 46
3.4 Case 2: Comparison of numerical L1-errors at different grid points. . . . . 47
3.5 Data used in the numerical simulation . . . . . . . . . . . . . . . . . . . . 52
3.6 Comparison of numerical errors (fractions) at different grid points . . . . . 52
4.1 Selected flux limiting functions in Eq. (4.112) . . . . . . . . . . . . . . . . 80
4.2 Parameters for Case 1 (EDM). . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3 Case 2: Errors and CPU times of schemes on 100 mesh points for EDM. . 86
4.4 Case 2: Errors and EOC for DG-method for EDM. . . . . . . . . . . . . . 86
4.5 Case 4: Parameters for linear isotherm (LKM). . . . . . . . . . . . . . . . 89
4.6 Errors and CPU times at 50 grid points for linear isotherm (LKM) . . . . 89
4.7 Errors and CPU times at 100 grid points for linear isotherm (LKM) . . . . 89
4.8 Parameters for nonlinear case (EDM). . . . . . . . . . . . . . . . . . . . . 90
4.9 Section 4.6.2: L1-errors and CPU time of schemes for nonlinear EDM. . . . 90
4.10 Parameters for two-component elusion (nonlinear isotherm, LKM). . . . . . 93
5.1 Problem 1: Comparison of errors at different grid points for K = 10−3. . . 116
5.2 Problem 1: Comparison of errors at different grid points for K = 10−2. . . 116
5.3 Problem 1: Comparison of errors at different grid points for K = 10−1. . . 116
5.4 Problem 1: Comparison of errors at different grid points for K = 0.5. . . . 116
xvii
5.5 Problem 2: Comparison of errors for K = 0 at different grid points and CPU
times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
xviii
LIST OF ABBREVIATIONS
Abbreviations
BC Boundary ConditionCDR Convection-Diffusion-Reaction SystemCE/SE Conservation Element and Solution ElementCE Conservation ElementCFD Computational Fluid DynamicsCFL Courant-Friedrichs-LewyDG Discontinues GalerkinEDM Equilibrium Dispersive ModelFCT Flux Corrected TransportFDM Finite Difference MethodFEM Finite Element MethodFVM Finite Volume MethodFV Ss Finite Volume SchemesGSR Gas Solid ReactionHHC Hyperbolic Heat ConductionKFV S Kinetic Flux Vector SplittingKS Kellar-SegalLKM Lumped Kinetic ModelMC Monotonized Central-DifferenceMUSCL Monotone Upstream Centered Schemes for Conservation LawsNT Nessyahu-TadmorODEs Ordinary Differential EquationsPDEs Partial Differential EquationsRHD Radiation HydrodynamicsRHEs Radiation Hydrodynamical EquationsRK Runge KuttaSE Solution ElementTV B Total Variation BoundedTV D Total Variation DiminishingWENO Weighted Essentially Non-Oscillatory
xix
Symbols units
x Space coordinate [m]t Time coordinate [s]w Vector of conservative variables [−]F(w) Flux vector [−]A(w) Jacobian matrix [−]∆Vi Finite volume [−]ρ Density [kg/m3]p Pressure [N/m2]E Energy [J ]T Temperature [k]c1 Concentration of specie 1 [kg/m3]c2 Concentration of specie 2 [kg/m3]k1 First order destruction rate of specie 1 [1/s]k2 First order destruction rate of specie 2 [1/s]D Diffusion or dispersion coefficient [m2/s]u Velocity constant [m/s]κp Heat-capacity, [J/kg.k]λ Averaged thermal-conductivity, [J/m.s.k]k0 Pre-exponential factor [−]∆H Reaction heat [J/kg]cg Gas concentration [kg/m3]cs Solid concentration [kg/m3]E Activation energy [J/kmol]R Universal gas constant [8314.472J/kmol.k]ϵ Porosity constant [−]
xx
symbols (continued)
ψ Dimensionless temperature [−]td Dimensionless time [−]cgd Dimensionless gas concentration [−]csd Dimensionless solid concentration [−]γg Inverse of dimensional heat of reaction for gas [−]γs Inverse of dimensional heat of reaction for solid [−]β Inverse of dimensionless activation energy [−]Pe Pecelet number [−]PeH Pecelet number for heat transfer [−]PeM Pecelet number for mass transfer [−]Le Lewis number [−]νk Courant number [−]ξk Diffusion number [−]ξ Sensitivity constant of chemotactic [−]φ Numerical solution [−]φref Refernce numerical solution [−]τC characteristic time [1/s]τD characteristic time [1/s]τMT characteristic time [1/s]Ij jth mesh interval [−]N Number of grid cells [−]ϕl local basis function of order l [−]z Axial coordinate [m]Nc Number of components [−]q∗ Solid concentration [−]Dapp Solid concentration [−]ai Henry coefficient [−]k Mass transfer coefficient [−]F Phase ratio [−]Ω Set of points of computational domain [−]
xxi
Chapter 1
Introduction
1
This chapter presents a brief introduction of convection-diffusion-reaction (CDR) models
considered in this thesis. A historical background of these models is also included for
the convenience of the reader. Furthermore, a discussion on the desired properties of a
numerical technique, brief overview of the proposed numerical schemes and a summary of
achievements are the contents of this chapter.
1.1 Problem and Motivation
Mathematically, convection-diffusion-reaction (CDR) models are the system of mixed type
non-linear partial differential equations (PDEs). These models describes the transport of
substances under three different processes, convection, diffusion and chemical reactions.
Convection is the movement of a substance within a medium, for example water or air.
Diffusion refers to the movement of a substance from denser region to a rare medium.
A chemical reaction process is the interconversion of chemical substances. Thus, CDR
systems are usually encountered in chemistry. However, such systems could also simulate
dynamical processes of non-chemical nature. The CDR type models have wide range of
applications, such as modeling the dynamics of fluids, chemical tubular reactors, aerody-
namics, biological systems, reactive chromatography, heat conduction and astrophysical
scenarios, etc. Such wide range of applications of CDR models give motivation to under-
stand and analyze the behavior of these systems qualitatively and quantitatively. Except
the simple linear systems, the CDR systems are generally not solvable analytically. There-
fore, accurate numerical techniques are the only tools to get physically reliable solutions in
acceptable computational time. The numerical solutions could help in deeper understand-
ing of the underlying physical process. However, the simulation of convection-dominated
CDR systems is a challenging task for the numerical schemes due to steep reaction gradi-
ents, sharp discontinuities and narrow peaks in the solutions. In such scenarios, reasonable
solutions can be obtained by performing numerical simulations on refined grids. But such
computations are highly expensive on large scale. On the other hand, the numerical results
on coarse grids have dissipated (smeared) wave fronts that can make the interpretation of
2
experimental data less effective and could produce negative affects on the design of indus-
trial process. For that reason, efficient and accurate numerical methods are needed to get
results of desired accuracy. In recent years, several numerical schemes were introduced for
solving these models. This work is a continuation of the previous research works which
have been carried out in the last few decades. Our goal is to extend different high resolu-
tion finite volume schemes (FVSs) and the finite element method (FEM) to the selected
nonlinear CDR models and to analyze their performance. Moreover, finding analytical
solutions of linear CDR models for different boundary conditions were also included in the
objective of this thesis project.
1.2 Investigated CDR-Type Models
This section briefly introduces the selected CDR-systems which are numerically investi-
gated during this thesis project. A thorough discussion about them will be presented in
their respective chapters.
1.2.1 Gas-solid reaction and chemotaxis models
Here, two types of models are numerically investigated which form complete CDR systems
involving all three types of transport phenomena, such as convention, diffusion and reaction.
Exothermic Heterogeneous Reaction of Gas and Solid:
This CDR system describes the heterogeneous gas-solid reactions (GSR). In such reac-
tions, substances of different phases react with each other through chemical reactions to
release heat. Mostly, such chemical reactions are of exothermic nature. Thus, steep so-
lution profiles of the concentration and temperature are developed due to the interaction
of mass diffusion, heat conduction and non-linear transient convection. Such models have
wide range of applications, such as modeling of heavy oil recovery, the pyrolysis of coal
and biomass, the reduction and roasting of ores, the incineration of waste, the burning
of solids, the acid gases absorption by lime, the deposition of reactive vapor phase, the
3
ceramic materials production, the gasification of coal, the manufacturing of catalysts, and
so on.
Chemotaxis Models:
The term chemotaxis refers to the movement of living species under the influence of a
chemical substance in the environment. Here, a classical Keller-Segel (KS) model of the
chemotaxis is considered in which solution profile becomes discontinuous in finite time even
if the initial data is smooth.
1.2.2 Chromatographic models
These are convection-diffusion type models in which chemical reactions are neglected.
Chromatography is one of the most versatile separation technique widely used for analysis
and purification of different mixtures which are not easily separable by traditional tech-
niques, such as extraction or distillation. It has numerous industrial applications, namely,
food, pharmaceutical, environmental, fine chemicals and biomedical. In this process the
mixture components (solute) are transported through a stationary phase by a carrier mo-
bile phase (solvent). Based on the process setup, the carrier phase can be either liquid or
a gas and the phase which is stationary is either a liquid film on the surface of immobile
support material or a rigid surface. In this study, only liquid chromatographic models will
be considered. In the liquid chromatographic process, a fluid mixture pulse is injected to
a cylinder shaped column which is packed with a porous solid. In doing so, a continuous
fluid flow is established. This flow enhances the interaction of mixture components with
the stationary solid phase and, thus, a distribution of solute is established between the
stationary and mobile phases. The solute component which has strong interactions with
the solid phase transports slowly than the components having less interactions. Because
of different speeds, each solute component form different band profiles within the column.
Thus, pure fractions of fluid components can be collected at the column outlet.
4
1.2.3 Radiation hydrodynamical models
This part of the thesis is concerned with the numerical investigation of radiation hydro-
dynamical (RHD) models in one and two space dimensions. The flow equations are the
set of non-linear mixed convection and convection-diffusion partial differential equations.
Such models have several applications in different scientific and engineering disciplines.
These applications include high-temperature hydrodynamics, modeling of gaseous stars in
astrophysics, accretion disks, radiatively driven outflows, supernovas, laser fusion physics,
combustion phenomena, reentry vehicles fusion physics, stellar convection and inertial con-
finement fusion. The branch of fluid mechanics that deals with the study of moving fluid
and changes in its state under diverse circumstances (e.g., internal and external forces) is
called hydrodynamics. The absorption or emission of radiations through matter produces
heating and cooling in a system, respectively. The existence of considerable radiation trans-
port intimate the presence of temperature gradients and energy density, which indicate the
existence of pressure gradients as well. When sufficient time is available, pressure gradients
generate considerable flow of fluid (radiation hydrodynamics) and changes in the density.
In zero diffusion limit, the radiation hydrodynamical equations (RHEs) could form a sys-
tem of hyperbolic conservation laws. But, they are different from the inviscid compressible
Euler’s equations of gas dynamics.
1.2.4 Hyperbolic heat conduction model
This model is a very special case of the CDR-type models where both diffusion and chemical
reactions are neglected. Many heat transfer situations are sufficiently described by the
parabolic thermal diffusion equation. This is due to Fourier’s Law of heat conduction
which is based on the assumption that a local thermal disturbance is instantaneously felt
at each point in the medium, which is a non-physical behavior. In many technological
processes, at normal room temperature (20oC − 23oC), the modes that do not propagate
with finite speed enduring a very large damping of heat pulses and are thus not discernible.
But, there exist some cases where either the damping of heat pulses is quite low or where
5
its moving distance is so small that the transit time is discernible quantity. In such cases
the predictions according to parabolic system fail miserably. Therefore, the hyperbolic
system is a better mathematical representation of such situations.
1.3 Historical Background
The history of numerical investigation of CDR type systems traces back in mid 19’s. Their
wide range applications and the unavailability of analytical solutions motivated researchers
and engineers of this field to develop efficient and accurate numerical techniques for solving
these systems. A variety of numerical techniques were introduced for solving CDR systems
during the last decades. In the following, a brief historical background of each selected
CDR system is presented.
The numerical simulation of GSR and KS models is a challenging task for the numerical
schemes. The major difficulties are the steep reaction gradients and narrow peaks in the
solutions. During the last decade, researchers in this field have given their attention to
introduce better numerical techniques for solving these models. In [44] different finite vol-
ume schemes (FVSs) were implemented to approximate heterogeneous GSR models. The
semi-discrete central-upwind and the discontinuous Galerkin (DG) methods were imple-
mented to solve the two-dimensional KS-system [21, 38]. Other articles on these models
are also available in the literature, see for example [4, 40, 44, 78, 123] and reference therein.
Chromatography is an old technique to purify substances and separate mixture compo-
nents, such as the extraction of dyes from plants. The word chromatography was intro-
duced by Michael Tswett [120, 121] and used this process to sperate plant pigments. In
the experiment, he used a vertical glass column packed with an adsorptive material, like
alumina or silica. Afterwards, he injected a solution of plant pigments at the upper end of
the column and washed the column with an organic solvent. As a result, a series of colored
pigment bands appeared in the column, separated by regions free from pigments. Due to
6
these color bands, he named this method chromatography which means color writing, de-
duced from the Greek for color-chroma and for write-graphein. This process got attention
after the works of Kuhn et al. [60] and Zechmeister and Cholnoky [130]. In the decades
of 1940s and 1950s, the present form of chromatography was developed. Martin [82] in-
troduced different chromatography methods, such as paper chromatography, gas-solid and
gas-liquid chromatography and various techniques of column liquid chromatography. Fur-
ther advances constantly improved the performance of chromatography for the separation
of more complex mixtures.
Radiation hydrodynamical models have wide range applications in different scientific and
engineering disciplines [27, 30, 81, 109, 119]. In zero diffusion limit, the radiation hydrody-
namical equations (RHEs) could form a system of hyperbolic conservation laws. One of the
major difficulties of standard numerical methods for solving RHEs is to resolve and deter-
mine the exact path of strong shocks. Some pioneering work on the problems of radiation
hydrodynamics can be found in [15, 70, 83, 84, 93]. Dai and Woodward [30] suggested the
Godunov scheme with linear and nonlinear Riemann solvers for the numerical solution of
the RHEs. The results showed that these methods have kept the key features of Godunov
schemes. But, they were found to be computationally expensive. For astrophysical prob-
lems, the method proposed in [113] was found to the most successful one where operator
splitting technique was combined with the Crank-Nicholson method. Further work on this
algorithm and several advantages and disadvantages of Godunov type schemes for solving
RHEs are exploited in [12, 52, 67, 68, 80, 93, 109, 128, 112]. Alhumaizi [5] compared differ-
ent numerical schemes for solving RHEs and showed that flux-corrected transport (FCT),
weighted essentially non-oscillatory (WENO) scheme and monotone upstream scheme for
conservation laws (MUSCL) are accurate for various parameters of radiation hydrody-
namics. Moreover, the work of Tang and Wu [119] on RHD is a remarkable addition to
the applications of kinetic flux-vector splitting (KFVS) methods. We have used the same
KFVS method to validate our numerical results.
7
The influence of non-Fourier heat conduction on solid phase was first recognized by Antaki
[7]. Abdel-Hamid [1] used finite integral transfer to analyze non-Fourier heat conduction
process with periodic thermal oscillations. Lor and Chu [79] presented the effect of inter-
face thermal resistance on heat transfer in a composite medium using the thermal wave
model. Liu et al. [75] discussed non-Fourier effects on the transient temperature response
in a semi-transparent medium caused by a laser pulse. Dreyer et al. [32, 33, 34, 35]
have developed kinetic solvers to study hyperbolic heat conduction (HHC) in crystalline
solids. Kunik et al. [62] have introduced a reduction of Bolzmann Peierals equation which
simplified its numerical solution. Sahoo and Roetzel [105] proposed the hyperbolic axial
dispersion model for heat exchangers. Yang [129] introduced the sequential method for
determining boundary conditions in HHC problems. Chen [20] proposed a new hybrid
method for investigating the effect of the surface curvature of a solid body on hyperbolic
heat condition. Sanderson [106] investigated hyperbolic heat equations in laser generated
ultrasound models. Lin [71] studied the non-Fourier effect on fin-performance under peri-
odically varying thermal conditions. Mullis [89] presented the rapid solidification and the
propagation of heat at a finite velocity. Roetzel et al. [102] analyzed the hyperbolic axial
dispersion model and its application to a plate heat exchanger [47], and so on.
1.4 Desired Performance of a Numerical Method
Mathematical modeling and simulation provides an alternative tool for investigating phys-
ical phenomena, instead of carrying out actual experiments which may be expensive, time-
consuming or even dangerous. They are often more informative than the actual experi-
ments.
The considered mathematical models are taken from different disciplines of science and en-
gineering. However, the types of governing equations and solution properties connect them
with each other. These connections provide a justification for dealing with them under
the same title. For the desirable performance from a numerical method, certain properties
8
should be satisfied which are discussed below.
The data, such as the velocity field, diffusion coefficients and reaction constants, are gen-
erally not much accurate. Therefore, the accuracy requirements for the numerical solution
are also low. On the other hand, from practical view point, the number of spatial grid
points may be very large. Thus, the computational cost of the problems could be very
large, needing fast and cheaper numerical methods for the solution.
One is often interested in long term effects, so that the equations have to be integrated over
long time intervals. Therefore, in spite of low accuracy demands, the numerical solutions
should be qualitatively accurate and should fulfill the properties like mass conservation,
positivity, and small phase errors.
We are considering CDR type PDEs with dominating convective terms and stiff reaction
source terms. Due to large reaction constants, some reactions take place on very small
time scales as compared to the overall time scale. Thus, steep spatial gradients and narrow
delta-type peaks could be observed in the solutions. For that reason, it is desired that
the numerical method should resolve these solution profiles efficiently and accurately. In
solving non-linear hyperbolic system, in which diffusion is neglected, capturing of shock
discontinuity at the correct location, efficiency and positively are the key demands from
applied numerical technique.
In this work, full CDR-systems, convection-diffusion systems, and nonlinear hyperbolic
systems are numerically studied. The performance of numerical schemes are quantitatively
and qualitatively analyzed in term of their simplicity, accuracy and efficiency.
9
1.5 Proposed Numerical Methods
In this work, the following numerical methods are proposed and analyzed. These methods
are applied for the first time to such models and it was found that these scheme are efficient
and accurate for solving these models.
1.5.1 The space time CE/SE-method
This central-type FVS is not an incrementally improved version of the previously existing
schemes and is substantially different from already established methods. Its formulation
is based on the integral formulation of conservation laws that produces better resolution
of sharp discontinuous profiles with minimized spurious oscillations. The main features of
the technique include the unified space-time treatment, the use of conservation-elements
(CEs) and solution-elements (SEs), as well as its ability to capture sharp discontinuities
without employing Riemann solvers. The technique was successfully used to solve different
physical and engineering problems [16, 17, 18, 19, 74, 76, 77, 78, 96, 97].
In this work, the CE/SE-method is extended to solve the models of gas-solid reaction,
chemotaxis and hyperbolic heat conduction. In the one space dimension, the CE/SE
scheme [16] is implemented, while in two-space dimensions a different version of CE/SE-
method introduced in [132] is employed.
1.5.2 The discontinuous Galerkin finite element method
The discontinuous Galerkin (DG) is a finite element method which allows discontinuities at
the cell interfaces. The method contains important features of both finite element and finite
volume schemes. Thus, it is an efficient and accurate numerical technique for convection
dominated and CDR-type models. In 1973, Reed and Hill [99] for the first time introduced
the DG finite element method for solving hyperbolic equations. Since then, various DG-
methods have been derived for solving hyperbolic, elliptic, and parabolic problems. The
technique was further developed by Cockburn and his coauthors by introducing the Runge-
10
Kutta (RK) DG scheme in a series of papers for solving hyperbolic conservation laws
[22, 23, 24, 25]. The scheme employs the DG scheme in space coordinates that convert
the given PDEs to the system of ordinary differential equations (ODEs). Then high order
nonlinearly stable and explicit RK-method is applied on the system of ODEs. The scheme
satisfies the total variation bounded (TVB) property that assure the positivity of the
scheme. Besides different classes of schemes, DG schemes are more stable and high-order
accurate. These schemes are capable of handling complex geometries and irregular meshes
with hanging nodes, and can incorporate arbitrary degree of polynomial approximations in
different elements. In this work, the same TVB RK-DG scheme is implemented for solving
multi-component liquid chromatographic models.
1.5.3 The central schemes
The central schemes could serve as a common numerical technique to solve several scientific
and engineering problems due to avoiding the specific eigenstructure of the problem [63].
The schemes have been successfully applied to solve problems in computational fluid dy-
namics, astrophysics, metrology, semiconductors, shallow flows and multi-component flows
[9, 95]. The first-order Lax-Friedrichs scheme is the back bone of such schemes. The cen-
tral Nessyahu-Tadmor (NT) scheme [91] is a Riemann-solver-free high resolution staggered
central scheme which do not need any information about the eignstructure of the prob-
lem. Thus, the method can be applied as a black box solver to any system of conservation
laws. However, this family of central schemes suffers from excessive numerical viscosity
when a sufficiently small time step is enforced, e.g., due to the presence of degenerate
diffusion terms. To overcome this deficiency, Kurganov and Tadmor [63, 64] improved the
NT scheme by using the correct information of local propagation speeds and obtained the
semi-discrete central upwind scheme. Similar to the staggered NT scheme, it enjoys the
benefits of high resolution, simplicity and robustness. However, the central upwind scheme
reduces large amount of numerical dissipation present in the NT central schemes.
11
1.5.4 The flux-limiting finite volume schemes (FVSs)
These are semi-discrete high resolution FVSs in which flux limiting functions are used
to ensure their local positivity (monotonicity). In these schemes, the space-coordinate
is discretized by using upwind finite volume scheme, while the time derivative remains
continuous. Afterward, ordinary differential equations (ODEs) solvers are employed to
solve the resulting systems of ODEs. To suppress numerical oscillations in the scheme
different limiters are used. In the literature, different limiting function exist namely, van
Leer, Superbee, minmod, monotonized central (MC), and Koren [55, 68]. Each limiter
gives a different numerical scheme.
1.6 The Laplace Transformation
The Laplace transformation is one of the frequently used tool to obtain analytical solutions
in many engineering applications [100]. The main idea is to transform a function from its
original domain into a transformed domain where certain operations can be carried out
more efficiently. After carrying out the operation in the transform domain, inverse Laplace
transformation is used to transform the result in the original domain.
In this work, the Laplace transformation is employed to transform the partial differential
equations (PDEs) of linear CDR models to ordinary differential equations (ODEs). After
solving the ODEs in transformed coordinates, the resulting solutions are transformed back
in the original coordinates. In the case of no analytical inverse Laplace transformation, the
numerical inversion is used to get back the solution in the original domain [100].
1.7 Achievements of the Project
This work is focused on the numerical solution of different CDR type models arising in
several areas of science and engineering. Furthermore, analytical solutions are obtained for
some linear CDR models. These analytical solutions are used to analyzed the performance
12
of suggested numerical schemes.
The first part of the thesis deals with numerical simulation of gas-solid reaction and chemo-
taxis models. For the first time, CE/SE-method is used to solve these models. The method
is based on the unique partitioning of the computational domain into conservation elements
and solution elements. It is derived from the integral formulation of conservation laws and
has capability to resolve discontinuous solution profiles. In order to reveal the accuracy
and efficiency of the method, several case studies are carried out. The performance of
this method is analyzed with that of NT schemes on staggered grids and other methods
available in the literature. Moreover, the accuracy of schemes is quantitatively analyzed by
comparing their results with analytical solutions of linear CDR models derived in Chapter
2. It is found that the current method produces less error as compared to the other methods.
In the second part of this thesis, two dynamic models of liquid chromatography are analyt-
ically and numerically studied. The Laplace transformation is applied to solve these mod-
els analytically for one component adsorption under linear conditions. The DG-method
is proposed to numerically approximate the linear and nonlinear chromatographic mod-
els. Higher order accuracy of the techniques is achieved even on coarse meshes and sharp
discontinuities in the solution profiles are well resolved. The numerical diffusion and dis-
persion are minimized. The results obtained from DG-method are compared with those of
flux-limiting finite volume approaches which revealed the accuracy of suggested numerical
method. A good agreement of the numerical and analytical solutions for simplified cases
verifies the robustness and accuracy of the proposed method. The method is also capa-
ble to solve chromatographic models for nonlinear and competitive adsorption equilibrium
isotherms. This work was performed in collaboration with Max Planck Institute, Germany.
In the third part of the thesis, the high resolution central-upwind schemes are applied to
radiation hydrodynamical equations (RHEs). The staggard central and kinetic flux vec-
tor splitting (KFVS) schemes are also implemented for solving RHEs up to two space
13
dimensions. A number of case studies are carried out and the accuracy of the schemes
are analyzed quantitatively and qualitatively. It was found that central-upwind schemes
have better resolved discontinuous profiles of the solutions as compared to the NT schemes.
Further, it is concluded that the proposed numerical scheme gives comparable results with
KFVS scheme at low CPU time.
Finally, the fourth part of the thesis is concerned with the implementation of CE/SE-
method to the one and two-dimensional hyperbolic heat conduction model. The KFVS
and central schemes are also implemented to the same model. Several case studies are
considered. The numerical results of CE/SE-method are found better than other schemes.
Therefore, it is concluded that the space-time CE/SE-method is an efficient and accurate
numerical technique for solving such models.
1.8 Thesis Layout
The remaining part of this thesis is organized as follows.
Chapter 2 provides a brief derivation of CDR scalar equation, a discussion on the major
numerical challenges, and the analytical solutions of single-specie and two-species linear
CDR models using Laplace transformation. Moreover, discussions on the fundamental no-
tions of conservation laws, hyperbolic systems, weak solutions and Riemann problems are
included in this chapter.
In Chapter 3, the one-dimensional gas solid reaction and two-dimensional chemotaxis mod-
els are briefly described. The numerical solutions are obtained by implementing the CE/SE-
method. For validation, the central schemes are used. The numerical analysis of suggested
schemes for linear CDR model is also included in the contents of this chapter.
Chapter 4 is concerned with analytical and numerical investigation of two established liq-
14
uid chromatographic models namely, the equilibrium dispersive and lump kinetic models.
A brief introduction and analytical and numerical solutions of the models are the contents
of this chapter. The Laplace transformation is applied to solve these models analytically
for single component adsorption under linear conditions. The discontinuous Galerkin finite
element method and a flux-limiting finite volume scheme are applied to approximate these
models for multi-component adsorption under nonlinear conditions.
In Chapter 5, the one and two-dimensional models of radiation hydrodynamics are pre-
sented. The numerical solutions of these models are sought out by employing the central-
upwind scheme. A comparison of the results with those of staggered central scheme and
KFVS method confirms the accuracy and efficiency of the suggested scheme.
Chapter 6 describes the implementation of CE/SE scheme for hyperbolic heat conduction
model in one and two space dimensions. Several numerical test problems are considered.
The results of CE/SE-method are compared with the staggard central and KFVS schemes.
Chapter 7 concludes the thesis and describes the potential areas of future work.
1.8.1 Published and accepted articles:
Most of the contents of this thesis are already published or accepted in international re-
search Journals of good impact factors.
The results presented in Chapter 3 appeared as
1. Shamsul Qamar and Waqas Ashraf, Application of space-time CE/SE-method for
solving gas-solid reaction and chemotaxis models, Industrial & Engineering Chemistry Re-
search, 51 (2012) 9173-9185. (Latest impact factor 2.237)
Parts of Chapter 4 appeared as
2. Shumaila Javeed, Shamsul Qamar, Waqas Ashraf, Gerald Warnecke and Andreas
15
Seidel-Morgenstern, Analysis and numerical investigation of two dynamic models for liq-
uid chromatography, accepted, Chemical Engineering Science, 2012. (Latest impact factor
2.431)
Chapter 5 contains
3. Shamsul Qamar and Waqas Ashraf, Application of central schemes for solving radia-
tion hydrodynamical models, accepted, Computer Physics Communications, 2012. (Latest
impact factor 3.268)
Chapter 6 appeared in
4. Shamsul Qamar and Waqas Ashraf, A space time CE/SE-method for solving hyper-
bolic heat conduction model, accepted, International Journal of Computational Methods
(IJCM), 2012 (Latest impact factor 1.167).
16
Chapter 2
Fundamentals ofConvection-Diffusion-ReactionSystems
17
This chapter presents the derivation of a scalar linear CDR equation. The Laplace trans-
formation is applied as a basic tool to obtain the analytical solutions of single-specie and
two-species linear CDR models. Moreover, discussions on the fundamental notions of con-
servation laws, hyperbolic systems, weak solutions and Riemann problems are included in
this chapter.
2.1 Derivation of Linear Scalar CDR Equation
Generally, the convection-diffusion-reaction (CDR) type equation describes the transporta-
tion of a chemical substance. This is a mixed type PDE which can be derived from the mass
balance law. Consider a concentration c(t, x) of a certain chemical substance depending
on the space variable x and time t. Let h > 0 be a small number and consider the average
concentration c(t, x) in a cell Ω(x) = [x− 12h, x+ 1
2h], then
c(t, x) =1
h
∫Ω
c(t, x)dx+1
24h2cxx(t, x) · · · . (2.1)
If the specie is carried along by a flowing medium with velocity u(t, x) then the mass
conservation law implies that the change in c(t, x) per unit time is the net balance of
inflow and outflow over the cell boundaries
∂c(t, x)
∂t=
1
h[u(t, x− 1
2h)c(t, x− 1
2h)− u(t, x+
1
2h)c(t, x+
1
2h)]. (2.2)
Here, u(t, x∓ 12h)c(t, x∓ 1
2h) are the fluxes over the left and right cell boundaries. Now, if
we let h→ 0, it follows that the concentration satisfies the following balance law
∂c(t, x)
∂t+∂(u(t, x)c(t, x))
∂x= 0. (2.3)
This equation is called the linear advection (or convection) equation. In above equation,
the second term inside a spatial derivative, f(t, x) := u(t, x)c(t, x), is called the flux term
and becomes nonlinear when f(t, x) is a nonlinear function of the dependent variable c(t, x).
According to Eq. (2.3), the net change in c in the given space-time domain is zero, i.e. c is
a conserved quantity.
18
In a similar manner, we can include the effect of diffusion. Then, the change in c(t, x) is
caused by gradients in the solution and the fluxes across the cell boundaries are given as
−D(t, x∓ 12h)cx(t, x∓ 1
2h). Here, D(t, x) represents the diffusion coefficient and cx :=
∂c∂x.
The corresponding diffusion equation can be expressed as
∂c(t, x)
∂t=
∂
∂x
(D(t, x)
∂c(t, x)
∂x
). (2.4)
Finally, c(t, x) may change due to sources, sinks and chemical reactions, leading to
∂c(t, x)
∂t= Q(c(t, x), t, x) . (2.5)
The overall change in concentration is described by combining all three effects, leading to
a convection-diffusion-reaction equation
∂c(t, x)
∂t+∂(u(t, x)c(t, x))
∂x=
∂
∂x
(D(t, x)
∂c(x, t)
∂x
)+Q(c(t, x), t, x) , (2.6)
with suitable initial and boundary conditions. The quantities u(t, x) that represent the
velocities of the transport medium, such as water or air. The diffusion coefficients D(t, x)
are constructed by the modelers and may include also parameterizations of turbulence.
The final term Q(c(t, x)), which gives a coupling between the various species, describes
the nonlinear chemistry together with emissions (sources) and depositions (sinks). For
simplicity, we will write the average value c(t, x) as c(t, x) onwards.
2.2 Analytical Solutions of Linear CDR Models
This section presents the analytical solutions of selected linear CDR equations which de-
scribe the movement of chemical species in porous mediums. The Laplace transformation
is employed to obtain solutions for different initial and boundary conditions. These solu-
tions will be used to analyze the accuracy of proposed numerical schemes in the coming
chapters.
19
2.2.1 A linear single CDR-equation
We consider a scaler CDR-equation of the form [2, 14, 41]
α∂c
∂t+ u
∂c
∂x= D
∂2c
∂x2− rc . (2.7)
The initial and boundary condition are given as
c(0, x) = cinit , c(t, 0) = c0 ,∂c
∂x(t,∞) = 0 . (2.8)
Here, α is the retardation factor, c denotes the solution concentration, D is the dispersion
coefficient and r represents a decay constant. The Laplace transformation is defined as
C(s, x) =
∞∫0
e−stc(t, x)dt, s > 0 . (2.9)
After applying the Laplace transformation on Eq. (2.7), we obtain
α
∞∫0
e−st∂c
∂tdt+ u
∂
∂x
∞∫0
e−stc(t, x)dt = D∂2
∂x2
∞∫0
e−stc(t, x)dt− r
∞∫0
e−stc(t, x)dt . (2.10)
By applying integration by parts on the first integral, the initial condition in Eq. (2.8) and
the definition (2.9), we get
D
α
∂2C
∂x2− u
α
∂C
∂x−(s+
r
α
)C = −cinit , (2.11)
with transformed boundary conditions
C(s, 0) =c0s,
∂C
∂x(s,∞) = 0 . (2.12)
The following Laplace transformed solution can be obtained by solving Eqs. (2.11) and
(2.12)
C(s, x) =c0sexp
[ux
2D− x
(u2
4D2+s+ r/α
D/α
) 12
]
− cinits+ r/α
exp
[ux
2D− x
(u2
4D2+s+ r/α
D/α
) 12
]+
cinits+ r/α
. (2.13)
20
The inverse Laplace transformation of the above equation is given as [2, 14, 41]
c(t, x) = cinitH1(t, x) + c0H2(t, x) , (2.14)
where
H1(t, x) = e−rtα
(1− 1
2erfc
[αx− ut
2(Dαt)12
]− 1
2e
uxD erfc
[αx+ ut
2(Dαt)12
]), (2.15)
H2(t, x) =1
2exp
[(u− v)x
2D
]erfc
[αx− vt
2(Dαt)12
]+
1
2exp
[(u+ v)x
2D
]erfc
[αx+ vt
2(Dαt)12
](2.16)
and
v = u
(1 +
4rD
u2
) 12
. (2.17)
If the left boundary condition, c(t, 0) = c0, in Eq. (2.8) is replaced by the following equation
c(t, 0) =
c0 , if 0 < t ≤ t00 , t > t0 .
(2.18)
Then, the analytical solution Eq. (2.14) becomes [41]
c(t, x) =
cinitH1(t, x) + c0H2(t, x) , if 0 < t ≤ t0cinitH1(t, x) + c0H2(t, x)− c0H(t− t0, x) , t > t0 .
(2.19)
2.2.2 A linear CDR-system of two equations
Here, we consider a linear CDR model which describes the transport of two species and an
irreversible reaction under which the species c1 converts to c2, i.e. c1 → c2. The Laplace
transformation is applied to solve this model analytically. The solution procedure is similar
to one presented in [28]. The governing equations of the model are given as
α1∂c1∂t
+ u∂c1∂x
−D∂2c1∂x2
= −r1c1 , (2.20)
α2∂c2∂t
+ u∂c2∂x
−D∂2c2∂x2
= r1c1 , (2.21)
where αi are constant coefficients, u is the transport velocity, D is the dispersion coefficient,
and r1 is the first order reaction rate constant of first species. In above equation, it
21
is assumed that both species have the same constant velocity and constant dispersion
coefficient. The initial and boundary condition are given as
ci(0, x) = 0 , ci(t, 0) = ci0 , ci(t,∞) = 0 , i = 1, 2 . (2.22)
By using the matrix notation, Eq. (2.20) and Eq. (2.21) can be re-written as[α1 00 α2
]∂
∂t
[c1c2
]+ u
∂
∂x
[c1c2
]−D
∂2
∂x2
[c1c2
]=
[−r1 0r1 0
] [c1c2
]. (2.23)
Applying Laplace transformation to the the above equation, we get
s
[α1 00 α2
] [C1
C2
]+ u
∂
∂x
[C1
C2
]−D
∂2
∂x2
[C1
C2
]=
[−r1 0r1 0
] [C1
C2
], (2.24)
where C1 and C2 are the concentration of species in the Laplace domain. The boundary
conditions of the problem can be transformed as
Ci(s, 0) = Ci0 =ci0s, Ci(s, x) = 0 , i = 1, 2. (2.25)
Eq. (2.24) can be rearranged as
u∂
∂x
[C1
C2
]−D
∂2
∂x2
[C1
C2
]=
[−r1 − sα1 0
r1 −sα2
] [C1
C2
](2.26)
From the above equation, one can identify the combined reaction coefficient matrix for the
system as
B =
[−r1 − sα1 0
r1 −sα2
]. (2.27)
The next step of the solution procedure is to compute the linear transformation matrix A.
Note that, the columns of A should be the eigenvectors of the combined reaction coefficient
matrix. The eigenvalues and eigenvector of B are given as
λ1 = −r1 − sα1, x1 =
[a11r1a11
s(α2−α1)−r1
]λ2 = −sα2, x2 =
[0a22
], (2.28)
where λ1 and λ2 are the eigenvalues and a11 and a22 are the arbitrary constants. For sim-
plicity, let us set the values of a11 and a22 to unity. Then, using (2.28), the transformation
matrix A and the diagonal matrix Λ can be written as
A =
[1 0r1
s(α2−α1)−r1 1
], Λ =
[−r1 − sα1 0
r1 −sα2
]. (2.29)
22
The matrix A can be used for the following linear transformation[C1
C2
]=
[1 0r1
s(α2−α1)−r1 1
] [b1b2
]. (2.30)
Applying the above linear transformation, Eq. (2.26) can be written in the b-domain as
u∂
∂x
[b1b2
]−D
∂2
∂x2
[b1b2
]=
[−r1 − sα1 0
0 −sα2
] [b1b2
]. (2.31)
Note that Eq. (2.31) represents two independent, steady state, advection-dispersion equa-
tions with a first order decay term. An explicit solution to these two independent equations
can be computed using the analytical expressions
b1(s, x) = b10 exp
(u−
√u2 + 4D(α1s+ r1)
2Dx
), (2.32)
b2(s, x) = b20 exp
(u−
√u2 + 4Dα2)
2Dx
), (2.33)
where bi0 is the boundary condition of the species i in the b-domain. Since the A matrix of
our problem has a lower triangular form, we can use Eq. (2.25) to write the transformed
boundary conditions in the b-domain as
b10 =c10s, b20 =
c20s
− A21b10 =c20s
− r1c10[s(α2 − α1)− r1]s
. (2.34)
After solving the problem in b-domain, we can use the transformation C = Ab to get back
the solution vector in the s-domain, where C = (C1, C2)T and b = (b1, b2)
T . Then, we
have
C1(s, x) =c10sM1 , C2(s, x) =
c20sM2 +
r1c10s[s(α2 − α1)− r1]
(M1 −M2) , (2.35)
where M1 and M2 are defined as
M1(s, x) = b10 exp
(u−
√(u2 + 4D(α1s+ r1))
2Dx
), (2.36)
M2(s, x) = b20 exp
(u−
√(u2 + 4Dα2s)
2Dx
). (2.37)
23
Finally, the solution in t-domain can be obtained by employing inverse Laplace transfor-
mation on the above equation. This step is summarized as
c1(t, x) = l−1(C1(s, x)) = c10 l−1(
M1
s) (2.38)
c2(t, x) = l−1(C2(s, x)) = c20 l−1(
M2
s) + r1c10 l
−1
((M1 −M2)
s[s(α2 − α1)− r1]
). (2.39)
Analytical expressions for the inverse laplace transformation can be obtained from standard
table [25]. The final expression for the solution is
c1(t, x) = c10H1 (2.40)
c2(t, x) = c20H2 −r1c10r1
(H1 −H2 − P1 + P2) , (2.41)
where,
Hi =exp( ux
2D)
2[exp(−aidi)erfc(
ai2t−
12 − dit
12 ) + exp(aidi)erfc(
ai2t−
12 + dit
12 )] (2.42)
ai =
√αiDx, d1 =
√u2
4Dα1
+r1α1
, d2 =
√u2
4Dα2
, (2.43)
Pi =exp( ux
2D)
2exp
(r1t
α2 − α1
)[exp(−eigi)erfc(
ei2t−
12 − git
12 ) + exp(eigi)erfc(
ei2t−
12 + git
12 )],
(2.44)
ei =
√αiDx
x, g1 =
√u2
4Dα1
+r1α1
+r1
(α2 − α1), g2 =
√u2
4Dα2
+r1
(α2 − α1). (2.45)
This completes the derivation of analytical solution for the linear CDR-system of two
species. In the Chapter 3, this analytical solution will be used to analyze the performance
of suggested numerical schemes.
Next, some discussions on the fundamental notions of conservation laws, hyperbolic sys-
tems, weak solutions and Riemann problems are presented. This theoretical background
is helpful for understanding the nonlinear convection dominated radiation hydrodynamical
and hyperbolic heat conduction models in Chapters 5 and 6, respectively. These systems
of conservation laws are the limiting cases of general nonlinear CDR models.
24
2.3 Conservation Laws
Conservation laws are the mathematical statements describing the transportation of prop-
erties of fluids which are needed to be conserved. This notion is fulfilled by considering
fluid enclosed in some fixed arbitrary volume V . The net change in the conserved quan-
tity is zero due to the flow of fluid across the surface S of this specific volume in a given
time interval. Generally, a vector of conserved variables, depending on space and time
coordinates (t, x, y, z), is represented as
w(t, x, y, z) = (w1(t, x, y, z), w2(t, x, y, z), · · · , wm(t, x, y, z))T , (2.46)
where m represents the number of vector components. The corresponding vector of flux
functions can be written as
f(w) = (f1(w), f2(w), · · · , fm(w))T . (2.47)
The flow over the surface S of arbitrary volume V within the time interval [t0, t0+dt] yields
∫∫∫V
[w(t0 + dt, x, y, z)−w(t0, x, y, z)] dV +
t0+dt∫t0
∫∫S
f(w(t, x, y, z))dSdt = 0, (2.48)
where the first volume integral is taken on the variables wi, i = 1, · · · ,m, at initial and final
times and the second integral represents fluxes across the surface S within time interval
[t0, t0 + dt]. The application of fundamental theorem of calculus to first integral and the
the divergence theorem on the second integral give the following expression
t0+dt∫t0
∫∫∫V
(∂w
∂t+∇ · f(w)
)dV dt = 0 . (2.49)
As the volume is chosen arbitrarily, we obtain the following form of the conservation laws
∂w
∂t+∇ · f(w) = 0 . (2.50)
According to this equation, the net change in w in the given space-time domain is zero.
Thus, w is a vector of conserved variables.
25
2.4 Hyperbolic Conservation Laws
The investigated models in fifth and sixth chapters of the thesis are hyperbolic in nature
and could be considered as special case of CDR systems. In one space dimensions, the
system of conservation laws can be written as
∂w
∂t+∂f(w)
∂x= 0, (2.51)
with w = (w1, ..., wm)T and f = (f1(w), ..., fm(w))T . The chain rule allows us to reduce
the system into quasi-linear form
∂w
∂t+ f ′
∂w
∂x= 0 , (2.52)
where f ′ = ∂f(w)∂w
. The Jacobian matrix of the flux function is given as
A(w) =f(w)
∂w=
(∂
∂w1
, ...,∂
∂wm
)T· (f1(w), ..., fm(w)) . (2.53)
The eigenvalues λi, i = 1, 2, ...,m, of a matrix A are the solutions of characterstic polyno-
mials
| A− λI |= det(A− λI) = 0. (2.54)
Here, we use the usual notation of identity matrix as I. A system is said to be hyperbolic at
a point (t, x) if the matrix A has m real eigenvalues λ1, λ2, ..., λm with a corresponding set
of m linearly independent right eigenvectors r1, r2, ..., rm. The system is called strictly hy-
perbolic when all eigenvalues λi are distinct and weakly hyperbolic when some eignevalues
are coinciding.
2.5 Weak Solutions
The integral form of the governed conservation laws can be used to handle discontinuities
in fluid flows. On integrating Eq. (2.51) over the finite volume ∆Vi with the domain
xi− 12≤ x≤ xi+ 1
2gives ∫
∆Vi
(∂w
∂t+∂f(w)
∂x
)dV = 0. (2.55)
26
By applying the Gauss divergence theorem, we get∫∆Vi
∂f(w)
∂xdV =
∮Si
f(w) · ds. (2.56)
where Si is surface enclosing ∆Vi. Thus, it follows
d
dt
∫∆Vi
wdV +
∮Si
f(w) · ds = 0. (2.57)
Eq. (2.57) demonstrates that the change in time of the quantity w in a volume ∆Vi is due
to the flux through the volume’s surface Si. Finally, the integration of Eq. (2.57) over a
time interval tn ≤ t ≤ tn+1 gives∫ tn+1
tn
(d
dt
∫∆Vi
wdV
)dt+
∫ tn+1
tn
(∮Si
f(w) · ds)dt = 0. (2.58)
This is more suitable form of conservation laws, since it does not depend on the conti-
nuity of the state vectors. Thus, integral formulation is also satisfied in the presence of
discontinuities.
2.6 Riemann Problem
A Riemann problem is a Cauchy problem with initial single jump discontinuity, for example
∂w
∂t+∂f(w)
∂x= 0 , (2.59)
with initial data
w(0, x) =
wL , x < 0 ,wR , x > 0 .
(2.60)
In this problem the system has discontinuous initial data given by two states, a left wL
and a right wR, separated by a plane across which at least one of the fields experiences
an instantaneous jump, i.e. for at least one wLi= wRi
. The study of a general Riemann
problems allows us to investigate the behavior of weak solutions.
27
Chapter 3
Models of Gas-Solid Reaction andKelar-Segal for Chemotaxis
28
In this chapter, two different CDR systems are numerically investigated. The first one is the
one-dimensional heterogeneous gas-solid reaction (GSR) model, while the second one is the
two-dimensional Keller-Segal (KS) model of chemotaxis. The governing equations of the
models form systems of mixed type nonlinear partial differential equations (PDEs) of CDR-
type. Both models are among the challenging models currently available in the literature.
The stiff source terms in the systems adds more complexities in the models and challenges
to the solution techniques. The space-time CE/SE-method [16, 132] are extended for the
first time to solve these models. Several test problems are carried out. The numerical
results are verified through comparison with those obtained from the staggered central
schemes [49, 91] which are also implemented for the first time to these models. Moreover,
the accuracy of numerical results are analyzed against the analytical solutions of linear
CDR models and the results of considered flux-limiting finite volume schemes [55, 68]. The
chapter is concluded with the summary of current contribution.
3.1 The Gas-Solid Reactions (GSRs) Model
This CDR system simulates the heterogeneous gas-solid reaction (GSR) which can be used
in the modeling of heavy oil recovery, the pyrolysis of coal and biomass, the roasting and
reduction of ores, the incineration of waste, the solids combustion, the acid gases absorption
by lime, the deposition of reactive vapor phase, the ceramic materials production, the
gasification of coal, the manufacturing of catalyst, and so on [6, 44]. The exothermic nature
of GSR produces large gradients of temperature which generates steep reaction fronts in
the concentration profile. The capturing of these fronts is very difficult for less accurate
numerical methods. Therefore, efficient and accurate numerical methods are essential for
producing physically realistic solutions. In this work, the space-time CE/SE-method of
Chang [16] and the staggered central scheme of Nessyahu and Tadmor [91] are extended
to solve this CDR system.
The contents of this chapter are published in Industrial & Engineering Chemistry Research, 51 (2012)9173-9185.
29
3.1.1 Governing equations
The model is based on the Rajaiah et al. [98] formulation which is a pseudo-homogeneous
two-phase model in one-space dimension. The reaction takes place between the solid phase
and a gaseous oxidizer (Nitrogen or Oxygen), the product of which is again a gas or solid.
In the solid phase, Fourier law of heat conduction holds. The sintering effects are neglected
and it is assumed that porosity of the system remains constant throughout the process.
The effects of radiation are included into thermal conductivity and a first order reaction is
assumed corresponding to solid and gas reactants. All physical properties of the system are
considered constant. The governing differential equation describing propagation of reaction
fronts form a set of nonlinear mixed parabolic-hyperbolic PDEs [44, 98]
ρκp∂T
∂t= λ
∂2T
∂x2− uρκp
∂T
∂x+ k0(−∆H)cgcse
− ERT , (3.1)
ϵ∂cg∂t
= Dϵ∂2cg∂x2
− u∂cg∂x
− k0cgcse− E
RT , (3.2)
(1− ϵ)∂cs∂t
= −k0cgcse−ERT . (3.3)
The initial and boundary condition are given as
T = T0 , t = 0 , x ∈ [0,∞] , (3.4)
cg = cg0 , t = 0 , x ∈ [0,∞] , (3.5)
cs = cs0 , t = 0 , x ∈ [0,∞] , (3.6)
−λ∂T∂x
= ρκpu(Tinlet − T ) at x = 0 , (3.7)
∂T
∂x= 0 at x = ∞ , (3.8)
−ϵD∂cg∂x
= u(cginlet− cg) at x = 0 , (3.9)
∂cg∂x
= 0 at x = ∞ . (3.10)
In the above model, ρκp = ρκpϵ+ ρsκps(1− ϵ) is the effective heat capacity, T denotes the
temperature, c represents the reactant concentration, ρ denotes the density, κp stands for
the heat-capacity, λ is the averaged thermal-conductivity, k0 is the pre-exponential factor,
30
∆H is the reaction heat, E represents the activation energy, R denotes the universal gas
constant, D is the coefficient of molecular diffusion, u is the velocity, ϵ is the porosity of
solid material, t is the time, and x represents the space coordinate. Further, the subscripts
g stands for the gas and s for the solid. The following substitutions are introduced to get
the dimensionless form of the model (3.1)-(3.3) [44, 98]
ψ =E(T − T ∗)
RT ∗2 , cdg =cg
cginlet, cds =
cscsinlet
, xd =
√ρκpλt∗
x , (3.11)
td =Ek0(−∆H)cgiρκpRT ∗2 exp(
−ERT ∗ )t , t∗ =
ρκpRT∗2
Ek0(−∆H)cginletcsinletexp(
E
RT ∗ ) , (3.12)
x∗ =
√λt∗
ρκp, P eH =
uρκpλ
√λt∗
ρκp, P eM =
u
ϵD
√λt∗
ρκp, Le =
λ
ρκpD, (3.13)
γg =ρκpRT
∗2
ϵE(−∆H)cginlet, γs =
ρκpRT∗2
(1− ϵ)E(−∆H)cginlet, β =
RT ∗
E. (3.14)
These are standard group of parameters available in the literature of combustion to get
dimensionless form. With the help of this group of scaling, the dimensionless equations are
given as
∂ψ
∂td=∂2ψ
∂x2d− PeH
∂ψ
∂xd+ cdgcds exp(
ψ
βψ + 1) , (3.15)
∂cdg∂td
=1
Le
∂2cdg∂x2d
− PeMLe
∂cdg∂xd
− γgcdgcds exp(ψ
βψ + 1) , (3.16)
∂cds∂td
= −γscdgcds exp(ψ
βψ + 1) . (3.17)
The dimensionless initial and boundary conditions are given as
ψ = ψ0 , x ∈ [0,∞] , t = 0 , (3.18)
cdg = cdg0 , x ∈ [0,∞] , t = 0 , (3.19)
cds = cds0 , x ∈ [0,∞] , t = 0 , (3.20)
∂cdg∂xd
= −PeM(1− cdg) , t ≥ 0 , x = 0 , (3.21)
∂ψ
∂xd= −PeH(ψ − ψ0) , t ≥ 0 , x = 0 , (3.22)
∂cdg∂xd
= 0 , x = ∞ . (3.23)
This completes the one-dimensional formulation of gas-solid reactions model.
31
3.1.2 The one-dimensional CE/SE-method
Here, the one dimensional CE/SE-method of Chang [16] is briefly reviewed. Let us first
re-write Eqs. (3.15)-(3.17) as
∂w
∂t+ a
∂w
∂x− b
∂2w
∂x2= Q(w), (3.24)
where
w =
ψcdgcds
, a =
PeH 0 00 PeM
Le0
0 0 0
, b =
1 0 00 1
Le0
0 0 0
, (3.25)
Q(w) =
cdgcds exp(ψ
βψ+1)
−γgcdgcds exp( ψβψ+1
)
−γscdgcds exp( ψβψ+1
)
. (3.26)
Now, Eq. (3.24) can be re-written as
∂w
∂t+
∂
∂x
(aw − b
∂w
∂x
)= Q(w). (3.27)
Therefore, the above equation has the following conservation form:
∂w
∂t+∂f
∂x= Q , (3.28)
where f = aw − bwx. Let x1 = t and x2 = x be the coordinates of Euclidean-space (E2)
and let for k = 1, 2, 3, Fk = [wk, fk,−Qk]T . After applying the Gauss divergence theorem,
Eq. (3.28) becomes: ∮s(U)
Fk.ds = 0, k = 1, 2, 3, (3.29)
where k stands for the total number of equations and s(U) denotes the boundary of a
space-time domain U . Eq. (3.29) is restricted to the conservation-element (CE) in the
space-time domain which permits the flow variables discontinuities. The actual numerical
integration is done directly on the solution elements (SEs).
Let Ω denote the set of grid points (j, n) with n = 0,±12,±1,±3
2, · · · and for every n, j =
n ± 12, n ± 1, n ± 3
2, · · · . With each (j, n) ∈ Ω one SE is associated. Assume that SE(j, n)
32
is the inner shaded space-time region in Figure 3.1. It includes segments of horizontal and
vertical lines and their close neighborhood. The actual size of the neighborhood is not a
concern. At point (t, x) in SE, the approximation of wk(t, x), fk(t, x) and Fk(t, x) are given
as w∗k(t, x; j, n), f
∗k (t, x; j, n) and F
∗k (t, x; j, n), respectively. Let
w∗k(t, x; j, n) = (wk)
nj + (wkt)
nj (t− tn) + (wkx)
nj (x− xj) , (3.30)
where (wk)nj , (wkx)
nj and (wkt)
nj are constant in SE. Similarly, we can define
f ∗k (t, x; j, n) = (fk)
nj + (fkt)
nj (t− tn) + (fkx)
nj (x− xj) . (3.31)
Figure 3.1: Staggered grid near SE(j, n), the CE−(j, n) and CE+(j, n), as well as theCE(j, n).
The chain rule gives
(fkx)nj =
3∑m=1
(fk,m)nj (wmx)
nj , (3.32)
(fkt)nj =
3∑m=1
(fk,m)nj (wmt)
nj , (3.33)
33
where
fk,m =∂fk∂wm
, k,m = 1, 2, 3 . (3.34)
The Jacobian matrix formed by fk,m, k,m = 1, 2, 3, is expressed as
A =
PeH 0 00 PeM
Le0
0 0 0
. (3.35)
Because Fk = (fk, wk,−Qk), one can write
F ∗k (t, x : j, n) = (f ∗
k (t, x : j, n), w∗k(t, x : j, n),−Q∗
k(t, x : j, n)) . (3.36)
Due to their definitions, (fk)nj are functions of (wk)
nj , (fkx)
nj are functions of (wk)
nj and
(wkx)nj , and (fkt)
nj of (wk)
nj and (wkt)
nj . Further, we assume that for any (t, x) in SE,w∗
k,
f ∗k and Q∗
k satisfy the Eq. (3.28), that is,
∂w∗k(t, x; j, n)
∂t+∂f ∗
k (t, x; j, n)
∂x= Qk(w
∗k) . (3.37)
Due to Eq. (3.28), Eq. (3.37) can be equivalently written as
(wkt)nj = −(fkx)
nj + (Qk)
nj . (3.38)
Eqs. (3.30) and (3.31) are first-order Taylor’s expansions, therefore Eq. (3.27) implies for
fkx = akwkx:
(wkt)nj = −ak(wkx)nj + (Qk)
nj . (3.39)
Thus, diffusion in Eq. (3.24) puts no impact in Eq. (3.39). Resultantly, there is no effect
of the diffusion term on the time variation of w∗k(t, x; j, n) within SE. Thus, one gets from
Eqs. (3.30) and (3.39)
w∗k(t, x; j, n) = (wk)
nj + (wkx)
nj ((x− xj)− ak(t− tn)) . (3.40)
Therefore, (wk)nj and (wkx)
nj are the independent variables of the scheme at every grid point
(j, n).
34
In Figure 3.1, the rectangular non-overlapped regions represent the CEs. The two CEs,
CE−(j, n) and CE+(j, n), correspond to every inner grid point (j, n). Thus, CE at point
(j, n) is the combination of CE−(j, n) and CE+(j, n). Moreover, the CE−(j, n) boundary
contains the subset of SE(j, n) and SE(j − 12, n − 1
2), and that of CE+(j, n) is obtained
from the subset of SE(j, n) and SE(j + 12, n− 1
2). Consider the integral representation of
Eq. (3.28)
∫∂Ω
wkdx− fkdt+
∫Ω
Qkdxdt = 0 , k = 1, 2, 3 , (3.41)
where, ∂Ω is the boundary of Ω. On integrating the rectangular region over the mesh
points (j ± 12, n − 1
2), we obtain (wk)
nj and (wkx)
nj . On integrating over the mesh point
(j − 12, n− 1
2), Eq. (3.41) gives
∫ xj
xj− 1
2
w∗k(t
n− 12 , x)dx−
∫ tn
tn− 12
f ∗k (t, xj)dt−
∫ xj
xj− 1
2
w∗k(t
n, x)dx
+
∫ tn
tn− 12
f ∗k (t, xj− 1
2)dt = −
∫ tn
tn− 12
∫ xj
xj− 1
2
Qkdxdt . (3.42)
Substituting the values of w∗ and f ∗ and using fk = akwk − bkwkx and fkt = akwkt =
ak(−akwkx) = −a2kkkx, we get after simplification
(wk)n− 1
2
j− 12
(∆x
2) + (wkx)
n− 12
j− 12
(∆x2
8)− ak(wk)
nj (∆t
2) + bk(wkx)
nj (∆t
2)
−a2k(wkx)nj (∆t2
8)− (wk)
nj (∆x
2) + (wkx)
nj (∆x2
8) + a(wk)
n− 12
j− 12
(∆t
2)
−bk(wkx)n− 1
2
j− 12
(∆t
2)− a2k(wkx)
n− 12
j− 12
(∆t2
8) = −∆x∆t
4(Qk)
n− 12
j . (3.43)
Now integrating Eq. (3.41) over the mesh point (j + 12, n− 1
2), we get
∫ xj+1
2
xj
w∗k(t
n− 12 , x)dx−
∫ tn
tn− 12
f ∗k (t, xj+ 1
2)dt−
∫ xj+1
2
xj
w∗k(t
n, x)dx
+
∫ tn
tn− 12
f ∗k (t, xj− 1
2)dt = −
∫ tn
tn− 12
∫ xj+1
2
xj
Qkdxdt . (3.44)
35
After some simplification, we obtain
(wk)n− 1
2
j+ 12
(∆x
2)− (wkx)
n− 12
j+ 12
(∆x2
8)− ak(wk)
n− 12
j+ 12
(∆t
2) + bk(wkx)
n− 12
j+ 12
(∆t
2)
+a2k(wkx)n− 1
2
j+ 12
(∆t2
8)− (wk)
nj (∆x
2)− (wkx)
nj (∆x2
8) + ak(wk)
nj (∆t
2)
−bk(wkx)nj (∆t
2) + a2k(wkx)
nj (∆t2
8) = −∆x∆t
4(Qk)
n− 12
j+1 . (3.45)
The summation of Eqs. (3.43) and (3.45) gives
(wk)nj =
(1 + νk)
2(wk)
n− 12
j− 12
+(1− νk)
2(wk)
n− 12
j+ 12
+ (1− ν2k − ξk)[(w+
kx)n− 1
2
j− 12
− (w+kx)
n− 12
j+ 12
]+
∆t
4
[(Qk)
n− 12
j + (Qk)n− 1
2j+1
], (3.46)
and the subtraction of Eqs. (3.43) and (3.45) implies
(w+kx)
nj =
−(1− ν2k)
2(1− ν2k + ξk)
[(wk)
n− 12
j− 12
− (wk)n− 1
2
j+ 12
](3.47)
+−1
2(1− ν2k + ξk)(1− ν2m − ξk)
[(1− νk)(w
+kx)
n− 12
j− 12
+ (1 + νk)(w+kx)
n− 12
j+ 12
]− ∆t
8(1− ν2k + ξk)
[(1− νk)(Qk)
n− 12
j − (1 + νk)(Qk)n− 1
2j+1
].
Here,
1− ν2k + ξk = 0 , (3.48)
and
νk =ak∆t
∆x, ξk =
4bk∆t
(∆x)2, (w+
kx)nj =
∆x
4(wkx)
nj . (3.49)
This completes the derivation of the one-dimensional CE/SE-method.
3.1.3 The one-dimensional central schemes
Here, the high-resolution central scheme [91] is briefly presented for solving Eq. (3.28).
The method is of predictor-corrector type and is applied in two steps. The predictor step
corresponds to the prediction of the midpoint values by considering the non-oscillatory
36
piecewise-linear reconstructions of the cell averages. In the second corrector step, stag-
gered averaging and the predicted mid-values are utilized to get the updated cell averaged
solution. In summary, the scheme can be presented as
Predictor: wn+ 1
2i =wn
i −ξ
2fx(wn
i ) , (3.50)
Corrector: wn+1i+ 1
2
=1
2(wn
i +wni+1) +
1
8(wx
i −wxi+1)
− ξ[fn+ 1
2i+1 − f
n+ 12
i
]+
∆t
2
[Qn+ 1
2i +Q
n+ 12
i+1
], (3.51)
where, ξ = ∆t/∆x. Moreover, 1∆x
fx(wi) is the approximated derivative of F(t, x = xi)
1
∆xfx(wi) =
∂
∂xf(w(t, x = xi) +O(∆x) . (3.52)
The derivatives of conservative variables are approximated as
wxi =MM
θ∆wi+ 1
2,θ
2
(∆wi+ 1
2+∆wi− 1
2
), θ∆wi− 1
2
, (3.53)
where 1 ≤ θ ≤ 2 is a parameter and ∆ denotes central differencing,
∆wi+ 12= wi+1 −wi .
Here, MM denotes the min-mod nonlinear limiter
MMx1, x2, ... =
minixi if xi > 0 ∀i ,maxixi if xi < 0 ∀i ,0 otherwise .
(3.54)
The fluxes fx(wi) are computed by the same manner as discussed for wx in Eq. (3.53).
3.2 The Keller-Segel (KS) model
This CDR system is a classical Keller-Segel (KS) model of the chemotaxis which admits
solutions that generate delta-type singularities within a finite simulation time [21, 38,
45, 46]. Chemotaxis is a phenomenon in which cells change their state of movement in
the presence of certain chemical substances and, thus, approaching chemically favorable
environments and avoid the unfavorable ones. The new chemotaxis model introduced by
37
Keller and Segel [53] is considered as a regularized KS system. This regularization is
based on a basic physical principle: boundedness of the chemotactic convective flux, which
should depend on the gradient of the chemoattractant concentration in a nonlinear way.
Solutions of the new system may develop spiky structures [21, 38]. A two-dimensional
CE/SE-method [132] is implemented to solve the model. The validity of the scheme is
verified by considering different numerical test problems and by comparing the results of
current method with those of central scheme [91].
3.2.1 Governing equations
The dimensionless KS-system is given as [21, 38]ρt +∇ · (ξρ∇c) = ρ ,ct = ∆c− c+ ρ .
(3.55)
Here, ρ(t, x, y) represent the cell density, c(t, x, y) denotes the concentration of chemoat-
tractant, and ξ is a sensitivity constant of chemotactic. The above model can be replaced
by a simpler model if it is assumed that the concentration c changes at smaller time scale
compared to the density ρ. Then, the equation for c becomes an elliptic equation [21, 38]
ρt +∇ · (ξρ∇c) = ∆ρ ,∆c− c+ ρ = 0 .
(3.56)
Let us consider the two-space dimensions and let (u, v) = (cx, cy). On differentiating the
second equation of Eq. (3.55) with respect to x and y and rewriting the system in equivalent
two-space dimensions, we obtain
ρt + (ξρu)x + (ξρv)y = ρxx + ρyy ,(cx)t = (cx)xx + (cx)yy − cx + ρx ,(cy)t = (cy)xx + (cy)yy − cy + ρy .
(3.57)
After substituting u = cx and v = cy, we get
ρt + (ξρu)x + (ξρv)y = ρxx + ρyy ,ut − ρx = uxx + uyy − u ,vt − ρy = vxx + vyy − v .
(3.58)
This system can be expressed in the form of a CDR system as
wt + f(w)x + g(w)y = ∆w +Q(w), (3.59)
38
where w = (ρ, u, v)T , f = (ξρu,−ρ, 0)T , g = (ξρv, 0,−ρ)T , and Q = (0,−u,−v)T . The
Jacobian matrices of f and g are given as:
∂f
∂w=
ξu ξρ 0−1 0 00 0 0
and∂g
∂w=
ξv 0 ξρ0 0 0−1 0 0
. (3.60)
The discussion of model is now complete. The next step is to solve this model by using
some numerical technique. In this work, the CE/SE-method is suggested which is derived
below.
3.2.2 The two-dimensional CE/SE-method
The details of this CE/SE-method are given in [132, 97]. Assume that t, x, and y are the
coordinates of a three-dimensional Euclidean space E3. The integral representation of Eq.
(3.59) is ∮s(U)
Fk · ds = 0 , k = 1, 2, 3 . (3.61)
Here, Fk = [wk, fk, gk,−(wx)k,−(wy)k,−Qk]T denotes the current density vectors in E3,
k denotes the number of equations and s(U) is the boundary of an arbitrary space-time
domain U .
The computational domain is partitioned into non-overlapped uniform rectangular cells as
given in Fig. 3.2. The centroid of each cell is denoted by a circle symbol that also represents
the grid point in this CE/SE-method, for instance the point Q in Fig. 3.2(b). The set of
these points is denoted by Ω. The method utilizes one conservation element (CE) and the
corresponding one solution element (SE) in each element.
In Fig. 3.2(b), the grid points Q, A1, A2, A3 and A4 are lying at time level t = tn at
which the new numerical solutions of flow variables are to be calculated. The points
Q′, A′1, A
′2, A
′3 and A
′4 are the corresponding points at time level t = tn−1/2 and the points
39
Figure 3.2: Space-time geometry of the modified CE/SE-method: (a) representative gridpoints in x-y plane, (b) the definitions of CE and SE.
Q′′, B′′1 , B
′′2 , B
′′3 and B′′
4 are located at time level t = tn+1/2. The same rule is applied
for all grid points to denote the time levels. The SE associated to point Q is defined
by the union of one horizontal plane segment A1A2A3A4 and two vertical plane segments
B′′1B
′1B
′3B
′′3 and B′′
2B′2B
′4B
′′4 . The CE belonging to point Q is represented by the cylin-
der A1B1A2B2A3B3A4B4 A′1B
′1A
′2B
′2A
′3B
′3A
′4B
′4. The centroid Q of the top surface of this
CE, denoted by polygon A1B1A2B2A3B3A4B4 is taken as the solution point. All the vari-
ables and their spatial derivative are stored at point Q denoting the set of solution points Ω.
Inside each SE, the flow variables are assumed smooth and the structure of the flow solution
is discretized by a prescribed function. Following Chang’s approach, the distribution of wm,
kk and gk is approximated by using first-order Taylor expansions about point Q. In other
40
words, for any (t, x, y) ∈ SE(Q), wk(t, x, y), fk(t, x, y) and gk(t, x, y) are approximated as
w∗k(t, x, y) = (wkt)Q(t− tn) + (wk)Q + (wkx)Q(x− xQ) + (wky)Q(y − yQ) , (3.62)
f ∗k (t, x, y) = (fkt)Q(t− tn) + (fk)Q + (fkx)Q(x− xQ) + (fky)Q(y − yQ) , (3.63)
g∗k(t, x, y) = (gkt)Q(t− tn) + (gk)Q + (gkx)Q(x− xQ) + (gky)Q(y − yQ) . (3.64)
Here tn, xQ, yQ are the space-time coordinates of Q. The variables wk,wkt, wkx and wky
on the left hand side of Eq (3.62) are the discretized variables. Once these variables are
known, the structure of the flow solution within SE is completely described. But these
variables are not fully independent. By using Eq. (3.59), we obtain
(wkt)Q = −(fkx)Q − (gky)Q − (wkxx)Q − (wkyy)Q − (Qk)Q . (3.65)
By applying chain rule, the x−derivatives of fluxes are calculated as
(fkx)Q = (Akl)Q(wlx)Q , (gkx)Q = (Bkl)Q(wlx)Q , k, l = 1, 2, 3 . (3.66)
where (Akl)Q and (Bkl)Q denote the elements of the Jacobian matrices of fk and gk at
point Q. Analogously, (fky)Q, (kky)Q, (fkt)Q and (gkt)Q are obtained. Thus, (wk)Q, (wkx)Q
and (wky)Q are the only independent discrete variables in each SE. For the derivation of
scheme, the continuous space-time flux vector Fk(t, x, y) is replaced by a discrete one
F∗k(t, x, y) = [w∗
k(t, x, y), f∗k (t, x, y), g
∗k(t, x, y)]
T (3.67)
and the Eq. (3.61) by its discrete counterpart∮S(CE(Q))
F∗k · ds = 0 . (3.68)
On substituting Eqs. (3.62)-(3.67) into Eq (3.68), the following algebraic equation can be
obtained
(wk)nQ =
(4∑l=1
R(l)k
)/s+
s∆t
2(Qk)
n− 12
Q , (3.69)
41
where
R(l)k =s(l)q
[(wk)
n−1/2Al
+ (x(l)q − xAl)(wkx)
n−1/2Al
+ (y(l)q − yAl)(wky)
n−1/2Al
]−
2∑m=1
n(l)mx
[(fk)
n−1/2Al
+ (x(l)m − xAl)(fkx)
n−1/2Al
+ (y(l)m − yAl)(fky)
n−1/2Al
+ ∆t/4 · (fkt)n−1/2Al
− (wkx)n−1/2Al
]−
2∑m=1
n(l)ky
[(gk)
n−1/2Al
+ (x(l)m − xAl)(gkx)
n−1/2Al
+ (y(l)m − yAl)(gky)
n−1/2Al
+∆t/4 · (gkt)n−1/2Al
− (wky)n−1/2Al
], (3.70)
for l = 1, 2, 3, 4 indicating the spatial flux contribution from the four neighboring points
and k = 1, 2, 3 denote the three flow equations. Here, (x(l)q , y
(l)q ) and s
(l)q are the spatial
coordinates of the centroids and the area of the four neighboring quadrilaterals A1B1QB4,
A2B2QB1, A3B3QB2 and A4B4QB3. Moreover, n(l)m = [n
(l)mx, n
(l)my, 0]T represent the eight
surface vectors of the eight lateral planes: A′1B
′4A1B4, A
′1B
′1A1B1, A
′2B
′1A2B1, A
′2B
′2A2B2,
A′3B
′2A3B2, A
′3B
′3A3B3, A
′4B
′3A4B3 and A
′4B
′4A4B4. Note that, the surface vector is defined
as the unit outward normal vector (outward from the interior of the CE) multiplied by its
area. Finally, s is the area of the polygon A1B1A2B2A3B3A4B4 that also represents the
top surface of the present CE. Because all flow conditions at the n − 1/2 time level are
known, Eqs. (3.69) and (3.70) represent an explicit method for calculating wnk at point Q.
To calculate (wkx)Q and (wky)Q, a central difference-type reconstruction procedure is em-
ployed. Due to Taylor series
(w′k)nAl
= (wk)n−1/2Al
+∆t
2(wkt)
n−1/2Al
, l = 1, 2, 3, 4 . (3.71)
This predicted value actually represents a linear expansion in time. By using the values
of (wm)nA1, (wm)
nA2
and (wm)nQ, the first pair of spatial derivatives of flow variables can be
obtained, i.e., w(1)kx , w
(1)ky at point Q
w(1)kx = Dkx/D, w
(1)ky = Dky/D , (3.72)
42
where
D =
∣∣∣∣∆x1 ∆y1∆x2 ∆y2
∣∣∣∣ , Dkx =
∣∣∣∣∣∆w(1)k ∆y1
∆w(2)k ∆y2
∣∣∣∣∣ , Dmy =
∣∣∣∣∣∆x1 ∆w(1)k
∆x2 ∆w(2)k
∣∣∣∣∣ , (3.73)
∆xl = (xAl− xQ) , ∆yl = (yAl
− yQ), ∆w(l)k = ((w′
k)nAl
− (wk)nQ) . (3.74)
Similarly, the solutions at A2, A3 and Q gives the second pair w(2)kx , w
(2)ky , the solutions at
A3, A4 and Q gives the third pair w(3)kx , w
(3)ky while the solutions at A4, A1 and Q gives the
fourth pair w(4)kx , w
(4)ky . Finally, re-weighting procedure is used to calculate wkx and wky at
Q as follows [16, 17, 132]
(wkx)nQ =
0 , if θkl = 0 , (l = 1, 2, 3, 4) ,4∑
m=1
[(w(m)k )β w
(m)kx ]/
4∑m=1
(w(m)k )β , otherwise ,
(3.75)
(wky)nQ =
0 , if θkl = 0 , (l = 1, 2, 3, 4) ,4∑
m=1
[(w(m)k )β w
(m)ky ]/
4∑m=1
(w(m)k )β , otherwise ,
(3.76)
where
w(m)k =
4∏l=1 ,l =m
θkl , θkl =
√(w
(l)kx)
2 + (w(l)ky)
2 . (3.77)
In Eqs. (3.75) and (3.76), the value of constant β can be either 1 or 2. The Eqs. (3.75)
and (3.76) have capability to preserve the local monotonicity (positivity) of the scheme.
This completes the scheme’s derivation.
3.2.3 The two-dimensional central scheme
Here, the two-dimensional central scheme of Jaing and Tadmor [49] is presented for solving
Eq. (3.59). The scheme has again a two-step predictor-corrector form. Starting with the
cell averages, wni,j, we use the first-order predictor step for the evolution of the midpoint
values, wn+ 1
2i,j , followed by the second-order corrector step for computation of the new cell
averages wn+1i,j . Like the one-dimensional case, no exact (approximate) Riemann solvers
are needed. The non-oscillatory behavior of the scheme is dependent on the reconstructed
discrete slopes, wx, wy, fx(w), and gy(w). At each time step the grid is staggered to avoid
the flux calculation at the cell interfaces. The scheme is summarized below.
43
In the predictor step one has to calculate the mid-point values
wn+ 1
2i,j = wn
i,j −ξ
2fx(wn
i,j)−η
2gy(wn
i,j) +ξ
2((wx)i,j)
x +η
2((wy)i,j)
y , (3.78)
where, ξ = ∆t/∆x and η = ∆t/∆y. This step is followed by a corrector step to get the
updated values at the next time step
wn+1i+ 1
2,j+ 1
2
=1
4(wn
i,j +wni+1,j +wn
i,j+1 +wni+1,j+1) +
1
16(wx
i,j −wxi+1,j)−
ξ
2
(fn+ 1
2i+1,j − f
n+ 12
i,j
)+
1
16(wx
i,j+1 −wxi+1,j+1)−
ξ
2
(fn+ 1
2i+1,j+1 − f
n+ 12
i,j+1
)+
1
16(wy
i,j −wyi,j+1)
− η
2
(gn+ 1
2i,j+1 − g
n+ 12
i,j
)+
1
16(wy
i+1,j −wyi+1,j+1)−
η
2
(gn+ 1
2i+1,j+1 − g
n+ 12
i+1,j
)+ξ
2
(((wx)
x)n+ 1
2i+1,j − ((wx)
x)n+ 1
2i,j
)+η
2
(((wy)
y)n+ 1
2i,j+1 − ((wy)
y)n+ 1
2i,j
)(3.79)
+ξ
2
(((wx)
x)n+ 1
2i+1,j+1 − ((wx)
x)n+ 1
2i,j+1
)+η
2
(((wy)
y)n+ 1
2i+1,j+1 − ((wy)
y)n+ 1
2i+1,j
)+
∆t
4
(Qn+ 1
2i,j +Q
n+ 12
i+1,j +Qn+ 1
2i,j+1 +Q
n+ 12
i+1,j+1
).
This completes the derivation of numerical schemes.
3.3 Numerical Case Studies
In order to verify the accuracy of proposed numerical schemes in the presence of exact
solutions, two test problems of linear CDR models are considered.
Case 1: In the first problem, a linear CDR model describing the transport of single specie
is solved. This model is given by Eqs. (2.7) and (2.8) of Chapter 2 and its analytical
solution is given by Eqs. (2.14)-(2.17). The parameters used in the test problem are given
in the Table 3.1. Fig. 3.3 shows the comparison of analytical and numerical solutions on
200 grid points. Moreover, L1-errors in Table 3.2 shows that CE/SE, Koren and Superbee
schemes [55, 69] give comparable results. The errors produced by central and upwind
schemes are large as compared to other methods. However, increase in the number of grid
points improves the performance of central and upwind schemes.
44
Table 3.1: Case 1: Data used in the numerical simulation of single CDR model.
Parameters c01 α r time u DValue 1 2 0.01 50 1 0.1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x−axis
con
cen
tra
tion
, c
ExactCE/SESuperbeekorencentralupwind
Figure 3.3: Case 1: Results of single specie test problem at td=50.
Case 2: In the second test problem, the two species transport model given by Eqs. (2.20)-
(2.22) is solved. The parameters used in the problem are given in the Table 3.3. The
analytical solution of this model is given by Eqs. (2.40)-(2.45). A comparison of analytical
and numerical solutions is given in Fig. 3.4 on 200 grid points. It can be observed from
the errors of Table 3.4 that CE/SE, Koren and Superbee schemes give comparable results,
while the errors produced by central and upwind schemes are large as compared to other
methods.
45
Table 3.2: Case 1: Comparison of numerical L1-errors at different grid points.N =200 N=500 N=103
Methods c c cCE/SE 2.15× 10−4 3.42× 10−5 1.24× 10−5
Superbee 2.05× 10−4 3.51× 10−5 1.10× 10−5
Koren 2.34× 10−4 3.63× 10−5 1.37× 10−5
Central 3.81× 10−4 4.16× 10−5 2.01× 10−5
Upwind 7.80× 10−3 1.42× 10−3 7.36× 10−4
Table 3.3: Case 2: Data used in the numerical simulationParameters c01 c02 α1 α2 r1 Time u D
Value 1 0.0 2 1 0.01 50 1 0.18
3.3.1 Test problems for gas-sold reaction model
Here, three case studies are considered and the numerical results of CE/SE-method are
compared with the staggered central scheme [91]. The initial data of all test problems
are given in Table 3.5. In all test problems, the computational domain 0 ≤ x ≤ 200 is
discretized into 500 grid points. The first two test problems were also considered in [44].
The numerical error is calculated by using the following expression
Error =
[∫(φ− φref)
2dx]
[∫φ2dx]
12
, (3.80)
where φ can be either temperature or concentration and the subscript ref denotes reference
solution. The reference solution was obtained at 5 × 103 grid points by using CE/SE-
method. Table 3.6 shows a comparison of numerical errors at different numbers of grid
points and CPU times at N = 200. In this table, CE/SE-method is compared with the
Superbee, Koren, central, and first order upwind schemes [55, 69]. It can be observed that
the proposed CE/SE-method has produced less errors as compared to other schemes at
low computational cost. Moreover, Fig. 3.5 shows errors in the schemes at different num-
bers of grid points. It can be observed that in the CE/SE-method errors reduce faster on
increasing the numbers of grid points. Fig. 3.6 gives the results of problem 1 at td = 0.6.
The numerical results of both schemes are in good agreement with each other. However,
46
Table 3.4: Case 2: Comparison of numerical L1-errors at different grid points.N =200 N=500 N=103
Methods c1 c2 c1 c2 c1 c2CE/SE 0.0225 0.0077 0.0198 0.0055 0.0195 0.0418Superbee 0.0105 0.0017 0.0256 0.0047 0.0301 0.0056Koren 0.0346 0.0065 0.0343 0.0065 0.0342 0.0065Central 0.0673 0.0264 0.0247 0.0098 0.0111 0.0044Upwind 0.0531 0.0088 0.0421 0.0074 0.0382 0.0069
Figure 3.4: Case 2: Results of two species test problem at t=50.
CE/SE-method has better resolved the peak in the temperature profile as compared to the
central scheme. The numerical results are in good agreement with available results in the
literature [44].
The results of problem 2 at td = 2.5 are shown in Fig. 3.7. The reaction peak in the gas
concentration obtained by CE/SE-method has larger height than those of central scheme
and other methods presented in the article of Hassanzadeh et al. [44]. Similarly, the results
of problems 3 at td = 1.0 are given in Fig. 3.8. An overshoot can be observed in the
temperature profile of the central scheme which is very small in the results of CE/SE-
method. In all test problems, the results of both schemes show good agreement. However,
the CE/SE-method has better resolved the sharp discontinuities and peaks.
47
100 200 300 400 500 600 700 800 900 10000
0.05
0.1
0.15
0.2
0.25
number of grid points (N)
erro
rs in
ψ
CE/SESuperbeeKorencentralupwind
Figure 3.5: Problem 1: errors at different numbers of grid points.
48
0 50 100 150 200
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
x−axis
solid
con
cent
ratio
n
CESEcentral
Figure 3.6: Results of problem 1 at td = 0.6.
49
0 50 100 150 200−6
−4
−2
0
2
4
6
8
10
12
x−axis
tem
pera
ture
CESEcentral
Figure 3.7: Results of problem 2 at td = 2.5.
50
0 50 100 150 200−1
0
1
2
3
4
5
x−axis
tem
pera
ture
CESEcentral
Figure 3.8: Results of problem 3 at td = 1.0.
51
Table 3.5: Data used in the numerical simulation
Parameters γs γg β ψ0 cdg0 cds0 PeH PeM LeProblem 1 2 0.76 0.15 -0.5 0 1 100 200 1Problem 2 0.214 0.1 0.0793 -4.667 0 1 50 500 10Problem 3 2.0 0.1 0.1 -1.0 0.2 10 10 200 1.2
Table 3.6: Comparison of numerical errors (fractions) at different grid points
N =200 N=500 N=103 CPU(s)Methods ψ cdg cds ψ cdg cds ψ cdg cds N=200CE/SE 0.12 0.03 0.008 0.08 0.01 0.003 0.02 0.009 0.001 0.57Superbee 0.12 0.04 0.009 0.08 0.02 0.003 0.02 0.016 0.001 0.63Koren 0.13 0.04 0.009 0.09 0.02 0.003 0.03 0.017 0.001 0.56Central 0.13 0.04 0.009 0.09 0.03 0.004 0.03 0.018 0.001 0.55Upwind 0.17 0.11 0.012 0.13 0.09 0.005 0.09 0.069 0.002 0.44
3.3.2 Test problems for KS-model
Here, three two-dimensional test problems are considered for the current KS-model.
Problem 4
The initial data inside a square region [−12, 12]× [−1
2, 12] with ξ = 1 are taken as [21]
ρ(0, x, y) = 103 exp−102(x2+y2), c(0, x, y) = 5× 102 exp−50(x2+y2) . (3.81)
A uniform grid of 100 × 100 points with outflow boundary conditions is used. The cell
densities computed at t = 1 × 10−4 are displayed in Fig. 3.9. For comparison, the same
problem was also solved by the central scheme [49] and the results of both schemes are
compared through the one-dimensional plots along y = 0.0 in Fig. 3.9. Further, the cell
densities are computed at different times and their comparison along y = 0.0 are presented
in Fig. 3.10. The blowup of density forming a narrow spike is visible in Fig. 3.9 and Fig.
3.10. The figures show that density peaks become narrow and longer with the passage of
time. Moreover, it can be observed that both schemes give comparable results. However,
CE/SE-method resolves the peaks better. The numerical results are also in good agree-
52
ment with the available results in the literature and the peaks heights are comparable [21].
Problem 5
In this problem, ξ = 1 and the same square region [−12, 12]× [−1
2, 12] is considered. The initial
data are
ρ(0, x, y) = 103 exp−102(x2+y2), c(0, x, y) = 0. (3.82)
Fig. 3.11 shows the results at different times. In this problem the density ρ blows up at
later time than in problem 4. Resultantly, the spreading of cell density occur and the
maximum value is lesser than that in Problem 4. The plots show the spreading of density
profiles and the hight of peaks reduces with the passage of time. The number of grid points
is the same as in Problem 4. Once again, the results of CE/SE-method are superior than
the central scheme.
Problem 6
This problem considers another initial data with ξ = 1 as given as
ρ(0, x, y) = 840 exp−84(x2+y2), c(0, x, y) = 420 exp−42(x2+y2) . (3.83)
Fig. 3.12 shows the results at t = 3 × 10−4. In this problem ρ and c blowup at the origin
in a finite time. A fixed mesh with 100 × 100 points is considered. Both schemes have
comparable results but CE/SE-method has better resolved the density peak. This can be
further observed in the density plots obtained at different times in Fig. 3.13. The figures
show that density peaks become narrow and longer with the passage of time. The CE/SE-
method captures the height of density profile much better than the central scheme. The
numerical results are comparable to those obtained by Epshteyn [38].
3.4 Summary
The space time CE/SE-method was implemented for solving one and two-dimensional CDR
type PDEs. The major difficulties associated with the current systems include the forma-
53
tion of steep reaction fronts and blowup of the solution in a finite time. The traditional
numerical approaches are not capable to capture such rapid variation of the physical quan-
tities. The CE/SE-method was successfully implemented in one and two space dimensions
and numerical results of the method were found in good agreement with those available in
the literature and with central schemes on staggered grids. The staggered central schemes
were also implemented for the first time to these models. Both methods captured the rapid
variations in the solutions more efficiently and accurately. However, CE/SE-method per-
formed better for the solutions with sharp discontinuities and narrow peaks. The proposed
method produced less errors in the results as compared to other available methods.
54
Figure 3.9: Results of problem 4 at t = 1× 10−4.
55
Figure 3.10: Results of problem 4 at t = 4.4 × 10−6, t = 4.4 × 10−5 and t = 4.4 × 10−4
(from top to bottom).
56
−0.5 0 0.50
200
400
600
800
1000
x−axisρ
CE/SEcentral
−0.5 0 0.50
200
400
600
800
1000
x−axis
ρ
CE/SEcentral
−0.5 0 0.50
200
400
600
800
1000
x−axis
ρ
CE/SEcentral
Figure 3.11: Results of problem 5 at t = 5× 10−6, t = 1× 10−4 and t = 4.5× 10−4 (fromtop to bottom)
57
Figure 3.12: Results of problem 6 at t = 3× 10−4.
58
Figure 3.13: Results of problem 6 at t = 1.5×10−5, t = 4.5×10−5 and t = 3.0×10−4(fromtop to bottom)
59
Chapter 4
Theoretical Investigation of LiquidChromatographic Models
60
This chapter presents the analytical and numerical solutions of two established models
for simulating liquid chromatographic processes namely, the equilibrium dispersive model
(EDM) and the lumped kinetic model (LKM). Such models contain systems of convection-
diffusion partial differential equations coupled with some algebraic or differential equations.
Thus, these models are special cases of a general CDR-system in which reaction terms are
neglected. The models are analyzed using continuous and discontinuous Dirichlet and
Robin (Danckwert’s) boundary conditions. The Laplace transformation is used as a ba-
sic tool to solve them analytically for one-component adsorption under linear conditions.
The discontinuous Galerkin (DG) finite element method is applied to numerically approx-
imate the more general nonlinear liquid chromatographic models. The results obtained by
this method are compared with some finite volume schemes which are accessible from the
literature in the relevant Journals.
4.1 Motivation and Background
Chromatography is a highly selective separation and purification process used in a broad
range of industries, including pharmaceutical and biotechnical applications. In recent years,
this technology emerged as a useful tool to isolate and purify chiral molecules, amino-acids,
enzymes, and sugars. It has capability to deliver high purity and yield with reasoned pro-
duction rates even for difficult separations, for instance to isolate enantiomers or to purify
proteins. During the migration of mixture components through tabular columns filled with
suitable particles forming the stationary phase, composition fronts develop and propagate
governed by the adsorption isotherms providing a characteristic retention behavior of the
species involved. Separated peaks of desired purity can be collected periodically at the
column ’s outlet.
Different models were introduced in the literature for describing chromatographic processes,
such as the general rate model, various kinetic models, and the equilibrium dispersive model
(EDM) [39, 42, 43, 104].
Some parts of this chapter are accepted in Chemical Engineering Science Journal, 2012.
61
In this work, the EDM and a non-equilibrium adsorption lumped kinetic model (LKM)
[66] are solved analytically and numerically studies. The Laplace transformation is utilized
as a basic tool to transform the partial differential equations (PDEs) of the models for
linear isotherms to ordinary differential equations (ODEs). The corresponding analytical
solutions of EDM and LKM are obtained along with Dirichlet and Robin boundary con-
ditions [43, 57, 58, 59, 85, 86, 87, 88, 104, 107, 115]. If no analytical inversion could be
performed, the numerical inversion is used to generate the time domain solution for both
types of boundary conditions [100].
For non-linear adsorption isotherms, analytical solutions of the model equations are not
existing. For that reason, numerical simulations are needed to accurately predict the dy-
namic behavior of chromatographic columns. Steep concentration fronts and shock layers
may occur due to the convection dominated PDEs of chromatographic models and, hence,
an efficient numerical method is required to obtain accurate and physically realistic solu-
tions.
The discontinuous Galerkin (DG) methods have been widely used in the computational
fluid dynamics and could be a good choice to solve non-linear convection-dominated prob-
lems [3, 10, 8]. This technique was initially introduced for hyperbolic equations [99] . After
that, different versions of DG-methods were published for nonlinear hyperbolic systems
[22, 23, 24, 25, 26]. The DG-methods are robust, high order accurate and stable. They
use discontinuous approximations which combine the concept of approximate fluxes and
limiters in order to avoid oscillations in the region of sharp variations. Their highly par-
allelizable nature make them easily applicable to complicated geometries and boundary
conditions. In this work, the Runge-Kutta discontinuous Galerkin method (RKDG) is ap-
plied to solve the more general equilibrium dispersive and lumped kinetic systems. The
proposed scheme converts the given PDEs of the models to system of ODEs. An explicit
and nonlinearly stable high order Runge-Kutta (RK) method is used to solve the ODEs
62
system. The considered RK-method is total variation bounded (TVB) which preserves the
non-negativity of the solutions in time-coordinate. On the other hand, the local mono-
tonicity in the space-coordinate is guaranteed by the corresponding limiting function, such
as minmod limiter in this study.
4.2 Chromatographic Models
This section introduces the considered models of chromatography. These models can
be used for both linear and nonlinear chromatography. They are based on several basic
assumptions which are listed below.
1. The chromatographic process is isothermal.
2. The bed is homogenous and no concentration gradient is assumed along radial coor-
dinates in the column.
3. The mobile phase is assumed to be an incompressible fluid.
4. The interaction between carrier and solid phase is neglected.
5. The packing material used in the stationary phase is made of porous spherical and
uniform size particles.
5. The dispersion coefficients of all components are constant, i.e. independent of the
concentrations.
4.2.1 The equilibrium dispersive model (EDM)
The EDM assumes that the mass transfer kinetics is of infinite rate. Moreover, all con-
tributions due to non-equilibrium and axial dispersion effects are aggregated into the cor-
responding apparent (lumped) dispersion coefficient Dapp. The multi-component EDM is
written as [42, 43, 104]
∂cj∂t
+ F∂q∗j∂t
+ u∂cj∂z
= Dapp∂2cj∂z2
, j = 1, 2, 3, · · · , Nc . (4.1)
63
In the above equations, Nc represents the number of mixture components in the sample, cj
denotes the j-th liquid concentration, q∗j is the j-th solid concentration, u is the interstitial
velocity, F = (1 − ϵ)/ϵ is phase ratio based on the porosity ϵ ∈ (0, 1), t is time, z stands
for the axial-coordinate, and Dapp represents the apparent axial dispersion coefficient. The
EDM predicts the chromatographic profiles accurately when the column efficiency is high
or when small particles are used in the stationary phase of the column. The frequently
applied convex nonlinear Langmuir isotherm is defined as
q∗j =ajcj
1 +Nc∑i=1
bici
, j = 1, 2, 3, · · · , Nc . (4.2)
For diluted systems or small concentrations, Eq. (4.2) reduces to linear isotherms
q∗j = ajcj, j = 1, 2, 3, · · · , Nc . (4.3)
In above equations, aj denotes the j-th Henry’s coefficient. The apparent dispersion coef-
ficient, Dapp, is linked with the Peclet number through relation Pe = Lu/Dapp. The initial
conditions for fully regenerated columns are given as
cj(0, z) = cj,init . (4.4)
Moreover, some suitable inlet and outlet boundary conditions are needed. The following
inflow boundary conditions are considered in this work.
1. The continuous Dirichlet boundary conditions at the column inlet:
cj|z=0 = cj,inj . (4.5)
2. The discontinuous Dirichlet inlet boundary conditions:
cj|z=0 =
cj,inj , if 0 < t ≤ tinj ,0 , t > tinj .
(4.6)
64
3. The Robin-type (Danckwerts) inlet boundary conditions: [31, 108]:
cj|z=0 = cj,inj +Dapp
u
∂cj∂z
. (4.7)
4. The discontinuous Robin-type inlet boundary conditions:
− Dapp
u
∂cj∂z
+ cj
∣∣∣∣z=0
=
cj,inj , if 0 < t ≤ tinj ,0 , t > tinj .
(4.8)
Neumann boundary conditions are used at the column ’s outlet which are given as:
∂cj(t, L)
∂z= 0 . (4.9)
4.2.2 The lumped kinetic model (LKM)
The LKM deals with the rate at which the local concentration of solute varies in the sta-
tionary phase and the local deviation from equilibrium concentration. This model combines
(lumps) the contribution coming from internal and external mass transport resistances with
a mass transfer coefficient k. Additionally, the axial dispersion coefficient D is considered
to be constant. The one-dimensional governing equations for multi-component LKM are
expressed as [39, 42, 43, 104]
∂cj∂t
= −u∂cj∂z
+D∂2cj∂z2
− k
ϵ
(q∗j − qj
), (4.10)
∂qj∂t
=k
1− ϵ
(q∗j − qj
), (4.11)
q∗j = F (cj) , j = 1, 2, · · · , Nc . (4.12)
In the above equations, q∗j denotes the solid concentration under equilibrium condition,
i.e. isotherms in the EDM given by Eqs. (4.2) and (4.3). This is the simple model that
accounts for the mass transfer kinetics and is more accurate compared to the EDM. The
three characteristic times in the model Eqs. (4.10)-(4.12) are defined as
τC =L
u, τD =
D
u2, τMT =
1
k. (4.13)
65
The ratios of these characteristic times provide three dimensionless quantities
τ1 =τCτD
=Lu
D, τ2 =
τCτMT
=Lk
u. (4.14)
Here, τ1 typically is frequently called the Peclet number Pe
Pe = τ1 =Lu
D, (4.15)
where L is the column length. The model has the same initial and boundary conditions
which are given above for the EDM model. Moreover, initially q(0, z) = q∗(0, z). The
solution of LKM converges to that of EDM when k is very large.
4.3 Analytical Solutions of EDM for Linear Isotherms
In this section one-component (Nc = 1) linear EDM model is considered which is given
by Eqs. (4.1) and (4.3). The Laplace transformation is used to derive analytical solutions
for linear isotherms (Eq. (4.3)) with different inlet boundary conditions given by Eqs.
(4.5)-(4.8) and outlet boundary condition (4.9). To simplify the notations, we consider
c(x, t) := c1(x, t). The Laplace transformation is defined as
C(s, x) =
∞∫0
e−stc(t, x)dt, s > 0. (4.16)
After normalizing Eq. (4.1) by defining
x = z/L , Pe = Lu/Dapp (4.17)
and using the linear isotherm of Eq. (4.3), we obtain
(1 + aF )PeL
u
∂c
∂t+ Pe
∂c
∂x=∂2c
∂x2. (4.18)
Case 1: Continuous Dirichlet inlet boundary condition (Eq. (4.5)).
Let us define
R = (1 + aF )PeL
u, v = Pe . (4.19)
66
Then Eq. (4.18) together with initial, inlet and outlet boundary conditions (4.4), (4.5) and
(4.9) in normalized form can be written as
R∂c
∂t+ v
∂c
∂x=∂2c
∂x2, (4.20)
c(0, x) = cinit , (4.21)
c(t, 0) = cinj , (4.22)
∂c(t, 1)
∂x= 0. (4.23)
By applying the Laplace transformation (4.16) to Eq. (4.20) and using the initial condition
(4.21), we obtain∂2C
∂x2− v
∂C
∂x− sRC = −Rcinit . (4.24)
The solution of this equation is given as
C(x, s) = A exp(λ1x) + B exp(λ2x) +cinits, (4.25)
λ1,2 =v
2∓ 1
2
√v2 + 4sR . (4.26)
After applying the boundary conditions in Eqs. (4.22) and (4.23), the values of A and B
are given as
A =(cinj − cinit)λ2e
λ2
s(λ2eλ2 − λ1eλ1), B =
(cinj − cinit)λ1eλ1
s(λ1eλ1 − λ2eλ2). (4.27)
Then, Eq. (4.25) takes the following simple form
C(s, x) =(cinj − cinit)λ2e
λ2
s(λ2eλ2 − λ1eλ1)eλ1x +
(cinj − cinit)λ1eλ1
s(λ1eλ1 − λ2eλ2)eλ2x . (4.28)
The solution in the time domain c(t, x) can be obtained by using the exact formula for the
back transformation as
c(t, x) =1
2π i
γ+i∞∫γ−i∞
e−tsC(s, x)ds , (4.29)
where, γ is a real constant that exceeds the real part of all the singularities of C(s, x). On
applying Eq. (4.29) to Eq. (4.28), we obtain
c(t, x) = cinit + (cinj − cinit)H(t, x) , (4.30)
67
where
H(t, x) =1
2erfc
[Rx− vt
2√Rt
]+
1
2evxerfc
[Rx+ vt
2√Rt
]+
1
2
[2 + v(2− x) +
v2t
R
]everfc
[R(2− x) + vt
2√Rt
](4.31)
−(v2t
πR
) 12
exp
[v − R
4t
(2− x+
vt
R
)2].
Here, erfc denotes the complementary error function.
Case 2: Discontinuous Dirichlet inlet boundary condition (Eq. (4.6)).
If we consider the second inlet boundary condition given by Eqs. (4.6) together with outlet
(4.23), the values of A and B take the following forms
A =(cinj(1− e−stinj)− cinit)λ2e
λ2
s(λ2eλ2 − λ1eλ1), B =
(cinj(1− e−stinj)− cinit)λ1eλ1
s(λ1eλ1 − λ2eλ2). (4.32)
With these values A and B, the back transformation of Eq. (4.25) is given as
c(t, x) =
cinit + (cinj − cinit)H(t, x) , 0 < t ≤ tinj ,cinit + (cinj − cinit)H(t, x)− cinjH(t− tinj, x) , t > tinj .
(4.33)
Here, the value of H(t, x) is given by (4.31).
Case 3: Continuous Robin-type inlet boundary condition (Eq. (4.7)).
The Robin-type (Danckwert’s) boundary condition in normalized form can be re-written
as
c(t, 0) = cinj +1
v
∂c
∂x, where v = Pe =
Lu
Dapp
. (4.34)
It has the Laplace transformation
C(s, 0) =cinjs
+1
v
∂C
∂x. (4.35)
If we consider the set of boundary conditions given by Eqs. (4.35) and (4.23), the values
of A and B take the following forms
A =cinjs
λ2 exp(λ2)
(1− λ1v)λ2 exp(λ2)− (1− λ2
v)λ1 exp(λ1)
, (4.36)
68
B =−cinjs
λ1 exp(λ1)
(1− λ1v)λ2 exp(λ2)− (1− λ2
v)λ1 exp(λ1)
. (4.37)
With these values A and B, the back transformation of Eq. (4.25) is given as
c(t, x) = cinit + (cinj − cinit)H1(t, x) , (4.38)
where
H1(t, x) =1
2erfc
[Rx− vt
2√Rt
]+
(v2t
πR
) 12
exp
[−(Rx− vt)2
4Rt
]− 1
2
(1 + vx+
v2t
R
)evxerfc
[Rx+ vt
2√Rt
]+
(4v2t
πR
) 12[1 +
v
4
(2− x+
vt
R
)]exp
[v − R
4t
(2− x+
vt
R
)2]
(4.39)
− v
[(2− x+
3vt
2R
)+v
4
(2− x+
vt
R
)2]exp(v) erfc
[R(2− x) + vt
2√Rt
].
Case 4: Discontinuous Robin-type inlet boundary condition (Eq. (4.8)).
In this case, the discontinuous Robin-type (Danckwert’s) boundary condition in normalized
form can be re-written as
− 1
v
∂c
∂x+ c(t, x)
∣∣∣∣x=0
=
cinj , 0 < t ≤ tinj ,0 , t > tinj .
(4.40)
It has the Laplace transformation
C(s, 0) =cinjs
(1− e−stinj
)+
1
v
∂C
∂x
∣∣∣∣x=0
. (4.41)
If we consider the set of boundary conditions given by Eqs. (4.41) and (4.23), the values
of A and B take the following forms
A =(cinj(1− e−stinj)
s
λ2 exp(λ2)
(1− λ1v)λ2 exp(λ2)− (1− λ2
v)λ1 exp(λ1)
, (4.42)
B =− (cinj(1− e−stinj)
s
λ1 exp(λ1)
(1− λ1v)λ2 exp(λ2)− (1− λ2
v)λ1 exp(λ1)
. (4.43)
With these values A and B, the back transformation of Eq. (4.25) is given as
c(t, x) =
cinit + (cinj − cinit)H1(t, x) , 0 < t ≤ tinj ,cinit + (cinj − cinit)H1(t, x)− cinjH1(t− tinj, x) , t > tinj .
(4.44)
where H1(t, x) is given by Eq. (4.39).
69
4.4 Analytical solutions of LKM for Linear Isotherms
Let us define
x = z/L , Pe = Lu/D . (4.45)
Then, in normalized form and for one-component case, Eqs. (4.10)-(4.12) can be written
as
PeL
u
∂c
∂t+ Pe
∂c
∂x=
∂2c
∂x2− L2k
Dϵ(q∗i − qi) , (4.46)
∂q
∂t=
k
1− ϵ(q∗ − q) , (4.47)
q∗ = F(c) . (4.48)
Let us choose the linear isotherm, F(c) = ac. By applying the Laplace transformation
(4.16) to Eqs. (4.46)-(4.48) and using the initial condition (4.21), we obtain
∂2C
∂x2− Pe
∂C
∂x− PeL
uCs− aL2k
DϵC +
L2k
ϵDQ = −PeL
ucinit, (4.49)
−Qinit + sQ =k
1− ϵ(aC −Q) ⇒ Q =
ka(1−ϵ)
s+ k(1−ϵ)
C +Qinit
s+ k(1−ϵ)
. (4.50)
On putting the value of Q in Eq. (4.49) and using Qinit = acinit and Pe =LuD, we obtain
∂2C
∂x2− Pe
∂C
∂x− PeL
uCs− aL2k
ϵDC +
L2
ϵD
k2a(1−ϵ)
s+ k(1−ϵ)
C = −PeLu
cinit
(1 +
akϵ
s+ k1−ϵ
),
(4.51)
or
∂2C
∂x2− Pe
∂C
dx− PeL
u
(s− 1
ϵ
k2a(1−ϵ)
s+ k(1−ϵ)
+ak
ϵ
)C = −PeL
ucinit
(1 +
akϵ
s+ k1−ϵ
). (4.52)
or
∂2C
∂x2− Pe
∂C
dx− PeL
us
(1 +
akϵ
s+ k1−ϵ
)C = −PeL
ucinit
(1 +
akϵ
s+ k1−ϵ
). (4.53)
Thus, the Laplace domain solution is given as
C(s, x) = A exp(λ1x) + B exp(λ2x) +cinits, (4.54)
70
where
λ1,2 =Pe
2∓
√√√√(Pe2
)2
+PeL
us
(1 +
akϵ
s+ k1−ϵ
). (4.55)
or
λ1,2 =Pe
2∓ 1
2
√√√√Pe2 +4PeL
us
(1 +
aF
1 + s(1−ϵ)k
). (4.56)
It can be seen that for k → ∞, Eq. (4.56) reduces to Eq. (4.26).
Case 1: Continuous Dirichlet inlet boundary condition (Eq. (4.5)).
If we consider the first inlet boundary condition given by Eqs. (4.5) together with outlet
(4.23), the values of A and B are given as
A =
( cinjs
− cinits
)λ2e
λ2
λ2eλ2 − λ1eλ1, B =
( cinjs
− cinits
)λ1e
λ1
λ1eλ1 − λ2eλ2. (4.57)
For sufficiently large values of k in Eqs. (4.54)-(4.57), i.e. when k → ∞, the transformed
solution C(s, x) for LKM reduces to the solution of EDM given by Eq. (4.28). No ana-
lytical inverse Laplace transformation is possible to bring back the solution in actual time
domain. Therefore, the numerical inverse Laplace transformation is employed to find the
original solution c(t, x) of Eqs. (4.54)-(4.57), see [100] for more details.
Case 2: Discontinuous Dirichlet inlet boundary condition (Eq. (4.6)).
If we consider the second inlet boundary condition given by Eqs. (4.6) together with outlet
(4.23), the values of A and B take the following forms
A =
(cinj(e−stinj−1)
s− cinit
s
)λ2e
λ2
λ2eλ2 − λ1eλ1, B =
(cinj(e−stinj−1)
s− cinit
s
)λ1e
λ1
λ1eλ1 − λ2eλ2. (4.58)
The numerical inverse Laplace transformation is employed to find c(t, x).
Case 3: Continuous Robin-type inlet boundary condition (Eq. (4.7)).
71
If we consider the set of boundary conditions given by Eqs. (4.35) and (4.23), the values
of A and B take the following forms
A =(cinjs
− cinits
) λ2 exp(λ2)
(1− λ1Pe)λ2 exp(λ2)− (1− λ2
Pe)λ1 exp(λ1)
, (4.59)
B = −(cinjs
− cinits
) λ1 exp(λ1)
(1− λ1Pe)λ2 exp(λ2)− (1− λ2
Pe)λ1 exp(λ1)
. (4.60)
The numerical inverse Laplace transformation is employed to find c(t, x).
Case 4: Discontinuous Robin-type inlet boundary condition (Eq. (4.8)).
If we consider the set of boundary conditions given by Eqs. (4.41) and (4.23), the values
of A and B take the following forms
A =
(cinj (1− e−stinj)
s− cinit
s
)λ2 exp(λ2)
(1− λ1Pe)λ2 exp(λ2)− (1− λ2
Pe)λ1 exp(λ1)
, (4.61)
B = −(cinj (1− e−stinj)
s− cinit
s
)λ1 exp(λ1)
(1− λ1Pe)λ2 exp(λ2)− (1− λ2
Pe)λ1 exp(λ1)
. (4.62)
Once again, the inverse Laplace transformation is not doable analytically. However, nu-
merical inverse transform is employed to get back c(t, x).
4.5 The Discontinuous Galerkin Scheme for Solving
LKM
In this section, the Runge Kutta discontinuous Galerkin method is applied to lumped
kinetic model (LKM) [22, 26]. Because LKM is the more general model, this scheme can
be analogously implemented to the EDM. The scheme is semi-discrete in nature i.e., in
space is discretized first followed by time discretization. A third-order Runge Kutta ODE
solver is used for the discretization of time variable. For simplicity, a one-component model
is considered to derive the numerical scheme (c.f. Eqs. (4.10)-(4.11))
∂c
∂t+
∂
∂z
(uc−D
∂c
∂z
)= −k
ϵ(q∗ − q), (4.63)
∂q
∂t=
k
1− ϵ(q∗ − q) . (4.64)
72
Let us define
G(c) :=√D∂c
∂z, f(c,G) := uc−
√DG(c) . (4.65)
Then, Eq. (4.63) takes the form
∂c
∂t= −∂f
∂z− k
ϵ(q∗ − q), (4.66)
G =√D∂c
∂z, (4.67)
∂q
∂t=
k
1− ϵ(q∗ − q) . (4.68)
The discretization of axial length variable z is as follows. For j = 1, 2, 3, ....N , let zj+ 12be
the cell divisions, Ij = [zj− 12, zj+ 1
2] be the domain of cell j, ∆zj = zj+ 1
2− zj− 1
2be the width
of cell j, and I = UIj is the division of entire domain. We look for an approximate solution
ch(t, z) to c(t, z) for which at each time t ∈ [0, T ], ch(t, z) belongs to the finite dimensional
space
Vh =v ∈ L1(I) : v|Ij ∈ Pm(Ij), j = 1, 2, 3, ....N
, (4.69)
where Pm(Ij) represents, up to m-order polynomials defined on the cell Ij. Note that in Vh,
the jumps of functions are permitted at the cell interface zj+ 12. To obtain the approximate
solution ch(t, z), a weak formulation is required. For this purpose, an arbitrary smooth
function v(z) is multiplied by Eqs. (4.66)-(4.68). After multiplication, the integration by
parts over the interval Ij yields∫Ij
∂c(t, z)
∂tv(z)dz =−
(f(cj+ 1
2,Gj+ 1
2)v(zj+ 1
2)− f(cj− 1
2,Gj− 1
2)v(zj− 1
2))
+
∫Ij
(f(c,G)∂v(z)
∂z
)dz − k
ϵ
∫Ij
(q∗ − q)v(z)dz, (4.70)
∫Ij
G(c)v(z)dz =√D(cj+ 1
2v(zj+ 1
2)− cj− 1
2v(zj− 1
2))−√D
∫Ij
c(z)∂v(z)
∂zdz, (4.71)
∫Ij
∂q
∂tv(z)dz =
k
1− ϵ
∫Ij
(q∗ − q)v(z)dz. (4.72)
73
One way to implement Eq. (4.69) is to employ Legendre polynomials Pl(z) of order l as
local basis functions. The L2-orthogonality property of these polynomials can be exploited,
namely
1∫−1
Pl(s)Pl′(s) =
(2
2l + 1
)δll′ . (4.73)
For each z ∈ Ij, the solutions ch and Gh can be expressed as
ch(t, z) =m∑l=0
c(l)j φl(z) , Gh(ch(t, z)) =
m∑l=0
G(l)j φl(z) , qh(t, z) =
m∑l=0
q(l)j φl(z), (4.74)
where
φl(z) = Pl (2(z − zj)/∆zj) , l = 0, 1, ...,m. (4.75)
If m = 0, the solution ch use the piecewise-constant basis function and a linear basis
function is utilized for m = 1. Similarly, for different values of m, the approximate solution
uses different order basis functions. Here, the linear basis functions are considered, therefore
l = 0, 1. By using Eqs. (4.73)-(4.75), it can be easily verified that
w(l)j (t) =
2l + 1
∆zj
∫Ij
wh(t, z)φl(z)dz , w ∈ c,G, q, q∗ . (4.76)
Then, the function v(z) is changed by the test function φl ∈ Vh. Further, c and g, which
are exact solutions are replaced by the approximate solutions ch and Gh. Moreover, the
function fj+ 12= f(c(t, zj+ 1
2),G(cj+ 1
2)) is not known at the cell interface zj+ 1
2. Therefore,
we must also replace the flux function f by a numerical flux depending on two different
values of ch at the interface, i.e.
fj+ 12≈ hj+ 1
2= h(c−
j+ 12
, c+j− 1
2
) . (4.77)
As G := G(c), we can drop it from the arguments of h for convenience. Due to Eq. (4.74)
c−j+ 1
2
=m∑l=0
c(l)j φl(zj+ 1
2) , c+
j− 12
=m∑l=0
c(l)j φl(zj− 1
2) . (4.78)
74
Using the just mentioned definitions, the weak formulations in Eqs. (4.70)-(4.72) simplify
to
dc(l)j (t)
dt=− 2l + 1
∆zj
(hj+ 1
2φl(zj+ 1
2)− hj− 1
2φl(zj− 1
2))
+2l + 1
∆zj
∫Ij
(f(ch,Gh)
dφl(z)
dz
)− k
ϵ(q
∗(l)j − q
(l)j ) , (4.79)
G(l)j (t) =
2l + 1
∆zj
√D
cj+ 12φl(zj+ 1
2)− cj− 1
2φl(zj− 1
2)−
∫Ij
ch(t, z)dφl(z)
dzdz
, (4.80)
dq(l)j (t)
dt=
k
1− ϵ(q
∗(l)j − q
(l)j ). (4.81)
The initial solution profile for this system is given (c.f. Eq. (4.76))
c(l)j (0) =
2l + 1
∆zj
∫Ij
c(0, z)φl(z)dz , G(l)j (0) = G(c(l)j (0)) , q
(l)j (0) = q(c
(l)j (0)) . (4.82)
It only remains to make a choice of the numerical flux function h. The Eqs. (4.71)-(4.81)
define a monotone scheme if the approximate flux function h(α, β) is consistent, Lipschitz
and monotone [69, 132]. The following Lax-Friedrichs flux is used that satisfy the above
properties [69, 132]
hLF (α, β) =1
2[f(α,G(α)) + f(β,G(β))− C(β − α)] , (4.83)
C = maxmin (α,β)≤s≤max (α,β)
|f ′(s, g(s))| , (4.84)
where C is the upper bound of f ′ over the whole domain. If C is calculated at the local
element edges then Eq. (4.83) is known as the local Lax-Friedrichs flux. The integral terms
in Eqs. (4.79) and (4.80) are approximated by the 10-th order Gauss-Lobatto quadrature
rule. Moreover, a limiting procedure has to be employed to get the total variation stability.
For that reason, the modification of c±j+ 1
2
in Eq. (4.77) is needed by using some local
projection [22]. Therefore, (4.78) can be re-written as
c−j+ 1
2
= c(0)j + cj , c+
j− 12
= c(0)j − cj , (4.85)
75
where
cj =m∑l=1
c(l)j φl(zj+ 1
2) cj = −
m∑l=1
c(l)j φl(zj− 1
2) l = 0, 1. (4.86)
If m = 0, cj = cj = 0 and when m = 1, cj = cj = 6c(1)j , etc. Next, the modified cj and cj
are given as
c(mod)j = mm
(cj,∆+c
(0)j ,∆−c
(0)j
), c
(mod)j = mm
(cj,∆+c
(0)j ,∆−c
(0)j
), (4.87)
where, ∆± := ±(cj±1 − cj) and mm is the minmod function which is
mm(α1, α2, α3) =
s · min
1≤j≤3|αj| if sign(α1) = s = sign(α2) = sign(α3) ,
0 otherwise .(4.88)
Thus, Eq. (4.85) becomes
c−(mod)
j+ 12
= c(0)j + c
(mod)j , c
+(mod)
j− 12
= c(0)j − c
(mod)j (4.89)
and (4.77) is replaced by
hj+ 12= h(c
−(mod)
j+ 12
, c+(mod)
j− 12
) . (4.90)
This local projection limiter is capable of maintaining the accuracy in the smooth regions
and convergence can be achieved without spurious oscillations in the vicinity of shocks [22].
Finally, a RK-method that preserves the TVB property is required for solving the resulting
system of ODEs. The Eqs. (4.79) and (4.81) can be concisely written as
dchdt
= Lh(ch, t) . (4.91)
For the approximation of Eq. (4.91), the TVB RK-method of order r can be applied
(ch)m =
m−1∑l=0
[α
′ml(ch)
(l) + β′
ml∆tLh((ch)(l), tn + γ
′
l∆t)], m = 1, 2, · · · , r , (4.92)
and due to Eq. (4.82)
(ch)(0) = (ch)
n , (ch)(r) = (ch)
n+1 . (4.93)
76
Here, n denotes the n-th time step. For m = 2, the constants involved are expressed as
α′10 = β
′
10 = 1 , α′20 = α
′21 = β
′
21 =1
2, β
′
20 = 0; γ′
0 = 0 , γ′
1 = 1 . (4.94)
While the coefficients for the m = 3 are
α′10 = β
′
10 = 1 , α′20 =
3
4, β
′
20 = 0 , α′= β
′
21 =1
4, α
′30 =
1
3
β′
30 = α′31 = β
′
31 = 0 , α′32 = β
′
32 =2
3; γ
′
0 = 0 , γ′
1 = 1 , γ′
2 =1
2. (4.95)
To guarantee the stability and numerical convergence of the scheme, the time step is chosen
according to the following Courant-Friedrichs-Lewy (CFL) condition
∆t ≤(
1
2η + 1
)∆zj|u|
, (4.96)
where, η = 1, 2 for second order and third order accurate schemes, respectively.
Boundary conditions: Consider the boundary at z− 12= 0. The left boundary condition
given by Eq. (4.7) can be implemented as
c−− 12
(t) = c(0)0 +
D
u
c(0)1 − c
(0)0
∆z, (4.97)
c(mod)0 = mm
(c0,∆+c
(0)0 , 2
(c(0)0 − cinj
)), c
(mod)0 = mm
(c0,∆+c
(0)0
). (4.98)
Neumann boundary condition is employed on the right end of the column, i.e., c(l)N+1 = c
(l)N .
4.6 The Flux-Limiting Finite Volume Schemes
In this section the finite volume schemes are implemented to solve the EDM. The extension
of this scheme to LKM is straightforward. In order to avoid the complexity of multi-
component EDM, a one-component EDM equation is taken into account (c.f. Eq. (4.1))
∂c
∂t+ F
∂q
∂t+ u
∂c
∂z= Dapp
∂2c
∂z2. (4.99)
Let us define w := w(c) = c+ Fq(c) and f(c) = uc, the Eq. (4.99) implies
∂w
∂t= −∂f(c)
∂z+Dapp
∂2c
∂z2. (4.100)
77
Next, we first dicretize the computational domain. Let N denote a large integer and
(zj− 12)j∈1,··· ,N+1 are the divisions of the given domain [0, L]. We take ∆z as uniform cell
width, zj as cell centers, and zj± 12as the boundaries of cells. Let us assign,
z1/2 = 0 , zN+1/2 = L , zj+1/2 = j ·∆z , . (4.101)
Further,
zj = (zj−1/2 + zj+1/2)/2 and ∆z = zj+1/2 − zj−1/2 =L
N + 1. (4.102)
Let Ωj :=[zj−1/2, zj+1/2
]for j ≥ 1. The cell averaged initial data w0(z) in each cell is
given as
wj(0) =1
∆z
∫Ωj
w0(z) dz , for j = 1, 2, · · ·N . (4.103)
By integrating Eq. (4.100) over Ωi =[zj−1/2, zj+1/2
], we obtain∫
Ωj
∂w
∂tdz = −
(fj+ 1
2− fj− 1
2
)−Dapp
((∂c
∂z
)j+1/2
−(∂c
∂z
)j−1/2
). (4.104)
In each Ωj, the averaged values of the conservative variable w(t) are given as
wj := wj(t) =1
∆z
∫Ωj
w(t, z) dz . (4.105)
Thus, Eq. (4.104) leads to the semi-discrete scheme of the form
dwjdt
= −fj+ 1
2− fj− 1
2
∆z− Dapp
∆z
((∂c
∂z
)j+ 1
2
−(∂c
∂z
)j− 1
2
), j = 1, 2, · · ·N , (4.106)
where (∂c
∂z
)j± 1
2
= ±(cj±1 − ci
∆z
). (4.107)
Different techniques are available for the approximation of the fluxes fj± 12in Eq. (4.106)
and each approximation gives a different numerical scheme [69, 55]. By exploiting the
inequalities u > 0 and f > 0, we have the following approximations of fj± 12.
78
dx2
L
x1 x2 x3
dx1
PSfrag replacements
zj−1
zjzj+1zj− 1
2
zj+ 12
hz
Figure 4.1: Cell centered finite volume grid
Backward difference scheme (first order): Here, the fluxes are given as
fj+ 12= fj = (uc)j , fj− 1
2= fj−1 = (uc)j−1 . (4.108)
As a result, the scheme has first order accuracy.
High resolution scheme of Koren: Koren [55] used the following approximation of the
fluxes at the cell interfaces together with Sweby-type flux-limiter [116] to guarantee the
positivity (monotonicity) of the scheme:
fj+ 12= fj +
1
2ϕ(rj+ 1
2
)(fj − fj−1) , (4.109)
where, rj+ 12is the ratio of consecutive flux gradients
rj+ 12=fj+1 − fj + η
fj − fj−1 + η. (4.110)
Here, η ≈ 10−10 and the limiting function ϕ is defined as
ϕ(rj+ 12) = max
(0,min
(2rj+ 1
2,min
(1
3+
2
3rj+ 1
2, 2
))). (4.111)
Due to the above limiter the scheme has second order accuracy [55].
Other approximations: In the literature, few other flux-limiting functions are available
[65, 101]. In these schemes, the cell interface fluxes are given as
fj+ 12= fj +
1
2φ(θj+ 1
2
)(fj+1 − fj) . (4.112)
79
Table 4.1: Selected flux limiting functions in Eq. (4.112)Flux limiter FormulaMC ([65]) φ(ζ) = max
(0,min
(2ζ, 1
2(1 + ζ), 2
))Superbee ([101]) φ(ζ) = max (0,min(2ζ, 1),min(ζ, 2))Minmod ([101]) φ(ζ) = max (0,min(1, ζ))
van Leer ([65]) φ(ζ) = |ζ|+ζ1+|ζ|
Similarly, fj− 12can be approximated. Here, θj+ 1
2is given as
θj+ 12=fj − fj−1 + η
fj+1 − fj + η, (4.113)
and the selected well known flux limiting functions are listed in Table 4.1.
Implementation of the schemes in the boundary cells: The Eqs. (4.109) and (4.112)
can not be directly applied to cells close to the boundaries. Let us assume that inflow
condition is imposed at the left boundary of the domain. Then, the left face z 12of the
boundary cell and the left inlet boundary are located at the same position. Since, z0 is
located outside the domain of computations, Eqs. (4.109) and (4.112) can not be applied
at z 32. To resolve this issue, the first order approximation in Eq. (4.108) is used at the
interval interfaces z 32and zN+ 1
2. Let finj be the inlet flux, then
f 12= finj , f 3
2= f1, fN+ 1
2= fN . (4.114)
xn2x1 x2
dx1 dx2
xn1
dxn1 dxn2
LL
PSfrag replacements
z1z2
zN−1
zNz 1
2
z 32
zN− 12
zN+ 12
z
a. Inflow. b. Outflow.
Figure 4.2: Grids near the boundaries
At the remaining cell interfaces, the fluxes are calculated through Eqs. (4.109) or (4.112).
The concentration c is required to update the isotherm q(c) and flux f(c) = uc at every
time step. However, Eq. (4.106) updates wj(t) = cj(t) + Fqj(c) in the interval Ωj. Thus,
80
by applying chain rule, Eq. (4.106) can be re-written as
dcjdt
= −
[1 + F
(dq
dc
)j
]−1 [fj+ 1
2− fj− 1
2
∆z− Dapp
∆z
((∂c
∂z
)j+ 1
2
−(∂c
∂z
)j− 1
2
)+Qj
],
(4.115)
for all j = 1, 2, · · · , N . The same TVB RK-method given in Eqs. (4.92)-(4.95) is used to
solve the given system of ODEs.
4.7 Numerical Test Problems
In this section, the proposed numerical method is implemented to solve various test prob-
lems. After discretizing the axial-coordinate with DG-scheme and FVSs, a third-order
RK-method given in Eqs. (4.92)-(4.95) is used for the evolution of solution in time. We
implemented all algorithms in C-language.
4.7.1 One-component elution with linear isotherm
Here, some case studies of the EDM and LKM models with linear and nonlinear isotherms
are presented. The analytical solutions obtained by the Laplace transformation for linear
isotherms are compared with the numerical solutions of DG-scheme and Koren method.
The accuracy of DG-scheme is analyzed by comparing its results with other schemes for
both EDM and LKM.
Case 1: Comparison of analytical and numerical solutions for EDM.
This part focuses on the comparison of analytical and numerical results obtained from the
EDM for different boundary conditions. The parameters of the problem are given in Table
4.2.
In Figure 4.3 (left), the analytical solution solution of EDM with continuous Dirichlet inlet
boundary conditions is compared with the numerical results of DG and Koren schemes.
Good agreement of these profiles verify the accuracy of proposed numerical schemes. In
Figure 4.3 (right), the results of the DG and Koren methods for the EDM with discontin-
uous Dirichlet boundary conditions are compared with analytical solution. The left and
81
Table 4.2: Parameters for Case 1 (EDM).Parameters valuesPorosity ϵ = 0.4
Column length L = 1.0 cmInterstitial velocity u = 1.0 cm/minDispersion coefficient D = 0.002 cm2/min
Injection time for discontinuous case tinj = 2 sTotal simulation time tmax = 10 minInitial concentration cinit = 0 g/lConcentration at inlet cinj = 1.0 g/l
Adsorption equilibrium constant a = 1.0
right plots of Figure 4.4 compares the numerical results of DG and Koren schemes with
the analytical solutions of EDM for continuous and discontinuous Danckwert’s boundary
conditions, respectively. These profiles show the high precision of the suggested numerical
schemes. Thus, it can also be concluded that the considered schemes give good approx-
imation of the EDM for linear isotherm. The results shown in Figure 4.5 illustrate the
importance of using more accurate boundary conditions for chromatographic model equa-
tions when the Peclet number is relatively small, e.g. Pe < 10. For such values, there are
visible differences between the results obtained by using Dirichlet and Danckwert’s bound-
ary conditions. On the basis of these results, one can conclude that the implementation
of Dirichlet boundary conditions is not adequate for large dispersion coefficients. For large
values of Peclet number (Pe >> 10) or small axial dispersion coefficients, there is no dif-
ference between Dirichlet and Danckwerts boundary conditions.
Case 2: Error Analysis of the DG-scheme for EDM with linear isotherm.
In this study, the accuracy of DG-scheme is analyzed by comparing its results with con-
sidered finite volume schemes. The initial data are given as
c(0, z) =
sin(π(z − 0.2)/0.2) , 0.2 ≤ z ≤ 0.4 ,0, otherwise
, (4.116)
with zero boundary condition at the inlet, c(0, t) = 0. This problem has the following
82
0 2 4 6 80
0.2
0.4
0.6
0.8
1
time [min]
c [g
/l]
continuous Dirichlet BC (EDM)
DGKorenAnalytical
2 2.2 2.4 2.6 2.8 30
0.2
0.4
0.6
0.8
1
0 2 4 6 80
0.2
0.4
0.6
0.8
1
time [min]
c [g
/l]
discontinuous Dirichlet BC (EDM)
DGKorenAnalytical
2.2 2.4 2.6 2.80
0.2
0.4
0.6
0.8
1
Figure 4.3: Case 1 (EDM): Comparison of results for Dirichlet boundary conditions.
0 2 4 6 80
0.2
0.4
0.6
0.8
1
time [min]
c [g
/l]
continuous Danckwert’s BC (EDM)
DGKorenAnalytical
0 2 4 6 80
0.2
0.4
0.6
0.8
1
time [min]
c [g
/l]
discontinuous Danckwert’s BC (EDM)
DGKorenAnalytical
Figure 4.4: Case 1 (EDM): Comparison of results for Danckwert’s boundary conditions.
83
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
1.2
time [min]
c [g
/l]
EDM solution
Dirichlet BC with Pe=1Danckwet’s BC with Pe=1Dirichlet BC with Pe=2Danckwer’t BC with Pe=2Dirichlet BC with Pe=10Danckwert’s BC with Pe=10
Dirichlet BC with Pe=102
Danckwert’s BC with Pe=102
Figure 4.5: Case 1 (EDM): Effect of boundary conditions for different values of Pecletnumbers (or diffusion coefficients).
analytical solution [55]
c(t, z) = 0.5 real (iep[erf(α)− erf(β)]) , (4.117)
where, erf represents the error function and
p = −0.5Dappt( π
0.2
)2+ i
π
0.2(0.2− z − 0.5t) , (4.118)
α =−0.2 + z − 0.5t
2√
0.5Dappt− iπ
√0.5Dappt
0.2, (4.119)
β =−0.4 + z − 0.5t
2√
0.5Dappt− iπ
√0.5Dappt
0.2. (4.120)
Here, a column of length 1 cm is considered. Moreover, we take a = 1, u = 1 cm/min,
ϵ = 0.5, and the final time is 0.6min. The L1-error in the axial-coordinate at the final
simulation time is
L1-error =N∑j=1
|cjexact − cjNumeric|∆z . (4.121)
84
Where cjexact denotes the exact solution at the midpoint of each discrete cell and cjNumeric is
the representation of corresponding numerical solution at the final simulation time. More-
over, N is the number of discretization points and ∆z represents the axial step size.
Table 4.3 gives a comparison of the L1-errors in schemes using 100 mesh points and with
different values of Dapp. It is evident that the DG scheme produces less errors in the solu-
tion compared to the considered finite volume schemes. Moreover, the errors produced by
the Koren scheme are also less at low computational cost. Table 4.4 displays the L1-error
and the experimental order of convergence (EOC) of the DG-scheme at different values of
N and Dapp. The EOC of the DG-method is approximately second order, see Table 4.4.
The DG-scheme is more effective for the convection dominated problems where the disper-
sion coefficient is small. The computational cost for the mimod limiter is larger than the
other schemes. The backward difference (first order) scheme has low computational cost
but produces large errors in the solution. The computational cost of the van Leer limiter is
comparable to the DG and Koren schemes, however its errors are larger. The rest limiters
are computationally expensive and low order accurate. Thus, one can conclude that the
DG and Koren schemes are better choices for solving EDM model.
Case 3: Comparison of analytical and numerical solutions for LKM.
Here, the analytical and numerical results of LKM model are compared for different bound-
ary conditions. The parameters of the problem are the same as given in Table 4.2. In Figure
4.6 (left), the analytical solution of LKM for continuous Dirichlet inlet boundary condition
is compared with the numerical solution of DG-scheme. In this figure, the adsorption rate
(mass transfer coefficient) is taken as k = 100. Thus, for this large value of k, the LKM
solution agrees with that of EDM in Figure 4.3. Good agreement of these profiles reveal
the accuracy of the proposed numerical method. In Figure 4.6 (right), the results of the
LKM and DG-method are compared with the each other for different values of k. It can
be observed that for small value of k, LKM produces different solutions from EDM. Fig-
ure 4.7 (left) compares the numerical results of DG-method with the analytical solutions
85
Table 4.3: Case 2: Errors and CPU times of schemes on 100 mesh points for EDM.
Limiter L1−error CPU (s)Dapp = 0.002 Dapp = 2× 10−4 Dapp = 2× 10−5 Dapp = 2× 10−6 Dapp = 0.002
MC 0.0023 0.0042 0.0050 0.0050 0.62Minmod 0.0062 0.0108 0.0118 0.0118 1.45Superbee 0.0034 0.0038 0.0046 0.0048 0.88van Leer 0.0027 0.0057 0.0068 0.0068 0.56Koren 0.0008 0.0028 0.0041 0.0042 0.56
First order 0.0402 0.0530 0.0655 0.0549 0.43DG 0.0001 0.0018 0.0028 0.0030 0.64
Table 4.4: Case 2: Errors and EOC for DG-method for EDM.
Mesh points Dapp = 2× 10−3 Dapp = 2× 10−4 Dapp = 2× 10−5
L1-error EOC L1-error EOC L1-error EOC50 0.0028 0.0062 0.0090100 9.31× 10−4 1.60 0.0018 1.78 0.0028 1.68200 3.38× 10−4 1.46 2.69× 10−4 2.73 6.40× 10−4 2.13400 1.07× 10−4 1.66 6.94× 10−5 1.96 1.34× 10−4 2.26800 3.05× 10−5 1.81 2.24× 10−5 1.63 2.43× 10−5 2.47
of the LKM for continuous Danckwert’s boundary conditions. While, Figure 4.7 (right)
shows the results of LKM for different values of Peclet number. Similar behaviors to the
EDM solutions in Figure 4.4 are observed. All these profiles show the high precision of the
suggested numerical scheme. Thus, it can also be concluded that the considered scheme
gives good approximation of the LKM for linear isotherm.
Case 4: Error analysis for LKM.
Here, different numerical schemes are comapred for lumped kinetic model. The parameters
of the problem are given in Table 4.5. The L1− errors at the column outlet are calculated
by
L1 − error =
NT∑n=1
|cnR − cnN |∆t . (4.122)
86
0 1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
LKM solution for continuous Dirichlet BC, k=100
t [min]
c [g
/l]
DGAnalytical
0 1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
t [min]
c [g
/l]
Effect of adsorption rate, k, on the solution in LKM
DG, k=100Analytical, k=100DG, k=50Analytical, k=50DG, k=10Analytical, k=10DG, k=1Analytical, k=1
Figure 4.6: Case 3 (LKM): Comparison of results for Dirichlet boundary conditions.
0 1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
t [min]
c [g
/l]
LKM solution for continuous BC with k=100
DG−schemeAnalytical
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
1.2
time [min]
c [g
/l]
LKM solution, k=100
Dirichlet BC with Pe=1Danckwet’s BC with Pe=1Dirichlet BC with Pe=2Danckwer’t BC with Pe=2Dirichlet BC with Pe=10Danckwert’s BC with Pe=10
Dirichlet BC with Pe=102
Danckwert’s BC with Pe=102
Figure 4.7: Case 3 (LKM): Comparison of results for Danckwert’s boundary conditions.
87
The relative error can be defined as
relative error =
NT∑n=1
|cnR − cnN |
NT∑n=1
|cnR|∆t , (4.123)
where cnR denotes the Laplace solution at the column outlet for time tn and cnN represents
the corresponding numerical solution. Further, NT is the total number of time steps and
∆t represents the increment in time. Comparisons of L1−errors, relative errors and com-
putational time of the schemes are given in Tables 4.6 and 4.7 for 50 and 100 grid points,
respectively. It can be seen that DG-scheme produces small errors compared to the other
schemes for both 50 and 100 grid cells, but efficiency (or CPU time) of the Koren scheme is
better than the other schemes. It can be noticed that relative errors of the DG and Koren
schemes are very low for 100 grid points. On the basis of these results, one can conclude
that the DG method could be an optimal choice to approximate chromatographic models.
4.7.2 One-component elution with nonlinear isotherm
In this subsection, the performance of DG-scheme is analyzed for a nonlinear model. The
EDM given by Eqs. (4.1) and the nonlinear isotherm q(c) = c/(1 + c) are considered.
The corresponding parameters of this problem are listed in the Table 4.8. The solution
of the model is obtained on 100 grid points. Moreover, 2000 grid points were used to get
the reference solution by using the same DG-scheme. A comparison of results is given
in Figure 4.8 at the outlet of the column, while L1-errors and CPU times of the schemes
are compared in Table 4.9. It is evident that the error produced by DG-scheme are less
as compared to other schemes. Further, DG and Koren methods give the most resolved
solutions with small errors. The CPU time of the DG-scheme, the koren scheme and other
finite volume schemes with van-Leer or minmod limiters are comparable. However, MC
and Superbee limiters have larger CPU time. The first order (backward difference) scheme
is computationally less expensive but its accuracy is lower. Moreover, Figure 4.9 shows
the solution of LKM model (c.f. Eqs. (4.10)-(4.12)) for different values of mass transfer
coefficient. It is evident from the results that LKM solution agrees with EDM solution for
88
Table 4.5: Case 4: Parameters for linear isotherm (LKM).Parameters valuesPorosity ϵ = 0.4
Column length L = 1.0 mInterstitial velocity u = 0.1 m/sCharacteristic time τC = 10 s
Dispersion coefficient for EDM Dapp = 10−4 m2/sPeclet no for EDM τ1 = Pe = 103
Characteristic time for EDM τD = 0.01 sDispersion coefficient for LKM D = 10−5 m2/s
Peclet no for LKM τ1 = Pe = 104
Characteristic time for LKM τD = 0.001 sMass transfer coefficient k = 100 1/s
Characteristic time τMT = 0.01 sDimensionless number τ2 = 103
Concentration at inlet c1,0 = 1.0 g/lAdsorption equilibrium constant a = 0.85
Table 4.6: Errors and CPU times at 50 grid points for linear isotherm (LKM)Limiter L1−error Relative error CPU (s)
DG Scheme 0.4011 0.0100 5.86Koren 0.5155 0.0137 4.90
Van Leer 0.9324 0.0248 8.26Superbee 1.0732 0.0286 9.34
MC 0.9762 0.0260 8.82
Table 4.7: Errors and CPU times at 100 grid points for linear isotherm (LKM)Limiter L1−error Relative error CPU (s)
DG Scheme 0.0139 3.71× 10−4 8.46Koren 0.0153 4.08× 10−4 7.82
Van Leer 0.2255 0.0060 14.90Superbee 0.2966 0.0079 17.73
MC 0.2477 0.0066 14.96
89
k = 300. Thus, for large value of k, the LKM solution agrees with the EDM solution. For
small values of k, the solution of LKM is away from the equilibrium condition and, hence,
from the solution of EDM. Thus, it can be concluded that DG and Koren schemes are
better choices for solving such nonlinear models.
Table 4.8: Parameters for nonlinear case (EDM).Parameters valuesPorosity ϵ = 0.5
Length of column L = 1.0 cmInterstitial velocity u = 1.0 cm/minDispersion coefficient D = 0.002 cm2/min
Injection time for discontinuous case tinj = 0.2 minTotal simulation time tmax = 3 minInitial concentration cinit = 0 g/lConcentration at inlet cinj = 1.0 g/l
Adsorption equilibrium constant a = 1.0
4.7.3 Two-component elutions
After validating the proposed numerical scheme for one-component linear and nonlinear
adsorptions, this subsection is intended to extend our study to nonlinear two-component
elutions. In this test problem, the two-component lumped kinetic model (c.f. Eqs. (4.10)-
(4.12)) along with nonlinear Langmuir isotherms (4.2) is considered for finite feed volumes.
A rectangular pulse of a liquid mixture of width tinj ∈ [0, 12] is injected to the column.
Table 4.9: Section 4.6.2: L1-errors and CPU time of schemes for nonlinear EDM.Limiter L1−error CPU (s)
N = 50 N = 100 N = 200 N = 50 N = 100 N = 200First order 0.115 0.072 0.041 0.12 0.14 0.70Minmod 0.065 0.029 0.012 0.22 0.48 1.57MC 0.059 0.027 0.013 0.013 0.32 1.76
Superbee 0.058 0.028 0.013 0.34 0.61 2.01van Leer 0.059 0.027 0.012 0.21 0.41 1.54Koren 0.050 0.022 0.010 0.22 0.42 1.62
DG Scheme 0.019 0.006 0.004 0.23 0.50 1.93
90
1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
c [g
/l]
solution at the column outlet
t [min]
referenceDGKorenvan LeerSuperbeeMCfirst order
Figure 4.8: Results comparison in the case of one-component nonlinear EDM.
0 1 2 30
0.1
0.2
0.3
0.4
0.5
LKM solution for k = 10
t [min]
c [g
/l]
0 1 2 30
0.1
0.2
0.3
0.4
0.5
LKM solution for k = 50
t [min]
c [g
/l]
0 1 2 30
0.1
0.2
0.3
0.4
0.5
LKM solution for k = 100
t [min]
c [g
/l]
0 1 2 30
0.1
0.2
0.3
0.4
0.5
LKM solution for k = 300
t [min]
c [g
/l]
Figure 4.9: Linear LKM solution for one-component elution.
91
The boundary conditions are given by Eqs. (4.4) and (4.9) for two components (Nc = 2).
The parameters of this problem are taken from [110] and are provided in Table 4.10. The
numerical results are shown in Figure 4.10 (left) by using 150 spatial grid points. The
figure elucidates that the proposed scheme gives better solution profile as compared to
Koren finite volume schemes. These results also agree with those obtained by [110], even
for coarse mesh cells. Due to large value of the adsorption rate k (c.f. Table 4.10), these
results of LKM are exactly similar to those obtained from EDM. The simulation results
validate the importance of suggested method for approximating nonlinear chromatographic
models. Figure 4.10 (right) describes nonequimolar injection concentrations with c1,0 =
4 mol/l and c2,0 = 2 mol/l, c1,0 = 2 mol/l and c2,0 = 1 mol/l, as well as, c1,0 = 1 mol/l
and c2,0 = 0.5 mol/l, respectively. The results in Figure 4.10 (right) illustrate the well-
known fact that strong nonlinearities produce overshoots in the profiles. The efficiency
and accuracy of the schemes can be graphically seen in Figure 4.11. These plots highlight
that errors of the DG-scheme are lower than the other schemes. It is probably worthwhile
to conclude that an increase in the number of grid points produces smaller errors, but the
computational time of the numerical schemes increases. The computational times of the
DG and the Koren scheme are comparable, while the time taken by the other schemes is
significantly higher. From the above observations, we conclude that the DG-scheme is a
reliable and better choice for solving such mathematical models.
4.8 Summary
In this study, The EDM and LKM were analytically and numerically investigated for
different types of boundary conditions. The Laplace transformation was used as a basic tool
to transform the one-component linear sub-models to linear ordinary differential equations
which were then solved analytically in the Laplace domain. The analytical or numerical
inversions were used to get back the time domain solutions. For the numerical solutions
of considered models, the DG-scheme was applied. This scheme and the applied RK-
method fulfill the TVB condition and, thus, avoid spurious oscillations in the vicinity of
92
Table 4.10: Parameters for two-component elusion (nonlinear isotherm, LKM).Parameters valuesPorosity ϵ = 0.4
Column length L = 1.0 mInterstitial velocity u = 0.1 m/sDispersion coefficient D = 10−4 m2/s
Adsorption rate k = 103 1/sCharacteristic time τC = 10 sCharacteristic time τD = 0.01 sCharacteristic time τMT = 0.001 s
Peclet no τ1 = Pe = 103
Dimensionless number τ2 = 104
Concentrations c1,0 = c2,0 = 10 mol/lAdsorption constants a1 = 0.5, a2 = 1, b1 = 0.05, b2 = 0.1
Injection time tinj = 12 s
discontinuities. For validation, this scheme is compared with the considered finite volume
schemes and analytical solutions for linear isotherms. It was found that DG-scheme is
efficient and accurate to solve chromatographic models.
93
1 1.5 2 2.5 3 3.5 40
2
4
6
8
10
12
14
16
18
t u / L
c i [g/l]
1.4 1.45 1.5 1.550
2
4
6
8
10
12
14
16
18
Solid lines: reference solutiondash lines: DG−schemedash−dot lines: Koren method
0 0.5 1 1.5 2 2.5 3 3.5 40
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
t u / L
c i [−]
solid lines: c1,0
=4 mol/l, c2,0
=2 mol/l, dashed lines: c1,0
=2 mol/l, c2,0
= 1 mol/ldashed−dotted lines: c
1,0=1 mol/l, c
2,0=0.5 mol/l
Figure 4.10: Left: Two component nonlinear elutions profile at the column outlet. Right:different injection volumes are used, first injection: c1,0 = 4 mol/l and c2,0 = 2 mol/l,second injection: c1,0 = 2 mol/l and c2,0 = 1 mol/l, third injection: c1,0 = 1 mol/l,c2,0 = 0.5 mol/l.
94
0 200 400 600 800 1000 12000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
number of grid points
L1 −er
ror
DG−schemeKoren methodVan LeerSuperbeeMC method
0 200 400 600 800 1000 12000
0.1
0.2
0.3
0.4
0.5
number of grid points
L1 −er
ror
DG−schemeKoren methodVan LeerSuperbeeMC method
0 200 400 600 800 1000 12000
200
400
600
800
1000
1200
1400
1600
number of grid points
Cpu
tim
e (s
econ
ds)
DG−schemeKoren methodVan LeerSuperbeeMC method
Figure 4.11: Error analysis for two component nonlinear elutions: top: L1-Error for com-ponent 1, middle: L1-Error for component 2, bottom: CPU time of different schemes.
95
Chapter 5
Numerical Approximations ofRadiation Hydrodynamics Model
96
The radiation hydrodynamical models [81, 109, 30, 119] are the set of conservation laws
which arise in several areas of science and engineering. One of the major difficulties of
standard numerical methods for solving these models is to resolve and determine the exact
path of strong shocks. This chapter deals with numerical study of one and two-dimensional
radiation hydrodynamical equations. Two types of high resolution central schemes are
extended to solve the models namely, the central upwind scheme [63] and the staggard
central (NT) schemes [49, 91]. For comparison, KFVS scheme [119] is also applied to
these models. Several case studies are carried out and the numerical results are analyzed
qualitatively and quantitatively. This chapter is concluded with the summary of current
work.
5.1 One-dimensional RHD model
In this section, the one-dimensional RHD model is presented. Afterwards, the model equa-
tions are expressed in analogous form of the compressible Euler equations. A radiatively
opaque gas of same radiative and fluid temperatures is considered and the mean free path
of photons is assumed narrow as compared to the length of flow. The one-dimensional
radiation hydrodynamical model is given as [81, 109, 30, 119]
∂ρ
∂t+∂(ρu)
∂x= 0 , (5.1)
∂(ρu)
∂t+∂(ρu2 + p+ 1
3arT
4)
∂x= 0 , (5.2)
∂(E + arT4)
∂t+∂[u(E + p+ 4
3arT
4)]
∂x= ∇ · [κ(T )∇T ] , (5.3)
where ρ, p, u T , E, ar are the mass density, thermal pressure, flow velocity, temperature,
total energy and a radiation constant, respectively. Moreover, p = (γ−1)ρe, E = ρe+ 12ρu2,
e is the specific internal energy and κ(T ) is the nonlinear heat diffusivity. Note that, κ(T )
includes two distinct physical effects. One is the ordinary (i.e. non-radiative) thermal
conductivity and the other one is radiation diffusion which is defined as 4arcλrT3/3. Here,
The contents of this chapter are accepted in Computer Physics Communications Journal, 2012.
97
c and λr denote the speed of light and the Rosseland mean free path for photons. The
above equations can be re-written as
∂ρ
∂t+∂(ρu)
∂x= 0 , (5.4)
∂(ρu)
∂t+∂(ρu2 + p∗)
∂x= 0 , (5.5)
∂E∗
∂t+∂[u(E∗ + p∗)]
∂x= ∇ · [κ(T )∇T ] , (5.6)
where p∗ = p+ 13arT
4 and E∗ = E + arT4. Here, e = T being the specific internal energy.
In the above model, κ(T ) = 0 corresponds to a restrictive case where photon diffusion and
energy exchange driven by differences in temperature between the gas and the radiation
field are negligible in comparison to radiation work and advection of radiation. This is
known as the dynamic diffusion limit [84]. Furthermore, the above system reduces to the
inviscid Euler equations of gas dynamics if κ = 0 and ar = 0. The eigenvalues have similar
structure to the Euler equations, i.e.
λ1 = u− c∗ , λ2 = u , λ3 = u+ c∗ , (5.7)
where,
c∗ =
√γ∗p∗
ρ, (5.8)
is the speed of sound in the radiation hydrodynamics case. Let µ = 13arT 4
pand Γ = γ − 1,
then
γ∗ =γ + [4Γ(2− 3Γ) + 12γΓ]µ+ 16Γµ2
(1 + µ)(1 + 12Γµ)=
(γγ−1
+ 20µ+ 16µ2)
(1
γ−1+ 12µ
)(1 + µ)
. (5.9)
Note that, the sound speed can also be re-written as
(c∗)2 = c2 + 4µp
ρ
Γ(2− 3Γ) + 4µΓ
1 + 12µΓ,
where, c =√
γpρ. If ar = 0 (i.e. if µ = 0) then c∗ = c. To express the temperature in
terms of conserved variables, we can write E = ρT + 12ρu2 + aRT
4 as
T 4 + βT + η = 0 , (5.10)
98
where β = ρaR
and η = − 1aR(E − 1
2ρu2). Let p = −η and q = −β2
8then
y =
[−q2+
√q2
4+p3
27
] 13
−
[q
2+
√q2
4+p3
27
] 13
. (5.11)
The quartic Eq. (5.10) has four roots from which the physically acceptable root is
T =1
2
(−√2y +
√−2y +
2β√2y
). (5.12)
After calculating T , the pressure can be obtained from p = (γ − 1)ρT .
5.1.1 One-dimensional central upwind scheme
Here, the semi discrete central-upwind scheme is derived [63]. The above radiation hydro-
dynamics model can be viewed as a system of convection diffusion radiation system of the
form
wt + f(w)x = Rx , (5.13)
wherew = (ρ, ρu, E∗)T , f(w) = (ρu, ρu2+p∗, u(E∗+p∗))T andR = (0, 0, κ(T )Tx)T . The ra-
diation effect is incorporated in convective term with eigenvalues given by Eq. (5.7). Before
applying the scheme, the discretization of the computational domain is required. Let N be
a large integer representing the total number of discretization points and (xi− 12)i∈1,··· ,N+1
denote the divisions of the given domain [0, xmax]. For each i = 1, 2, · · · , N , ∆x represents
the constant width of each mesh cell, xi denotes the cell centers, and xi± 12refer to the cell
boundaries. We allocate,
x1/2 = 0 , xN+1/2 = xmax , xi+1/2 = i ·∆x . (5.14)
Moreover,
xi = (xi−1/2 + xi+1/2)/2 and ∆x = xi+1/2 − xi−1/2 =xmax
N + 1. (5.15)
99
Let Ωi :=[xi−1/2, xi+1/2
]for i ≥ 1. In each Ωi, the average values of conserved variables
w are defined as
wi := wi(t) =1
∆x
∫Ωi
w(x, t) dx . (5.16)
By integrating Eq. (5.13) over the interval Ωi =[xi−1/2, xi+1/2
], we obtain
dwi
dt= −
Si+ 12(t)− Si− 1
2(t)
∆x+
Ri+ 12(t)−Ri− 1
2(t)
∆x. (5.17)
The numerical fluxes are defined as
Si+ 12=
f(w+i+ 1
2
) + f(w−i+ 1
2
)
2−ai+ 1
2
2
(w+i+ 1
2
−w−i+ 1
2
), (5.18)
and for R = (R1, R2, R3)T , we have
(Rk)i± 12= 0 , for k = 1, 2 , (5.19)
(R3)i± 12=κ(T−
i± 12
) + κ(T+i± 1
2
)
2·T+i± 1
2
− T−i± 1
2
∆x. (5.20)
Here, w+ andw− are the point values of the piecewise linear reconstruction w = (ρ, ρu, E∗)
for w, namely:
w+i+ 1
2
= wi+1 −∆x
2wxi+1, w−
i+ 12
= wi +∆x
2wxi . (5.21)
The numerical derivatives wxi are at least first-order approximations of wx(xi, t) and are
computed using a nonlinear limiter that would ensure a non-oscillatory nature of the re-
construction (5.21). One possible way to compute the derivatives of conserved variables is
given as
wxi =MM
θ∆wi+ 1
2,θ
2
(∆wi+ 1
2+∆wi− 1
2
), θ∆wi− 1
2
, (5.22)
where 1 ≤ θ ≤ 2 is a parameter and ∆ denotes central differencing,
∆wi+ 12= wi+1 −wi .
Here, MM represents the min-mod non-linear limiter
MMx1, x2, ... =
minixi if xi > 0 ∀i ,maxixi if xi < 0 ∀i ,0 otherwise .
(5.23)
100
Further, the local one sided speed at xi+ 12is given as:
ai+ 12(t) = max
ρ
(∂f
∂w(w−
i+ 12
(t))
), ρ
(∂f
∂w(w+
i+ 12
(t))
). (5.24)
To achieve the second order accuracy in time coordinate, a second order TVD RK-method
is employed to solve Eq. (5.17). Denoting the right-hand side of Eq. (5.17) as L(w), a
second order TVD Runge-Kutta scheme updates w through the following two stages
w(1) = wn +∆tL(wn) , (5.25a)
wn+1 =1
2
(wn +w(1) +∆t L(w(1))
), (5.25b)
where wn represents the solution at time step tn and wn+1 denotes the updated solution
at tn+1. Moreover, ∆t represents the time step which is calculated under the following
Courant-Friedrichs-Lewy (CFL) condition
∆t ≤ 0.5min
(∆x
max(|u|+ c∗),
∆x2
max(2κ(T ))
), (5.26)
where c∗ is given by Eq. (5.8). The same CFL condition is also used for the other considered
schemes.
5.1.2 One-dimensional central schemes
Here, the high-resolution central scheme [91] is briefly presented. The method is of
predictor-corrector type and is applied in two steps. The predictor step corresponds to
the prediction of the midpoint values by considering the non-oscillatory piecewise-linear
reconstructions of the cell averages. In the second corrector step, staggered averaging and
the predicted mid-values are utilized to get the updated cell averaged solution. In summary,
the scheme can be presented as
Predictor: wn+ 1
2i =wn
i −ξ
2fx(wn
i ) , (5.27)
Corrector: wn+1i+ 1
2
=1
2(wn
i +wni+1) +
1
8(wx
i −wxi+1)
− ξ[fn+ 1
2i+1 − f
n+ 12
i
]+ ξ
[Rn+ 1
2i+1 −R
n+ 12
i
], (5.28)
101
where, ξ = ∆t/∆x. Moreover, 1∆x
fx(wi) is the approximated derivative of F(t, x = xi)
1
∆xfx(wi) =
∂
∂xf(w(t, x = xi) +O(∆x) . (5.29)
The fluxes fx(wi) are computed by the same manner as discussed for wx in Eq. (5.22).
5.2 Two-dimensional RHD model
In this section, we describe the RHEs in two space dimensions and the corresponding
Euler-like form. The model can be written as [48, 127]
∂ρ
∂t+∂(ρu)
∂x+∂(ρv)
∂y= 0 , (5.30)
∂(ρu)
∂t+∂(ρu2 + p+ 1
3arT
4)
∂x+∂(ρuv)
∂y= 0 , (5.31)
∂(ρv)
∂t+∂(ρuv)
∂x+∂(ρv2 + p+ 1
3arT
4)
∂y= 0 , (5.32)
∂(E + arT4)
∂t+∂[u(E + p+ 4
3arT
4)]
∂x+∂[v(E + p+ 4
3arT
4)]
∂y= ∇ · [κ(T )∇T ] , (5.33)
where p = (γ − 1)ρe, E = ρe + 12ρ(u2 + v2) and u and v are fluid velocities in x and
y-directions. The above system can be re-written as
∂ρ
∂t+∂(ρu)
∂x+∂(ρv)
∂y= 0 , (5.34)
∂(ρu)
∂t+∂(ρu2 + p∗)
∂x+∂(ρuv)
∂y= 0 , (5.35)
∂(ρv)
∂t+∂(ρuv)
∂x+∂(ρv2 + p∗)
∂y= 0 , (5.36)
∂(E∗)
∂t+∂[u(E∗ + p∗)]
∂x+∂[v(E∗ + p∗)]
∂y= ∇ · [κ(T )∇T ] , (5.37)
where p∗ = p+ 13arT
4 and E∗ = E + arT4.
5.2.1 Central upwind scheme
This subsection presents the extension of suggested central upwind scheme in two space
dimensions [63]. The above model can be expressed as:
wt + f(w)x + g(w)y = Rxx +Ry
y , (5.38)
102
where, w = (ρ, ρu, ρv, E∗)T , f(w) = (ρu, ρu2+p∗, ρuv, u(E∗+p∗))T , g(w) = (ρu, ρuv, ρu2+
p∗, v(E∗ + p∗))T , Rx = (0, 0, 0, κ(T )Tx)T and Ry = (0, 0, 0, κ(T )Ty)
T . The eigenvalues in
two space dimension is straightforward extension of Eq. (5.7).
Let Nx and Ny be the large integers in x and y-directions, respectively. A cartesian grid
with a rectangular domain [x0, xmax] × [y0, ymax] is considered which is covered by cells
Cij ≡[xi− 1
2, xi+ 1
2
]×[yj− 1
2, yj+ 1
2
]for 1 ≤ i ≤ Nx and 1 ≤ j ≤ Ny. The representative
coordinates in the cell Cij are denoted by (xi, yj). Here,
(x1/2, x1/2) = (0, 0), xi =xi−1/2 + xi+1/2
2, yj =
yj−1/2 + yj+1/2
2(5.39)
and
∆xi = xi+1/2 − xi−1/2 , ∆yj = yj+1/2 − yj−1/2 . (5.40)
The cell averaged values of wi,j(t) at any time t are given as
wi,j := wi,j(t) =1
∆xi∆yj
∫Cij
w(x, y, t) dydx . (5.41)
Now construct a piecewise linear interpolant
w(x, y, t) =∑i,j
[wi,j + (wx)i,j(x− xi) + (wy)i,j(y − yj)]χi,j, (5.42)
where χi,j is the characteristic function for the corresponding cell (xi− 12, xi+ 1
2)×(yj− 1
2, yj+ 1
2),
(wx)i,j and (wy)i,j are the approximations of x and y-derivatives of w at the cell centers
(xi, yj). The generalized MM limiter is used for the computation of these partial derivatives
to avoid oscillations
(wx)ni,j =MM
(θwi+1,j −wi,j
∆x,wi+1,j −wi−1,j
2∆x, θ
wi,j −wi−1,j
∆x
),
(wy)i,j =MM
(θwi,j+1 −wi,j
∆y,wi,j+1 −wi,j−1
2∆y, θ
wi,j −wi,j−1
∆y
), (5.43)
where 1 ≤ θ ≤ 2 and MM is given by Eq. (5.22). After integrating Eq. (5.38) over the
103
control volume Cij, the two-dimensional extension of the scheme can be scripted as
dwi,j
dt=−
Sxi+ 1
2,j− Sx
i− 12,j
∆x−
Syi,j+ 1
2
− Syi,j− 1
2
∆y
+Rxi+ 1
2,j−Rx
i− 12,j
∆x+
Ry
i,j+ 12
−Ry
i,j− 12
∆y. (5.44)
Here
Sxi+ 1
2,j=
f(w−i+ 1
2,j) + f(w+
i+ 12,j)
2−axi+ 1
2,j
2
(w+i+ 1
2,j−w−
i+ 12,j
),
Syi,j+ 1
2
=g(w−
i,j+ 12
) + g(w+i,j+ 1
2
)
2−ayi,j+ 1
2
2
(w+i,j+ 1
2
−w−i,j+ 1
2
). (5.45)
Similarly, Rxk = 0 = Ry
k for k = 1, 2, 3 and
(Rx4)i± 1
2,j =
κ(T−i± 1
2,j) + κ(T+
i± 12,j)
2·T+i± 1
2,j− T−
i± 12,j
∆x,
(Ry4)i,j± 1
2=κ(T−
i,j± 12
) + κ(T+i,j± 1
2
)
2·T+i,j± 1
2
− T−i,j± 1
2
∆y. (5.46)
The intermediate values are expressed as
w−i+ 1
2,j= wi,j +
∆x
2(wx)i,j , w+
i+ 12,j= wi+1,j −
∆x
2(wx)i+1,j
w−i,j+ 1
2
= wi,j +∆y
2(wy)i,j , w+
i,j+ 12
= wi,j+1 −∆y
2(wy)i,j+1 . (5.47)
Here, axi+ 1
2,jand ay
i,j+ 12
are the local speeds which can be calculated as
axi+ 1
2,j= max
±ρ
(∂f
∂w(w±
i+ 12,j)
), ay
i,j+ 12
= max±
ρ
(∂g
∂w(w±
i,j+ 12
)
). (5.48)
For complete derivation of the scheme the reader is referred to [63].
5.2.2 Two-dimensional central scheme
This central scheme was introduced in [49] for solving two-dimensional hyperbolic systems
of conservation laws. This scheme is formed by two steps to evolve the solution at the
next time level namely, predictor step and a corrector step. The predictor step is used to
104
calculate the midpoint values wn+ 1
2i,j from the cell average values wn
i,j. This step is followed
by the second-order corrector step for computation of the new cell averages wn+1i,j . Similar
to the one-dimensional case, the scheme do not need any Riemann solver for computing
the cell interface fluxes. Once again, the local monotonicity of the scheme is guaranteed by
applying the minmod limiters to compute the discrete slopes, wx, wy, fx(w), and gy(w).
The calculation of fluxes at the cell interfaces is avoided by staggering the grid at each
time level. The scheme is briefly presented below.
The predictor step corresponds to the calculation of midpoint values
wn+ 1
2i,j = wn
i,j −ξ
2fx(wn
i,j)−η
2gy(wn
i,j) , (5.49)
where, ξ = ∆t/∆x and η = ∆t/∆y. Afterwards, the corrector step is used to get the
updated solution at the next time level
wn+1i+ 1
2,j+ 1
2
=1
4(wn
i,j +wni+1,j +wn
i,j+1 +wni+1,j+1)
+1
16(wx
i,j −wxi+1,j)−
ξ
2
(fn+ 1
2i+1,j − f
n+ 12
i,j
)+
1
16(wx
i,j+1 −wxi+1,j+1)−
ξ
2
(fn+ 1
2i+1,j+1 − f
n+ 12
i,j+1
)+
1
16(wy
i,j −wyi,j+1)−
η
2
(gn+ 1
2i,j+1 − g
n+ 12
i,j
)+
1
16(wy
i+1,j −wyi+1,j+1)−
η
2
(gn+ 1
2i+1,j+1 − g
n+ 12
i+1,j
)+ξ
2
((Rx)
n+ 12
i+1,j − (Rx)n+ 1
2i,j
)+η
2
((Ry)
n+ 12
i,j+1 − (Ry)n+ 1
2i,j
). (5.50)
This completes the derivation of numerical schemes.
5.3 Numerical Case Studies
This section presents six test problems to analyze the accuracy and performance of the
proposed central schemes. For comparison, the KFVS scheme is also applied to this model
[119].
105
Problem 1
This is a one-dimensional shock-tube problem involving two rarefaction waves moving in
the opposite directions [48]. The diaphragm is placed at x = 0.5. The initial data are given
as
(ρ, T, u) =
(1, 1,−1), x ≤ 0.5 ,(1, 1, 1) x ≥ 0.5 .
(5.51)
The computational domain is [0, 1] which is subdivided into 50 cells and the final simulation
time is t = 0.2. The nonlinear heat diffusivity is taken as κ(T ) = K(1+ 10T 3), where K is
a scaling factor. To analyze the performance of numerical schemes for transport and diffu-
sion dominated cases, different values of K are considered from the interval [0, 0.5]. Figure
5.1 show the numerical results of central upwind scheme for K = 0, 10−3, 10−2, 10−1, 0.5.
One can observe that for K ≥ 10−2 the effect of diffusion term becomes visible. Tables
5.1, 5.2, 5.3, 5.4 show the L1-errors between the reference and numerical solutions of the
central and KFVS schemes for different values of the scaling factor K. The reference so-
lution was obtained at 2000 mesh points. The error plots of density, pressure and velocity
for different values of the scaling factor are given in Figures 5.2, 5.3, 5.4. Moreover, Figure
5.5 shows the comparison of results for K = 0. It can be observed that the results of
central-upwind scheme are in good agreement with KFVS and NT central schemes. All
figures and tables show that the central upwind and the KFVS schemes give compara-
ble results for the whole range of K while the staggered central (NT) scheme produces
large errors in the solutions. In overall, the KFVS scheme has little edge over the central
schemes. From these results it can be concluded that the proposed schemes have uniform
behavior over the whole range of scaling factor K. Moreover, minor changes can be seen
in the magnitudes of errors over the whole range of scaling factor K. Thus, the consid-
ered numerical schemes behave uniformly in both transport and diffusion dominated limits.
Problem 2
This one dimensional shock-tube problem was considered in [48]. The initial data are given
106
as
(ρ, T, u) =
(1, 0.5, 50), x ≤ 0.6 ,(2, 1,−40) x ≥ 0.6 .
(5.52)
The computational domain is [0, 1] which is partitioned into 400 grid cells. The simulation
results at t = 0.04 for the density, velocity, pressure and temperature are shown in Figure
5.6. Two shocks of Mach 82 and 39 and a contact discontinuity are produced from the ini-
tial discontinuities. The numerical results of central upwind and KFVS schemes are better
than the staggered central (NT) scheme. The central upwind scheme seems to be superior
in all schemes. To analyze the accuracy of proposed scheme quantitatively, we have calcu-
lated the L1-errors at different grid points given in Table 5.5. The reference solution was
obtained at 2000 grid points. It is clear from the results that the central-upwind scheme
produces less errors in the solution as compared to the KFVS and NT central schemes.
Moreover, Figure 5.7 shows the plots of L1-errors which justify the results of Table 5.5.
Problem 3
This problem is given by Jiang and Sun [48] that involves a strong shock in one-dimensional
Riemann problem. Initially the values of (ρ, T, u) for x0 ≤ 0.5 are (1, 0.5, 150) and for
x0 > 0.5 are (2, 1,−100). The number of cells are 400 and the computational domain is
[0, 1]. The simulation results at t = 0.018 for the density, velocity, pressure and tempera-
ture are shown in Figure 5.8. The solution consists of two shocks with Mach numbers of
about 227.4 and 107.7 and a contact discontinuity. Once again the results of KFVS and
central upwind scheme are comparable and NT scheme gives diffusive results. However,
central upwind scheme is superior in all three schemes.
Problem 4
We consider another test problem where the initial data (ρ, T, u) for x0 ≤ 0.5 are (5, 1.5, 4)
and for x0 > 0.5 are (5, 1.5,−4). The number of cells are 400 and the domain is [0, 1]. The
simulation results at t = 0.18 for the density, velocity, pressure and temperature are shown
in Figure 5.9. The solution consist of two shocks with Mach numbers of about 227.4 and
107
107.7 and a contact discontinuity. Once again the results of KFVS and central upwind
schemes are comparable and NT scheme gives diffusive results. However, the central up-
wind scheme is superior in all three schemes.
Problem 5
This is a two-dimensional problem describing the interaction of a wind and a denser circular
cloud. In the rectangular domain [0, 2]× [0, 1], there is a 25 times denser cylindrical bubble
at (0.3, 0.5) whose radius is r = 0.15 and the number of grids are 256 × 128. The state
of ambient gas for (ρ, T, u, v) is (1, 0.09, 0, 0) and the wind state is (1, 0.09, 6(1− e−10t), 0)
which is introduced through the left boundary and ar = 1. The outflow boundary condi-
tions are applied at the right, lower and upper boundaries of the domain. We computed
the solution without and with diffusion limit, where, κ(T ) = 10−3(1 + 10T 3) [30]. It is ob-
served that the incoming shock produces further shocks inside the bubble and a refracted
shock. The contours for ρ, T and u at t = 0.6 are shown in Figures 5.10, 5.11, 5.12, 5.13,
5.15, and 5.16. The comparisons show good agreements between the proposed schemes
and the results available in the literature. It can be observed that KFVS and central up-
wind schemes produce comparable solutions and the NT scheme is diffusive. However, The
KFVS scheme produces more resolved solution.
Problem 6
Here, a square box of length 2.0 is considered. Further, a circular bubble of radius 0.15
is placed at the center of this box. The values for (ρ, T, u, v) are (1, 0.9, 0, 0) inside the
circle and (25, 0.9, 0, 0) outside the circle. The simulation results are obtained at t = 0.5 on
128×128 grid points. Figures 5.17 and 5.18 show the contour plots and comparison among
the central-upwind, NT central and KFVS schemes. Once again, a good performance of
the central upwind scheme can be seen. However, KFVS scheme shows better performance
in all three schemes.
108
5.4 Summary
We focused on the numerical solution of one- and two-dimensional radiation hydrodynami-
cal equations (RHEs). Two different types of central finite volume schemes were applied to
solve these equations, the central-upwind and the staggered central schemes. The proposed
numerical schemes preserves monotonicity due to using MUSCL-type reconstruction and
also avoid the detailed knowledge of complicated exact/approximate Riemann solver. For
validation, the KFVS scheme was also applied to solve the model equations. A number
of case studies were carried out and the accuracy of the schemes were analyzed quantita-
tively and qualitatively. It was found that central-upwind schemes have better resolved
discontinuous profiles as compared to the NT schemes. Further, it was concluded that the
proposed numerical scheme gives comparable results with KFVS scheme at low CPU time.
109
Figure 5.1: Problem 1: The central upwind scheme results for different values of K on 50grid points at t = 0.2.
110
Figure 5.2: Problem 1: L1-error plots of density for different values of K.
111
Figure 5.3: Problem 1: L1-error plots of pressure for different values of K.
112
Figure 5.4: Problem 1: L1-error plots of temperature for different values of K.
113
Figure 5.5: Problem 1: Results on 400 mesh cells at t = 0.2 for κ(T ) = 0.
114
Figure 5.6: Problem 2: Results on 400 mesh cells at t = 0.04.
115
Table 5.1: Problem 1: Comparison of errors at different grid points for K = 10−3.N =80 N=160 N=320
Methods ρ T p ρ T p ρ T pcentral upwind 0.013 0.007 0.014 0.005 0.003 0.006 0.002 0.001 0.002
KFVS 0.012 0.007 0.013 0.006 0.003 0.006 0.002 0.001 0.002NT central 0.020 0.013 0.027 0.014 0.008 0.015 0.009 0.005 0.009
Table 5.2: Problem 1: Comparison of errors at different grid points for K = 10−2.N =80 N=160 N=320
Methods ρ T p ρ T p ρ T pcentral upwind 0.007 0.005 0.009 0.003 0.002 0.004 0.001 0.001 0.002
KFVS 0.007 0.005 0.009 0.003 0.002 0.004 0.001 0.001 0.002NT central 0.009 0.009 0.014 0.004 0.004 0.006 0.002 0.002 0.004
Table 5.3: Problem 1: Comparison of errors at different grid points for K = 10−1.N =80 N=160 N=320
Methods ρ T p ρ T p ρ T pcentral upwind 0.015 0.003 0.011 0.007 0.001 0.005 0.003 0.001 0.002
KFVS 0.014 0.003 0.010 0.006 0.001 0.005 0.003 0.001 0.002NT central 0.030 0.005 0.019 0.016 0.002 0.010 0.012 0.001 0.007
Table 5.4: Problem 1: Comparison of errors at different grid points for K = 0.5.N =80 N=160 N=320
Methods ρ T p ρ T p ρ T pcentral upwind 0.014 0.002 0.012 0.008 0.001 0.005 0.003 0.000 0.002
KFVS 0.016 0.002 0.011 0.008 0.001 0.005 0.003 0.000 0.002NT central 0.050 0.003 0.030 0.036 0.002 0.021 0.021 0.001 0.012
Table 5.5: Problem 2: Comparison of errors for K = 0 at different grid points and CPUtimes.
N =200 N=800 N=1600 CPU (s)Methods ρ T p ρ T p ρ T p N=200
central upwind 0.21 0.13 23.16 0.06 0.02 4.16 0.02 0.01 1.32 1.7KFVS 0.27 0.12 23.50 0.09 0.02 4.24 0.05 0.01 1.49 3.6
NT central 0.40 0.23 43.90 0.14 0.05 8.84 0.07 0.02 3.66 2.3
116
Figure 5.7: Problem 2: L1-Error plots at κ(T ) = 0.
117
Figure 5.8: Problem 3: Results on 400 mesh cells at t = 0.02.
118
Figure 5.9: Problem 4: Results on 400 mesh cells at t = 0.18.
119
x−axis
y−ax
is
temperature, T
0 0.5 1 1.5 2
0.5
1
x−axis
y−ax
is
velocity, u
0 0.5 1 1.5 2
0.5
1
Figure 5.10: Problem 5: 2D results on 256× 128 mesh cells at t = 0.6, κ(T ) = 0.
Figure 5.11: Problem 5: 2D results on 256 × 128 mesh cells at t = 0.6, κ(T ) = 10−3(1 +10T 3).
120
Figure 5.12: Problem 5: 1D results along x = 0.25 at t = 0.6, κ(T ) = 0.
121
Figure 5.13: Problem 5: 1D results along y = 0.5 at t = 0.6, κ(T ) = 0.
122
Figure 5.14: Problem 5: Density and temperature contour at different time steps, κ(T ) = 0.
Figure 5.15: Problem 5: 1D results along y = 0.55 at t = 0.5, κ(T ) = 0.
123
Figure 5.16: Problem 5: 1D results along x = 0.30 at t = 0.5, κ(T ) = 0.
Figure 5.17: Problem 6: 2D results on 128× 128 mesh cells at t = 0.5, κ(T ) = 0.
124
Figure 5.18: Problem 6: 1D results along y = 1.0 at t = 0.5, κ(T ) = 0.
125
Chapter 6
Application of CE/SE Method toHyperbolic Heat Conduction Model
126
In 1929, Peierls [92] has introduced his distinguished theoretical model developed from
the Boltzmann kinetic equation. According to his theoretical model, the vibrations of
lattices are the main sources of heat transports which can be expressed as an interacting
Phonons gas. This kinetic approach of Peierls is a landmark achievement in the theoretical
investigation of heat transport inside the solids at low temperatures. Peierls overcame the
failure of Fourier theory of heat flow at low temperatures in the heat conduction processes.
Further studies on the heat conduction models can be found in Dreyer et al. [32, 33, 34, 35].
This chapter deals with the numerical study of hyperbolic heat conduction models in one
and two space dimensions. The CE/SE method of Chang et al. [16, 132] is applied to
solve the models. For analyzing the accuracy, the results of CE/SE method are compared
with the central schemes [91] and kinetic flux vector splitting (KFVS) schemes. A brief
summary of the current contribution is presented at the end of this chapter.
6.1 The Hyperbolic Heat Conduction
In this section, the three dimensional model of hyperbolic heat conduction is derived [32,
33, 34, 35, 95]. The kinetic Boltzmann Peierls equation (BPE) describes the evolution of
phase density F (t, x, k). The three-dimensional BPE is by [32, 33]
∂F
∂t+∂ω
∂kk
∂F
∂xi= ζ(F ), (6.1)
where ω denotes the frequency of phonon, t denotes time, k stands for the phonon wave
vector, and ζ represents the collision operator. A real crystal has three phonon modes.
Thus, three phase densities exist namely, one for longitudinal mode and two for transversal
modes. For simplicity, the actual crystal can be replaced by a so called Debye solid which
is characterized by a single mode only [34]. The assumed dispersion relation between the
phonon frequency ω and the wave vector k is given as
ω = c|k|. (6.2)
The contents of this chapter are accepted in International Journal of Computational Methods (IJCM),2012.
127
Here, c is known as the Debye velocity that corresponds to the mean of two transversal
and longitudinal sound speeds of the actual crystal. Therefore, the BPE is given by
∂F
∂t+ c
3∑i=1
ki
|k|∂F
∂xi= ζ(F ). (6.3)
The most important moments of the phase density F , reflecting the kinetic processes on
the scale of continuum physics, are given as
e(t,x) = ~c+∞∫
−∞
|k|F (t,x,k)d3k, (6.4)
Qi(t,x) = ~c2+∞∫
−∞
kiF (t,x,k)d3k, (6.5)
N ij(t,x) = ~c
+∞∫−∞
kikj
|k|F (t,x,k)d3k, i, j = 1, 2, 3 , (6.6)
where ~ is Planck’s constant. The fields e and Q = (Q1,Q2,Q3)T represent the energy
density and the heat flux, respectively. The matrix (N ij) denotes the momentum flux.
Phonons are treated as Bose particles and the corresponding entropy density entropy flux
pair (h, φ) is given as [33, 34, 35, 92]
h(F ) := y
∫R3
[(1 +
F
y
)ln
(1 +
F
y
)− F
yln
(F
y
)]d3k , (6.7)
φi(F ) := yc
∫R3
ki
|k|
[(1 +
F
y
)ln
(1 +
F
y
)− F
yln
(F
y
)]d3k, (6.8)
where y = 38π3 . In ordinary gas atoms, the phonons may interact by two different collision
processes, called the R- and the N -processes. The R-processes include interactions of
phonons with lattice impurities which destroy the periodicity of the crystal, while the N -
processes can be interpreted as phonon-phonon interactions which are due to the deviations
from harmonicity of the crystal forces. Both energy and momentum are conserved in the
N -processes, however, in the R-processes energy is conserved only. The actual interaction
process is simplified by the Callaway approximation of the collision operator [34] and may
128
be written in the form of the sum of two relaxation operators modeling the R and the
N -processes separately
ζ(F ) = ζR(F ) + ζN(F ), ζα =1
τα(PαF − F ), α ∈ R,N . (6.9)
The constants τR and τN are the relaxation times, while Pα are two non linear projectors.
Here PαF represent the phase densities in the limiting case when the relaxation time tends
to zero. Therefore, we define PRF and PNF as the solutions of two optimization problems,
namely
h(PRF ) = maxF ′
h(F ) : e(F ′) = e(F ) , (6.10)
h(PNF ) = maxF ′
h(F ) : e(F ′) = e(F ), Q(F′) = Q(F) , (6.11)
where e(F ) and Q(F ) are given by Eqs. (6.4) and (6.5). When the thermodynamic state is
described by four fields e and Qi only, then we can derive the following balance equations
from the Boltzmann Peierls Eq. (6.1) and the maximum entropy principle [32, 33]
∂e
∂t+∂Qi
∂xi= 0, (6.12)
∂Qi
∂t+∂(c2N ij)
∂xi= − 1
τRQi, i, j = 1, 2, 3, (6.13)
N ij =1
3eδij +
1
2e(3χ− 1)
(QiQj
|Q|2− 1
3δij), (6.14)
where χ is the Eddington-factor
χ =5
3− 4
3
√1− 3
4
(| Q |ce
)2
, |Q| =√(Q1)2 + (Q2)2 + (Q3)2. (6.15)
It should be noted that the term τN do not exists on the righthand side of the above
equation. Therefore, these equations are applicable only when the relaxation limit τN −→ 0
[33, 95]. The Eqs. (6.12)-(6.14) can be written in conservative form as
∂w
∂t+
3∑i=1
∂f i(w)
∂xi= P(w), (6.16)
where
w =
eQ1
Q2
Q3
, f i(w) =
Qi
c2N1i
c2N2i
c2N3i
, (6.17)
129
and
P(w) = − 1
τR
eQ1
Q2
Q3
. (6.18)
This complete the derivation of three dimensional model for hyperbolic heat conduction.
The next step is to implement CE/SE method for solving the above model up to two space
dimensions.
6.2 One-Dimensional CE/SE Method
The CE/SE-method was developed by Chang [16] to solve the hyperbolic systems of con-
servation laws. The method was developed to overcome some of the drawbacks of well
established methods. Here, the CE/SE scheme is derived for solving the one dimensional
moment system. In one space dimension, Eqs. (6.16)-(6.18) reduce to
∂w
∂t+∂f(w)
∂x= P(w), (6.19)
where
w =
(e
Q
), f(w) =
(Q
c2N
), P(w) = − 1
τR
(0
Q
). (6.20)
Let x1 = t and x2 = x be the coordinates of a two-dimensional Euclidian space E2 and for
i = 1, 2, hi = [wi, fi,−Pi]T be the current density vectors in E2. By applying the Gauss
divergence theorem, the Eq. (6.19) is equivalent to the integral equation [16]∫S(V )
hi·dS = 0, i = 1, 2 , (6.21)
where i indicates the number of equations and S(V ) is the boundary of an arbitrary space-
time domain V . The Eq. (6.21) is restricted to the space-time domain, the conservation-
element (CE), that permits the flow variables discontinuities. The actual numerical inte-
gration is done directly on the solution elements (SEs). An SE is a different space-time
region and the flow variables are considered to be smooth inside this region. Therefore,
one can perform the discretization of the flow variables with a prescribed order of accuracy.
130
Let Ω denotes the set of grid points (j, n) with n = 0,±12,±1,±3
2, · · · and for every n, j =
n ± 12, n ± 1, n ± 3
2, · · · . With each (j, n) ∈ Ω one SE is associated. Assume that SE(j, n)
is the inner shaded space-time region in Figure 6.1. It include segments of horizontal and
vertical lines and their close neighborhood. The actual size of the neighborhood is not a
concern. At point (t, x) in SE, the approximation of wk(t, x), fk(t, x) and hk(t, x) are given
as w∗k(t, x; j, n), f
∗k (t, x; j, n) and h
∗k(t, x; j, n), respectively. Let
w∗i (t, x; j, n) = (wi)
nj + (wit)
nj (t− tn) + (wix)
nj (x− xj) , (6.22)
where (wi)nj , (wit)
nj and (wix)
nj are constants in SE(j, n). They can be considered as the
numerical analogues of the values of wi,∂wi
∂xand ∂wi
∂tat (tn, xj), respectively. Let
fi,k =∂fi∂wk
, i, k = 1, 2, (6.23)
where (fi)nj and (fi,k)
nj denote the values of fi and fi,k respectively. By applying chain rule,
we obtain
(fix)nj =
3∑k=1
(fi,k)nj (wkx)
nj = (Jii)
nj (wkx)
nj , (6.24)
(fit)nj =
3∑k=1
(fi,k)nj (wkt)
nj = (Jii)
nj (wkt)
nj , (6.25)
where J is the Jacobian matrix given as follows
J =
(0 1
c2χ− (Qe)2[1− 3
4(Qce)2]−1/2 Q
e
[1− 3
4(Qce)2]−1/2
). (6.26)
Now, in Eqs. (6.24) and (6.25), (fix)nj and (fit)
nj can be considered as the numerical ana-
logues of the values of ∂fi∂x
and ∂fi∂t
at (tn, xj) respectively. Therefore, it is assumed that
f ∗i (t, x; j, n) = (f i)
nj+(f it)
nj (t− tn) + (f ix)
nj (x− xn). (6.27)
Because of hi = (fi, wi), one can write
h∗i (t, x; j, n) = (f ∗i (t, x; j, n), w
∗i (t, x; j, n)− P ∗
i (t, x; j, n)). (6.28)
131
By definitions, (fi)nj are the functions of (wi)
nj , (fix)
nj are the functions of (wi)
nj and (wix)
nj ,
and (fit)nj are the functions of (wi)
nj and (wit)
nj . Moreover, it is assumed that for any
(t, x) ∈ SE(j, n), w∗i (t, x; j, n) and f
∗i (t, x; j, n) satisfy the Eq. (6.19), i.e.
∂w∗i (t, x; j, n)
∂t+∂f ∗
i (t, x; j, n)
∂x= P i(w). (6.29)
According to Eqs. (6.22) and (6.27), Eq. (6.29) is equivalent to
(wit)nj = −(fix)
nj + Pi(w) . (6.30)
As (fix)nj are functions of (wi)
nj and (wix)
nj , Eq. (6.30) implies that (wit)
nj are also the
functions of (wi)nj and (wix)
nj . Due to this result and Eq. (6.30), one conclude that (wit)
nj ,
(fi)nj , (fix)
nj and (fit)
nj are the functions of (wi)
nj and (wix)
nj . As a result (wi)
nj and (wix)
nj
are needed to be solved in the current marching scheme and which are the only independent
discrete variables.
In Figure 6.1, the rectangular non-overlapped regions represent the CEs. Consider the
integral representation of Eq. (6.19)∫∂Ω
w∗i dx− f ∗
i dt = 0 , i = 1, 2, 3, (6.31)
where, ∂Ω is the boundary of Ω. On integrating over the rectangular regions Ω1 =
[xj−1/2, xj+1/2] × [tn, tn+1/2] and Ω2 = [xj−1/2, xj+1/2] × [tn, tn+1/2], where Ω = Ω1UΩ2,
we obtain
(wi)nj − (wi)
n− 12
j± 12
± ∆x
4
[(wix)
n− 12
j± 12
+ (wix)nj
](6.32)
± ∆t
∆x
[(fi)
n− 12
j± 12
− (fi)nj
]± ∆t2
4∆x
[(fit)
n− 12
j± 12
− (fit)nj
]=
∆x∆t
4(Pi)
n− 12
j− 12
.
Note that, we have have two equations for two unknowns wni and wnix. By adding and then
subtracting the above two equations gives
(wi)nj =
1
2
((wi)
n− 12
j− 12
+ (wi)n− 1
2
j+ 12
+ (Si)n− 1
2
j− 12
− (Si)n− 1
2
j+ 12
)+
∆t
4
((Pi)
n− 12
j− 12
+ (Pi)n− 1
2
j+ 12
)(6.33)
132
Figure 6.1: Illustration of the one-dimensional staggered grid.
and
∆x
2(wix)
nj =
((wi)
n− 12
j+ 12
− (wi)n− 1
2
j− 12
)−((Si)
n− 12
j+ 12
+ (Si)n− 1
2
j− 12
)(6.34)
− ∆t
∆x
(∆t
2(fit)
nj − 2(fi)
nj
)+
∆t
2
((Pi)
n− 12
j− 12
− (Pi)n− 1
2
j+ 12
),
where
(Si)n− 1
2
j± 12
=∆x
4(wix)
n− 12
j± 12
+∆t
∆x(fi)
n− 12
j± 12
+∆t2
4∆x(fit)
n− 12
j± 12
. (6.35)
In the presence of steep fronts or discontinuities, approximated spatial derivatives (wix)nj
in Eq. (6.34) can be inaccurate. In this case, the solution can be very oscillatory and for a
small Courant-Friedrichs-Lewy (CFL) condition the solution may become overly diffusive.
However, to overcome this diffusive nature of the solution, some limiting formulations for
133
the slopes of conservative variables has been introduced by Chang [16].
(wix)nj = Ui((wix−)
nj , (wix+)
nj ;α) , i = 1, 2, 3 . (6.36)
Here 1 ≤ α ≤ 2 and
Ui(x−, x+;α) =|x+|αx− + |x−|αx+
|x+|α + |x−|α. (6.37)
Moreover,
(wix±)nj = ±
(w′i)nj± 1
2
− (wi)nj
∆x/2, (w′
i)nj± 1
2= (wi)
n− 12
j± 12
+∆t
2(wit)
n− 12
j± 12
. (6.38)
Thus, Eq. (6.34) is replaced by Eqs. (6.36)-(6.38) in our numerical scheme. This concludes
the derivation of the space-time CE/SE method for the current model.
6.3 Two-Dimensional CE/SE Method
In this section, the extension of suggested method to two space dimensions is briefly pre-
sented.∂w
∂t+∂f(w)
∂x+∂g(w)
∂y= P(w), (6.39)
where
w =
eQ1
Q2
, f(w) =
Q1
c2N11
c2N12
(6.40)
and
g(w) =
Q2
c2N21
c2N22
, P(w) =−1
τR
0Q1
Q2
. (6.41)
Here, w = [w1, w2, w3]T is the vector of conserved variables, f and g are their corresponding
fluxes along each direction.
The details about the techniques are given in Zhang et al. [132]. Here, we give the final
formulation of the scheme on regular rectangular grid. Let N and M denote the larger
integers in the space directions x and y, respectively. Consider a cartesian grid on a
134
rectangular domain covered by cells Ωij :=[xi− 1
2, xi+ 1
2
]×[yj− 1
2, yj+ 1
2
]for i = 1, 2, · · · , N
and j = 1, 2, · · · ,M . The representative coordinates of Ωij are given by (xi, yj). The
scheme has the following formulation for Eq. (6.39)
wni,j =
1
4
(wn− 1
2i−1,j +w
n− 12
i+1,j +wn− 1
2i,j−1 +w
n− 12
i,j+1
+ (Sx)n− 1
2i−1,j − (Sx)
n− 12
i+1,j + (Sy)n− 1
2i,j−1 − (Sy)
n− 12
i,j+1
)(6.42)
+λx2
((wx)
n− 12
i,j − (wx)n− 1
2i−1,j
)+λy2
((wy)
n− 12
i,j − (wy)n− 1
2i,j−1
)+
∆t
2Pn− 1
2i,j ,
where λx = ∆t/∆x, λy = ∆t/∆y and
(Sx)n− 1
2i,j =
∆x
2(wx)
n− 12
i,j + λx
(fn− 1
2i,j + (ft)
n− 12
i,j
), (6.43)
(Sy)n− 1
2i,j =
∆y
2(wy)
n− 12
i,j + λy
(gn− 1
2i,j + (gt)
n− 12
i,j
). (6.44)
To obtain (wx)ni,j and (wy)
ni,j, a central difference-type reconstruction procedure is em-
ployed. Let
u1,x = u2,x =
(wni,j −w
n− 12
i−1,j −∆t
2(wt)
n− 12
i−1,j
)/∆x , (6.45)
u3,x = u4,x =
(wn− 1
2i+1,j +
∆t
2(wt)
n− 12
i+1,j −wni,j
)/∆x . (6.46)
Similarly
u1,y = u4,y =
(wn− 1
2i,j+1 +
∆t
2(wt)
n− 12
i,j+1 −wni,j
)/∆y , (6.47)
u2,y = u3,y =
(wni,j −w
n− 12
i,j−1 −dt
2(wt)
n− 12
i,j−1
)/∆y . (6.48)
Then
ϕl =√(ui,x)2 + (ui,y)2 , for i = 1, . . . , 4 (6.49)
and ω1
ω2
ω3
ω4
=
ϕ2ϕ3ϕ4
ϕ1ϕ3ϕ4
ϕ1ϕ2ϕ4
ϕ1ϕ2ϕ3
. (6.50)
135
Using the above relations, finally we obtain
(wα)ni,j =
4∑i=1
ui,α|ωi|θ
4∑i=1
|ωi|θ, α ∈ x, y . (6.51)
where 1 ≤ θ ≤ 2. This completes the derivation of two-dimensional CE/SE method on
rectangular grids.
6.4 Application of Boundary Conditions
As we compute on some finite set of grid cells covering a bounded domain, the current
model is an initial and boundary value problem requiring a set of boundary conditions
along with the initial data. The implementation of boundary conditions are simpler in
the conservative schemes. Here, we discuss the left boundary condition (x = 0). This
procedure is analogous for the right boundary and for the multidimensional problems. Let
us denote the auxiliary cell by a subscript A and the time step by a superscript n. The
following boundary conditions are used for the considered numerical test problems.
Reflecting boundary conditions: In such boundary conditions, the heat flux is needed to
be zero at the wall, therefore we take enA = en1 and QnA = −Qn
1 .
Outflow boundary conditions: In this type of boundary conditions, the values of variables
are the same on the both sides of the wall, enA = en1 and QnA = Qn
1 .
Inflow boundary conditions: If the values of enW and QnW , are given then we can find the
values of enA and QnA. However, from experimental point of view enW and Qn
W cannot be
known simultaneously. Either the energy density enW is controlled at the wall, or the
wall is equipped with a procedure of Joule’s heat and thus the heat flux is prescribed
[35, 34, 33, 32]. Here, we consider the case that only enW is given. Therefore, QnW is
obtained from the shock condition [35].
136
6.5 Numerical Test Cases
In this section, one and two-dimensional case studies are carried out to analyze the accuracy
of suggested numerical technique. Moreover, the results of central and KFVS schemes
[49, 91, 95] are also compared with those obtained from the proposed CE/SE method.
These numerical case studies were also presented in Dreyer et al. [32, 33, 95].
6.5.1 One-dimensional test problems
Here, three one-dimensional test problems are presented.
Problem 1: Two heat pulses interacting each other
In this test problem, we discuss two heat pulses interacting each other. At the point of
collision, the density jumps to a high value during a short period of time. The initial
conditions are given as
e(0, x) =
1, x ≤ 0.3 ,2, x ∈ [0.3, 0.4] ,1, x ∈ [0.4, 0.6] ,2, x ∈ [0.6, 0.7] ,1, x ≤ 1.0 ,
, Q(0, x) =
0, x ≤ 0.3 ,1, x ∈ [0.3, 0.4] ,0, x ∈ [0.4, 0.6] ,−1, x ∈ [0.6, 0.7] ,0, x ≤ 1.0 .
(6.52)
We solve the system (6.19) for τR = 1.0 at times t = 0.2 and t = 0.3. The computational
domain is discretized into 200 mesh points and the numerical results are presented in Fig-
ure 6.2. The outflow boundary conditions are used at both boundaries. A comparison
of the initial data and the solution at t = 0.2 reveals that a large increase in the density
at the collision point x = 0.5 produces a narrow peak. Afterwards, that peak splits into
two-peaks moving in the opposite directions which can be observed in the energy density
plot at t = 0.5. The results of CE/SE method are better resolved as compared to the
results of central and KFVS schemes. The central scheme gives a most diffusive solution.
Problem 2: Single shock
A single shock solution is considered here. The problem has reflecting boundary conditions
at x = 0. The outflow boundary conditions are used at the right boundary. The initial
137
data are
(e,Q)(0, x) =
(1, 0), x ≤ 0.5 ,
2,− 1√3
√3√2−1√2+1
, x ≥ 0.5 .(6.53)
This data for a single shock is taken from [35]. x ∈ [0, 1] is the computational domain and
is discretized into 200 mesh points. Figure 6.3 shows the results. The comparison shows
that the CE/SE method has better accuracy than the other implemented schemes and the
position of the shock is more accurate.
Problem 3: Inflow periodic boundary conditions
In this numerical problem the inflow boundary conditions are used and the following phe-
nomena are observed:
(a) The formation and steeping of shock fronts.
(b) The speed of shock front is apparently larger than c/√3.
(c) The broadening of initial heat pulses at later times.
A periodic energy pulse is created at the left boundary
ew(t) = 2− cos(8πt) . (6.54)
The corresponding value of the heat flux is obtained from the shock conditions given in
[35]
Qw(t) = (ew(t)− 1)1√3
√3√ew(t)− 1√ew(t) + 1
. (6.55)
The outflow boundary conditions are taken on the right boundary of the domain. The
initial data are e0 = 1 and Q0 = 0 and x ∈ [0, 1.2] is the computational domain which is
discretized into 200 mesh points. Figure 6.4 gives the results at t = 0.5. The formation and
steepening of shock fronts is clearly visible in the solution. The proposed CE/SE method
has better accuracy compared to the central and KFVS schemes. Once again the results
of the central scheme are the most diffusive ones.
138
6.5.2 Two-dimensional test problems
Here, three two-dimensional problems are presented. The first test problem is related to
the explosion in square box, the second one is a heat pulse inside a square box, while the
third one is an initial value problem with inflow left boundary conditions. The numeri-
cal results of CE/SE method are compared with the central and KFVS schemes [49, 91, 95].
Problem 4: Explosion inside a square box
This problem presents the evolution of energy pulse. A small square box of sides length
0.5 is placed in the center of a larger box of side length 2. Initially there are no heat
fluxes. The energy density is 2.0 inside the small box and it is unity outside. The reflecting
boundary conditions are considered at the walls of outer box. The domain is discretized
into 300 × 300 grid points and τR = ∞. The numerical results at times t = 1.0, t = 1.5
and t = 2.0 are shown in Figures 6.5 and 6.6. The energy and heat pulses move towards
the boundaries of the computational domain and reflect back from the solid walls of the
domain. In Figure 6.6, the one dimensional plots at y = 1 show that the proposed CE/SE
method gives the most resolved solutions.
Problem 5: Heat pulse in 2D
Once again, the evolution of energy pulse is analyzed in a box of equal sides. The length
of each side of the box is considered 2.0 with periodic boundaries. Initially there are no
heat fluxes i.e. the fluxes are zero. Another square box of sides length 0.11 is taken in the
center of the large box. The value of energy density is 1.5 inside the small square box while
it is unity elsewhere. The outflow boundary conditions are used at both boundaries. The
domain is discretized into 200 × 200 grid points and τR = ∞. The numerical results are
given in Figures 6.7 and 6.8 at times t = 0.5 and t = 1.2. In Figure 6.8, the one-dimensional
plots at y = 1 demonstrate the high order accuracy of the proposed CE/SE method.
139
Problem 6:. Inflow boundary conditions in 2D
The propagation of heat pulse is considered here
ew(t) =
1, t ≤ 0.0 ,3, 0 < t ≤ 0.5 ,1, t > 0.5 ,
(6.56)
which is generated at the lower xy-boundaries and the heat fluxes are zero at this boundary.
The initial data are e0 = 2 and Q10 = 0, Q2
0 = 0. The outflow boundary conditions are
used at the remaining boundaries. The computational domain is 0 ≤ x, y ≤ 1.5 which is
discretized into 200× 200 grid points. The Figure 6.9 shows the numerical results at time
t = 1.5. It can be observed that CE/SE-method has given more resolved solution.
6.6 Summary
The space-time CE/SE-method was applied for solving the one and two-dimensional initial
and BVP of hyperbolic heat conduction. One of the major difficulty for such system is the
occurrence of discontinuities in the solutions. A less accurate method fails to capture such
systems. The implemented method has successfully captured the sharp solution profile.
For comparison, the central and KFVS schemes were also implemented to the same model.
It was found that the results of CE/SE-method are better than these schemes. Therefore,
it is concluded that the space-time CE/SE-method is an efficient and accurate numerical
technique for solving such engineering problems.
140
Figure 6.2: Problem 1: Interaction of two pulses.
Figure 6.3: Problem 2: A single shock reflection.
141
Figure 6.4: problem 3: Inflow periodic boundary conditions.
142
x−axis
y−ax
is
energy density at t=1.0
0 0.5 1 1.5 20
0.5
1
1.5
2
x−axisy−
axis
heat flux Q2 at t=1.0
0.5 1 1.5 20
0.5
1
1.5
2
x−axis
y−ax
is
energy density e at t=1.5
0 0.5 1 1.5 20
0.5
1
1.5
2
x−axis
y−ax
is
heat flux Q2 at t=1.5
0 0.5 1 1.5 20
0.5
1
1.5
2
x−axis
y−ax
is
energy density at t=2.0
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x−axis
y−ax
is
heat flux Q2 at t=2.0
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Figure 6.5: Problem 4: Evolution of energy density and heat flux (explosion in a box).
143
0 0.5 1 1.5 2
0.7
0.8
0.9
1
1.1
1.2
1.3
x−axis
ener
gy d
ensi
ty, e
t=1.0
CE/SE
KFVS
central
0 0.5 1 1.5 2−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
x−axishe
at fl
ux, Q
2
t=1.0
CE/SE
KFVS
central
0 0.5 1 1.5 20.7
0.8
0.9
1
1.1
1.2
1.3
1.4
x−axis
ener
gy d
ensi
ty, e
t=1.5
CE/SE
KFVS
central
0 0.5 1 1.5 2
−0.1
−0.05
0
0.05
0.1
x−axis
heat
flux
, Q2
t=1.5
CE/SE
KFVS
central
0 0.5 1 1.5 2
0.9
0.95
1
1.05
1.1
1.15
x−axis
ener
gy d
ensi
ty, e
t=2.0
CE/SE
KFVS
central
0 0.5 1 1.5 2−0.15
−0.1
−0.05
0
0.05
0.1
0.15
x−axis
heat
flux
, Q2
t=2.0
CE/SEKFVScentral
Figure 6.6: Problem 4: One-dimensional plots along x-axis at y = 1.
144
x−axis
y−ax
is
heat flux Q2 at t=0.5
0.5 1 1.5 20
0.5
1
1.5
2
Figure 6.7: Problem 5: Evolution of energy density and heat flux (interacting heat pulses).
145
Figure 6.8: Problem 5: One-dimensional plots along x-axis at y = 1.
146
Figure 6.9: Problem 6: Evolution of energy density, heat flux, and 1D plots at y = 0.75(inflow boundary conditions in 2D).
147
Chapter 7
Conclusions and FutureRecommendations
148
7.1 Conclusions
In this thesis, five selected convection-diffusion-reaction (CDR) type models, coupled with
some algebraic equations, were theoretically investigated. The considered models include
the gas-solid reaction, chemotaxis, liquid chromatography, radiation hydrodynamical, and
hyperbolic heat condition models. These systems have wide range applications in different
areas of science and engineering, such as chemical engineering, biological systems, astro-
physics, heat transfer and fluid dynamics.
The Laplace transformation was used as a basic tool to solve the single and two-species
linear CDR models analytically. These analytical solutions were used to quantitatively
analyze the performance of suggested numerical schemes in Chapter 3.
For the chosen nonlinear CDR models, numerical solution techniques are the only tools
to get physically realistic solutions. The strong nonlinearity of transport terms and the
stiffness of reaction terms in the systems pose major difficulties for the applied numerical
schemes. For that reason, computational efficiency and accuracy of a numerical method
are of large relevance.
For the first time, the space-time conservation element and solution element (CE/SE)
methods were applied to solve the gas-solid-reaction model and the Kelar-Segel model of
chemotaxis. For the one-dimensional gas-solid-reaction, the original CE/SE method was
applied, while a variant CE/SE method on regular rectangular meshes was employed for
solving the two-dimensional Keller-Segel model of chemotaxis. The methods are derived
from integral formulation of the given CDR systems and have capabilities to accurately
capture the steep gradients and narrow peaks of the solutions. Several test cases were car-
ried out. The performance of the methods were analyzed by comparing their results with
those obtained from the staggered central schemes and selected flux-limiting finite volume
schemes (FVSs). It was observed that space time CE/SE methods have better capabilities
149
to handle the steep variation and delta type singularities in the solutions.
Two types of non-reactive liquid chromatographic models were theoretically studied un-
der isothermal condition namely, the equilibrium dispersive model (EDM) and the lumped
kinetic model (LKM). Both models were analyzed using continuous and discontinuous
Dirichlet and Robin inlet boundary conditions. These models are formed by the systems
of convection-diffusion partial differential equations with dominated convective terms and
are coupled with algebraic equations. They were investigated analytically and numeri-
cally. For linear adsorption isotherms, the Laplace transformation was used to solve the
single-component models analytically. In the case of no analytical Laplace inversion, the
numerical inversion was used to get the solution in actual domain. Good agreements of the
analytical and numerical results were observed. A close connection between equilibrium
dispersive and lumped kinetic models was pointed out. For nonlinear EDM and LKM,
the FVS of Koren [55] and the DG-method were proposed to approximate the considered
chromatographic models. The second order accurate Koren scheme is a flux-limiting FVS
in which fluxes are limited by using a nonlinear minmod limiter. This limiting proce-
dure preserves the local monotonicity (positivity) of the scheme by suppressing numerical
oscillations, usually encountered in the numerical schemes of second and higher orders.
The suggested DG-scheme satisfies the TVB property in time coordinate and gives overall
second order accuracy. High order basis functions and better limiters can be utilized to
extend the accuracy of the scheme to high orders. A better option for limiters can be the
WENO limiters introduced in [111]. Analytical solutions obtained in the Laplace domain
were used to validate the numerical results. The accuracy of the scheme was analyzed
by comparing its results with the considered FVSs. The suggested methods were found
to be efficient, accurate and suited to carry out simulations of dynamic column of chro-
matographic processes. The numerical test problems showed that DG-scheme produces
more accurate solutions compared to the FVS of Koren, especially at sharp discontinuities.
However, desirable results can also be achieved by using the Koren scheme.
150
The high resolution central upwind and staggered central schemes were applied to the one
and two-dimensional radiation hydrodynamical equations. The major difficulty in such
equations is to resolve and track the shock involved at right location. The KFVS schemes
were also implemented for comparison. It was observed that central upwind scheme pro-
duces accurate results as compared to staggered central schemes. Moreover, the results of
central upwind and KFVS schemes were found to be comparable and have low computa-
tional costs.
Finally, space-time CE/SE methods were implemented for solving hyperbolic heat conduc-
tion models up to two space dimensions. The staggard central and KFVS schemes were
also implemented for comparison. The space-time CE/SE methods performed better as
compared to the KFVS and the staggered central schemes.
7.2 Future Recommendations
It can be stated that the research work of this thesis lays down a foundation for further
extensions in different directions.
Although this study was restricted to one and two space dimensions, the extension of the
numerical methods to higher dimensions is analogous. In this study the selected CDR
models were solved by using the space-time CE/SE method, the central schemes and the
DG method. The present experience with the numerical methods paved the way for some
future work. The space-time CE/SE method can be applied to hypotaxis model which
is closely related to current model of chemotaxis. The DG-method could be applied to
solve the radiation hydrodynamical and hyperbolic heat transfer models. Moreover, work
is in progress on the application of DG-method for solving gas-Solid reaction model. In
this work, the equilibrium dispersive and lumped kinetic models were studied. This work
can be extended to solve more complicated models, such as the general rate model which
is quite complicated. In addition, the operating conditions of chromatographic process
151
have to be optimized to improve yield, productivity and to reduce the operational cost.
Optimization of this complex process is a challenging task due to the nonlinearity of chro-
matographic models and the large number of operating variables. Thus, it is necessary to
use an effective mathematical tool for global optimization of nonlinear chromatographic
problems.
152
Chapter 8
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