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Computer Physics Communications
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Application of central schemes for solving radiation hydrodynamicalmodelsShamsul Qamar a,b,∗, Waqas Ashraf aa Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistanb Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany
a r t i c l e i n f o
Article history:Received 29 July 2012Received in revised form21 November 2012Accepted 19 December 2012Available online 16 January 2013
This paper is concerned with the numerical investigation of radiation hydrodynamical models in oneand two space dimensions. The flow equations are the set of nonlinear mixed type partial differentialequations. The semi-discrete second order central upwind scheme is applied to solve the models. Theproposed numerical scheme uses precise information of the local speeds of propagation and, thus, avoidsthe excessive numerical diffusion which is normally observed in the staggered central scheme. Thesecond order accuracy of the scheme is achieved by using MUSCL-type reconstruction and Runge–Kuttatime stepping method. Several case studies are carried out. For validation, the numerical results ofthe suggested scheme are qualitatively and quantitatively compared with the staggered central (NT)and kinetic flux-vector splitting (KFVS) schemes. The accuracy, efficiency and simplicity of the methoddemonstrate its potential to solve radiation hydrodynamical models.
Radiation hydrodynamical (RHD) models have wide rangeapplications in different scientific and engineering disciplines[1–4]. The application areas include high-temperature hydrody-namics, modeling of gaseous stars in astrophysics, accretion disks,radiatively driven outflows, supernovas, laser fusion physics, com-bustion phenomena, reentry vehicles fusion physics, stellar con-vection and inertial confinement fusion [5,6]. The branch of fluidmechanics that deals with the study of moving fluid and changesin its state under diverse circumstances (e.g. internal and externalforces) is called hydrodynamics. The absorption or emission of ra-diations through matter produces heating and cooling in a system,respectively. The existence of considerable radiation transport in-timate the presence of temperature gradients and energy density,indicating the existence of pressure gradients as well. When suf-ficient time is available, pressure gradients generate considerableflowof fluid (radiation hydrodynamics) and changes in the density.
In zero diffusion limit, the radiation hydrodynamical equations(RHEs) could form a system of hyperbolic conservation laws.However, they are different from the inviscid compressible Euler
∗ Corresponding author at: Max Planck Institute for Dynamics of ComplexTechnical Systems, Magdeburg, Germany. Tel.: +92 51 9235952; fax: +92 519247006.
equations of gas dynamics. One of themajor difficulties of standardnumerical methods for solving RHEs is to resolve and determinethe exact path of strong shocks. Some pioneering work on theproblems of radiation hydrodynamics can be found in [6–10].Dai and Woodward [3] proposed the Godunov scheme includinglinear and nonlinear Riemann solvers for the solution of the RHEs.The numerical results showed that their methods have kept thekey features of Godunov schemes. However, they were foundto be computationally expensive. For astrophysical problems,the method proposed in [11] was found to be more successfulone where operator splitting technique was combined with theCrank–Nicolson method. Further work on this algorithm andseveral advantages and disadvantages of Godunov type schemesfor solving RHEs are exploited in [2,10,12–18]. Alhumaizi [19]compared different numerical schemes for solving RHEs andshowed that flux-corrected transport (FCT), weighted essentiallynon-oscillatory (WENO) scheme and monotone upstream schemefor conservation laws (MUSCL) are accurate for various parametersof radiation hydrodynamics. Moreover, the work of Tang and Wu[4] on radiation hydrodynamics is a remarkable addition to theapplications of kinetic flux-vector splitting (KFVS) methods. In thisarticle, the same KFVS scheme is used for the validation of ournumerical results.
Central schemes are the simple ones in the sense that theyavoid the eigenstructure of the problem. These schemes havebeen successfully applied in various areas including fluid me-chanics, astrophysics, metrology, semiconductors, shallow flowand multicomponent flows [20,21]. The first-order Lax–Friedrichs
1350 S. Qamar, W. Ashraf / Computer Physics Communications 184 (2013) 1349–1363
Fig. 1. Problem 1: The central upwind scheme results for different values of K on 50 grid points at t = 0.2.
scheme is the forerunner for such central schemes. The centralNessyahu–Tadmor (NT) scheme [22] offers higher resolutionwhileretaining the simplicity of the Riemann-solver-free approach. Thenumerical viscosity present in the central scheme is of the or-der O(∆x2r/∆t). In the convective regime where ∆t ∼ ∆x, theimproved resolution of the NT scheme and its generalizations isachieved by lowering the amount of numerical viscosity with in-creasing r . At the same time, this family of central schemes suffersfrom excessive numerical viscosity when a sufficiently small timestep is enforced, e.g., due to the presence of degenerate diffusionterms. Kurganov and Tadmor [23,24] improved these schemes byusing the correct information of local propagation speeds and ob-tained the semi-discrete central upwind scheme. Analogously tothe staggered non-oscillatory central (NT) scheme, it enjoys thebenefits of high resolution, simplicity and robustness. However,the central upwind scheme reduces large amount of numerical dis-sipation present in the NT central scheme.
The KFVS scheme is based on the splitting of macroscopic fluxfunctions of the system of equations of the RHDmodel [4]. The up-winding bias in numerical flux function can be naturally obtainedby considering fluid as a collection of particles. The movements ofparticles in the forward or backward directions automatically splitthe fluxes of mass, momentum and energy into forward and back-ward fluxes across the cell interface. In this scheme, we start with acell averaged initial data of conservative variables and get back thecell averaged values of the conservative variables in the same cellat the next time step. In the two-dimensional case, the flux splittingis done in a usual dimensionally split manner, that is, formula for
the fluxes can be used along each coordinate direction. In order toget second-order accuracy, the MUSCL-type initial reconstructionand the Runge–Kutta time stepping method are employed.
In this manuscript, the central upwind scheme is proposed forsolving the system of RHEs. The NT central scheme is also appliedfor the first time to such models. Moreover, the results of theKFVS scheme are also presented for validation and comparison [4].Several numerical case studies are considered in one and twospace dimensions. It was found that central upwind scheme givescomparable solutions to the KFVS scheme.
This article is organized as follows. In Section 2, the one-dimensional RHD model is introduced. Afterwards, the one-dimensional central upwind scheme is derived for the numericalapproximation of these equations. We also present the one-dimensional staggered central (NT) scheme very briefly. In Sec-tion 3, the RHD model and the corresponding numerical schemesare extended to the two-dimensional case. In Section 4, numericaltest problems are presented. Finally, Section 5 gives conclusionsand remarks.
2. One-dimensional RHDmodel
In this section, the one-dimensional RHD model is presented.Afterwards, the model equations are expressed in analogous formof the compressible Euler equations. A radiatively opaque gas ofsame radiative and fluid temperatures is considered and the meanfree path of photons is assumed narrower compared to the lengthof flow. The one-dimensional radiation hydrodynamical model is
S. Qamar, W. Ashraf / Computer Physics Communications 184 (2013) 1349–1363 1351
Fig. 2. Problem 1: L1-error plots of density for different values of K .
given as [1–4]
∂ρ
∂t+
∂(ρu)∂x
= 0, (1)
∂(ρu)∂t
+∂ρu2
+ p +13arT
4
∂x= 0, (2)
∂(E + arT 4)
∂t+
∂uE + p +
43arT
4
∂x= ∇ · [κ(T )∇T ], (3)
where ρ, p, uT , 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 +
12ρu
2, e is thespecific internal energy and κ(T ) is the nonlinear heat diffusivity.Note that, κ(T ) includes two distinct physical effects. One is theordinary (i.e. non-radiative) thermal conductivity and the otherone is radiation diffusion which is defined as 4arcλrT 3/3. Here, cand λr denote the speed of light and the Rosseland mean free pathfor photons. The above equation can be re-written as
∂ρ
∂t+
∂(ρu)∂x
= 0, (4)
∂(ρu)∂t
+∂(ρu2
+ p∗)
∂x= 0, (5)
∂E∗
∂t+
∂[u(E∗+ p∗)]
∂x= ∇ · [κ(T )∇T ], (6)
where p∗= p +
13arT
4 and E∗= E + arT 4. Here, e = T
being the specific internal energy. In the above model κ(T ) = 0
corresponds to a restrictive case where photon diffusion andenergy exchange driven by differences in temperature between thegas and the radiation field are negligible in comparison to radiationwork and advection of radiation. This is known as the dynamicdiffusion limit [6]. Furthermore, the above system reduces to theinviscid Euler equations of gas dynamics if κ = 0 and ar = 0. Theeigenvalues have a similar structure to the Euler equations, i.e.
λ1 = u − c∗, λ2 = u, λ3 = u + c∗, (7)
where
c∗=
γ ∗p∗
ρ, (8)
is the speed of sound in the radiation hydrodynamics case. Letµ =
13ar T4p and Γ = γ − 1, then
γ ∗=
γ + [4Γ (2 − 3Γ ) + 12γΓ ]µ + 16Γ µ2
(1 + µ)(1 + 12Γ µ)
=
γ
γ−1 + 20µ + 16µ2
1
γ−1 + 12µ
(1 + µ). (9)
Note that, the sound speed can also be re-written as
(c∗)2 = c2 + 4µpρ
Γ (2 − 3Γ ) + 4µΓ
1 + 12µΓ,
1352 S. Qamar, W. Ashraf / Computer Physics Communications 184 (2013) 1349–1363
Fig. 3. Problem 1: L1-error plots of density for different values of K .
where, c =
γ pρ. If ar = 0 (i.e. if µ = 0) then c∗
= c. Toexpress the temperature in terms of conserved variables, we canwrite E = ρT +
12ρu
2+ aRT 4 as
T 4+ βT + η = 0, (10)
where β =ρ
aRand η = −
1aR
(E −12ρu
2). Let p = −η and q = −β2
8then
y =
−
q2
+
q2
4+
p3
27
13
−
q2
+
q2
4+
p3
27
13
. (11)
The quartic equation (10) has four roots fromwhich the physicallyacceptable root is
T =12
−2y +
−2y +
2β√2y
. (12)
After calculating T , the pressure can be obtained from p = (γ −
1)ρT .
2.1. One-dimensional central upwind scheme
In this section, the semi-discrete central-upwind scheme isderived [24]. The above radiation hydrodynamics model can beviewed as a system of convection–diffusion radiation system of theformWt + F(W)x = Rx, (13)
and R = (0, 0, κ(T )Tx)T . The radiation effect is incorporated inconvective term with eigenvalues given by Eq. (7).Before applying the scheme, it is required to discretize thecomputational domain. Let N be the number of discretizationpoints and (xi− 1
2)i∈1,...,N+1 are partitions of the given interval
[0, xmax]. For each i = 1, 2, . . . ,N , ∆x is a constant width of eachmesh interval, xi denote the cell centers, and xi± 1
2refer to the cell
boundaries. We assign,
x1/2 = 0, xN+1/2 = xmax, xi+1/2 = i · ∆x,
for i = 1, 2, . . . ,N. (14)
Moreover,
xi = (xi−1/2 + xi+1/2)/2 and
∆x = xi+1/2 − xi−1/2 =xmax
N + 1.
(15)
Let Ωi :=xi−1/2, xi+1/2
for i ≥ 1. In each Ωi, the averaged values
of the conservative variableW(t) are given as
Wi := Wi(t) =1
∆x
Ωi
W(x, t) dx. (16)
By integrating Eq. (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. (17)
S. Qamar, W. Ashraf / Computer Physics Communications 184 (2013) 1349–1363 1353
Fig. 4. Problem 1: L1-error plots of density for different values of K .
The numerical fluxes are defined as
Si+ 12
=
F(W+
i+ 12) + F(W−
i+ 12)
2−
ai+ 12
2
W+
i+ 12
− W−
i+ 12
, (18)
and for R = (R1, R2, R3)T , we have
(Rk)i± 12
= 0, for k = 1, 2, (19)
(R3)i± 12
=
κ(T−
i± 12) + κ(T+
i± 12)
2·
T+
i± 12
− T−
i± 12
∆x. (20)
Here, W+ and W− are the point values of the piecewise linearreconstruction W = (ρ, ρu, E∗) for W, namely:
W+
i+ 12
= Wi+1 −∆x2
Wxi+1, W−
i+ 12
= Wi +∆x2
Wxi . (21)
The numerical derivatives Wxi are at least first-order approxima-
tions of Wx(xi, t) and are computed using a nonlinear limiter thatwould ensure a non-oscillatory nature of the reconstruction (21).A possible computation of these slopes is given by the family ofdiscrete derivatives parameterized with 1 ≤ θ ≤ 2, for example
Wxi = MM
θ∆Wi+ 1
2,θ
2
∆Wi+ 1
2+ ∆Wi− 1
2
, θ∆Wi− 1
2
, (22)
where 1 ≤ θ ≤ 2 is a parameter and ∆ denotes centraldifferencing,
∆Wi+ 12
= Wi+1 − Wi.
Here,MM denotes the min-mod nonlinear limiter
MMx1, x2, . . . =
min
ixi if xi > 0∀i,
maxi
xi if xi < 0∀i,
0 otherwise.
(23)
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))
. (24)
To obtain the second order accuracy in time, we use a second orderTVD Runge–Kutta scheme to solve Eq. (17). Denoting the right-hand side of Eq. (17) as L(W), a second order TVD Runge–Kuttascheme updateW through the following two stages
W(1)= Wn
+ ∆tL(Wn), (25a)
Wn+1=
12
Wn
+ W(1)+ ∆t L(W(1))
, (25b)
where Wn is a solution at previous time step tn and Wn+1 isupdated solution at next time step tn+1. Moreover, ∆t repre-sents the time step which is calculated under the followingCourant–Friedrichs–Lewy (CFL) condition
∆t ≤ 0.5min
∆xmax(|u| + c∗)
,∆x2
max(2κ(T ))
, (26)
where c∗ is given by Eq. (8). The same CFL condition is also used forthe other considered schemes.
1354 S. Qamar, W. Ashraf / Computer Physics Communications 184 (2013) 1349–1363
Fig. 5. Problem 1: Results on 400 mesh cells at t = 0.2 for κ(T ) = 0.
2.2. One-dimensional central schemes
Her, the high-resolution non-oscillatory central scheme ofNessyahu and Tadmor [22] is briefly presented. These predictor–corrector type methods are applied in two steps. In the predictorstep, the midpoint values are predicted by using the non-oscillatory piecewise-linear reconstructions of the cell averages. Inthe second corrector step, staggered averaging, together with thepredictedmid-values, are used to obtain the updated cell averagedsolution. In summary, the scheme can be presented as
Predictor: Wn+ 1
2i = Wn
i −ξ
2Fx(Wn
i ), (27)
Corrector: Wn+1i+ 1
2=
12(Wn
i + Wni+1) +
18(Wx
i − Wxi+1)
− ξ
Fn+ 1
2i+1 − F
n+ 12
i
+ ξ
Rn+ 1
2i+1 − R
n+ 12
i
, (28)
where, ξ = ∆t/∆x. Moreover, 1∆xF
x(Wi) stands for an approxi-mate numerical derivatives of the flux F(t, x = xi)
1∆x
Fx(Wi) =∂
∂xF(w(t, x = xi)) + O(∆x). (29)
The fluxes Fx(Wi) are computed by the same manner as discussedfor Wx in Eq. (22).
3. Two-dimensional RHDmodel
In this section, we describe the RHEs in two space dimensionsand the corresponding Euler-like form. The model can be written
as [25,26]
∂ρ
∂t+
∂(ρu)∂x
+∂(ρv)
∂y= 0, (30)
∂(ρu)∂t
+∂ρu2
+ p +13arT
4
∂x+
∂(ρuv)
∂y= 0, (31)
∂(ρv)
∂t+
∂(ρuv)
∂x+
∂ρv2
+ p +13arT
4
∂y= 0, (32)
∂(E + arT 4)
∂t+
∂uE + p +
43arT
4
∂x+
∂vE + p +
43arT
4
∂y
= ∇ · [κ(T )∇T ], (33)where p = (γ − 1)ρe, E = ρe+
12ρ(u2
+ v2) and u and v are fluidvelocities in x and y-directions. The above system can be re-writtenas∂ρ
∂t+
∂(ρu)∂x
+∂(ρv)
∂y= 0, (34)
∂(ρu)∂t
+∂(ρu2
+ p∗)
∂x+
∂(ρuv)
∂y= 0, (35)
∂(ρv)
∂t+
∂(ρuv)
∂x+
∂(ρv2+ p∗)
∂y= 0, (36)
∂(E∗)
∂t+
∂[u(E∗+ p∗)]
∂x+
∂[v(E∗+ p∗)]
∂y= ∇ · [κ(T )∇T ], (37)
where p∗= p +
13arT
4 and E∗= E + arT 4.
S. Qamar, W. Ashraf / Computer Physics Communications 184 (2013) 1349–1363 1355
Fig. 6. Problem 2: Results on 400 mesh cells at t = 0.04.
3.1. Central upwind scheme
Here, the semi-discrete central upwind scheme is presented intwo space dimensions [24]. The above model can be expressed as:
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. (7).Let Nx and Ny be the large integers in x and y-directions, respec-tively. We assume a Cartesian grid with a rectangular domain[x0, xmax] × [y0, ymax] 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
The cell averaged values ofWi,j(t) at any time t are given as
Wi,j := Wi,j(t) =1
∆xi∆yj
Cij
W(x, y, t) dydx. (41)
Now construct a piecewise linear interplant
W(x, y, t) =
i,j
Wi,j + (Wx)i,j(x − xi)
+ (Wy)i,j(y − yj)χi,j, (42)
where χi,j is the characteristic function for the corresponding cell(xi− 1
2, xi+ 1
2) × (yj− 1
2, yj+ 1
2), (Wx)i,j and (Wy)i,j are the approxima-
tions of x and y-derivatives ofW at the cell centers (xi, yj). The gen-eralized MM limiter is used for the computation of these partialderivatives 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
, (43)
where 1 ≤ θ ≤ 2 and MM is given by Eq. (22). After integratingEq. (38) over the control volume Cij, the two-dimensional exten-sion of the scheme can be scripted as
dWi,j
dt= −
Sxi+ 1
2 ,j− Sx
i− 12 ,j
∆x−
Syi,j+ 1
2− Sy
i,j− 12
∆y
+
Rxi+ 1
2 ,j− Rx
i− 12 ,j
∆x+
Ryi,j+ 1
2− Ry
i,j− 12
∆y. (44)
1356 S. Qamar, W. Ashraf / Computer Physics Communications 184 (2013) 1349–1363
Fig. 7. Problem 2: L1-error plots at κ(T ) = 0.
Here
Sxi+ 1
2 ,j=
F(W−
i+ 12 ,j
) + F(W+
i+ 12 ,j
)
2−
axi+ 1
2 ,j
2
W+
i+ 12 ,j
− W−
i+ 12 ,j
,
Syi,j+ 1
2=
G(W−
i,j+ 12) + G(W+
i,j+ 12)
2
−
ayi,j+ 1
2
2
W+
i,j+ 12
− W−
i,j+ 12
. (45)
Similarly, Rxk = 0 = Ry
k for k = 1, 2, 3 and
(Rx4)i± 1
2 ,j =
κ(T−
i± 12 ,j
) + κ(T+
i± 12 ,j
)
2·
T+
i± 12 ,j
− T−
i± 12 ,j
∆x,
(Ry4)i,j± 1
2=
κ(T−
i,j± 12) + κ(T+
i,j± 12)
2·
T+
i,j± 12
− T−
i,j± 12
∆y. (46)
The intermediate values are expressed as
W−
i+ 12 ,j
= Wi,j +∆x2
(Wx)i,j, W+
i+ 12 ,j
= Wi+1,j −∆x2
(Wx)i+1,j
W−
i,j+ 12
= Wi,j +∆y2
(Wy)i,j,
W+
i,j+ 12
= Wi,j+1 −∆y2
(Wy)i,j+1. (47)
Here, axi+ 1
2 ,jand ay
i,j+ 12are the local speeds which can be calculated
as
axi+ 1
2 ,j= max
±ρ
∂F∂W
(W±
i+ 12 ,j
)
,
ayi,j+ 1
2= max
±ρ
∂G∂W
(W±
i,j+ 12)
.
(48)
For complete derivation of the scheme the reader is referredto [24].
3.2. Two-dimensional central scheme
The two-dimensional central scheme was proposed by Jaingand Tadmor [27]. The scheme has again a two-step predic-tor–corrector form. Starting with the cell averages, Wn
i,j, we usethe first-order predictor step for the evolution of themidpoint val-
ues, Wn+ 1
2i,j , followed by the second-order corrector step for com-
putation 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 recon-structed discrete slopes, Wx, Wy, Fx(W), and Gy(W). At each timestep the grid is staggered to avoid the flux calculation at the cellinterfaces. The scheme is summarized below.In the predictor step one has to calculate the midpoint values
Wn+ 1
2i,j = Wn
i,j −ξ
2Fx(Wn
i,j) −η
2Gy(Wn
i,j), (49)
S. Qamar, W. Ashraf / Computer Physics Communications 184 (2013) 1349–1363 1357
Fig. 8. Problem 3: Results on 400 mesh cells at t = 0.02.
where, ξ = ∆t/∆x and η = ∆t/∆y. This step is followed by acorrector step to get the updated values at the next time step
Wn+1i+ 1
2 ,j+ 12
=14(Wn
i,j + Wni+1,j + Wn
i,j+1 + Wni+1,j+1)
+116
(Wxi,j − Wx
i+1,j) −ξ
2
Fn+ 1
2i+1,j − F
n+ 12
i,j
+
116
(Wxi,j+1 − Wx
i+1,j+1) −ξ
2
Fn+ 1
2i+1,j+1 − F
n+ 12
i,j+1
+
116
(Wyi,j − Wy
i,j+1) −η
2
Gn+ 1
2i,j+1 − G
n+ 12
i,j
+
116
(Wyi+1,j − Wy
i+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
. (50)
This completes the derivation of numerical schemes.
4. Numerical case studies
In this section, six test problems are considered to validate theaccuracy and performance of the proposed central schemes. Forcomparison, the KFVS scheme is also applied to this model [4].
Problem 1. This is a one-dimensional shock-tube problem involv-ing two rarefaction waves moving in the opposite directions [25].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.
(51)
The computational domain is [0, 1] which is subdivided into 50cells and the final simulation time is t = 0.2. The nonlinear heatdiffusivity is taken as κ(T ) = K(1+10T 3), where K is a scaling fac-tor. To analyze the performance of numerical schemes for transportand diffusion dominated cases, different values of K are consideredfrom the interval [0, 0.5]. Fig. 1 show the numerical results of cen-tral upwind scheme for K = 0, 10−3, 10−2, 10−1, 0.5. One canobserve that for K ≥ 10−2 the effect of diffusion term becomesvisible. Tables 1–4 show the L1-errors between the reference andnumerical solutions of the central and KFVS schemes for differentvalues of the scaling factor K . The reference solution was obtainedat 2000mesh points. The error plots of density, pressure and veloc-ity for different values of the scaling factor are given in Figs. 2–4.Moreover, Fig. 5 shows the comparison of results for K = 0. It canbe observed that the results of central-upwind scheme are in goodagreement with KFVS and NT central schemes. All figures and ta-bles show that the central upwind and the KFVS schemes give com-parable results for thewhole range of K while the staggered central(NT) scheme produces large errors in the solutions. In overall, theKFVS scheme has little edge over the central schemes. From theseresults it can be concluded that the proposed schemes have uni-form behavior over the whole range of scaling factor K . Moreover,
1358 S. Qamar, W. Ashraf / Computer Physics Communications 184 (2013) 1349–1363
Fig. 9. Problem 4: Results on 400 mesh cells at t = 0.18.
density, ρ temperature, T
Pressure, p velocity, u
1
0.5
0 0.5 1 1.5 2x–axis
0 0.5 1 1.5 2x–axis
0 0.5 1 1.5 2x–axis
0 0.5 1 1.5 2x–axis
y–ax
is
1
0.5
y–ax
is
1
0.5
y–ax
is
1
0.5
y–ax
is
Fig. 10. Problem 5: 2D results on 256 × 128 mesh cells at t = 0.6, κ(T ) = 0.
minor changes can be seen in the magnitudes of errors over thewhole range of scaling factor K . Thus, the considered numerical
schemes behave uniformly in both transport and diffusion domi-nated limits.
S. Qamar, W. Ashraf / Computer Physics Communications 184 (2013) 1349–1363 1359
density, ρ temperature, T
Pressure, p velocity, u
1
0.5
0
0
0.5 1 1.5 2x–axis
00.5 1 1.5 2
x–axis
0 0.5 1 1.5 2x–axis
0.5 1 1.5 2x–axis
y–ax
is
1
0.5
y–ax
is
1
0.5
y–ax
is
1
0.5
y–ax
isFig. 11. Problem 5: 2D results on 256 × 128 mesh cells at t = 0.6, κ(T ) = 10−3(1 + 10T 3).
Table 1Problem 1: Comparison of numerical errors at different grid points for K = 10−3 .
1360 S. Qamar, W. Ashraf / Computer Physics Communications 184 (2013) 1349–1363
Fig. 12. Problem 5: 1D results along x = 0.25 at t = 0.6, κ(T ) = 0.
Problem 2. This one dimensional shock-tube problemwas consid-ered in [25]. The initial data are given as
(ρ, T , u) =
(1, 0.5, 50), x ≤ 0.6,(2, 1, −40) x ≥ 0.6. (52)
The computational domain is [0, 1] which is partitioned into 400grid cells. The simulation results at t = 0.04 for the density, veloc-ity, pressure and temperature are shown in Fig. 6. Two shocks ofMach 82 and 39 and a contact discontinuity are produced from theinitial discontinuities. The numerical results of central upwind andKFVS scheme are better than the staggered central (NT) scheme.The central upwind scheme seems to be superior in this case. To
Fig. 13. Problem 5: 1D results along y = 0.5 at t = 0.6, κ(T ) = 0.
quantitatively analyze the accuracy of proposed scheme, we havecalculated the L1-errors at different grid points given in Table 5. Thereference solutionwas obtained at 2000 grid points. It is clear fromthe results that the central-upwind scheme produces less errorsin the solution as compared to the KFVS and NT central schemes.Moreover, Fig. 7 shows the plots of L1-errors which justify the re-sults of Table 5.
Problem 3. This problem is given by Jiang and Sun [25] that in-volves a strong shock in one-dimensional Riemann problem. Ini-tially the values of (ρ, T , u) for x0 ≤ 0.5 are (1, 0.5, 150) and for
S. Qamar, W. Ashraf / Computer Physics Communications 184 (2013) 1349–1363 1361
1
0.5
y–ax
is
1
0.5
y–ax
is
1
0.5
y–ax
is
1
0.5
y–ax
is
0 0.5 1 1.5 2x–axis
0 0.5 1 1.5 2x–axis
0 0.5 1 1.5 2x–axis
0 0.5 1 1.5 2x–axis
1
0.5
y–ax
is
1
0.5
y–ax
is
0 0.5 1 1.5 2x–axis
0 0.5 1 1.5 2x–axis
density, ρ at t=0.25 temperature, T at t=0.25
temperature, T at 0.5
temperature, T at t=0.75
density, ρ at t=0.5
density, ρ at t=0.75
Fig. 14. Problem 5: Density and temperature contour at different time steps, κ(T ) = 0.
Fig. 15. Problem 5: 1D results along y = 0.55 at t = 0.5, κ(T ) = 0.
x0 > 0.5 are (2, 1, −100). The number of cells are 400 and thecomputational domain is [0, 1]. The simulation results at t = 0.018for the density, velocity, pressure and temperature are shown inFig. 8. The solution consists of two shocks with Mach numbers ofabout 227.4 and 107.7 and a constant discontinuity. Once again theresults of KFVS and central upwind scheme are comparable andtheNT scheme gives diffusive results. However, the central upwindscheme is superior in all three schemes.Problem 4. We consider another test problem where the initialdata (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, pres-sure and temperature are shown in Fig. 9. The solution consists oftwo shocks with Mach numbers of about 227.4 and 107.7 and aconstant discontinuity. Once again the results of KFVS and centralupwind schemes are comparable andNT scheme gives diffusive re-sults. However, the central upwind scheme is superior in all threeschemes.Problem 5. This is a two-dimensional problem describing the in-teraction of a wind and a denser circular cloud. In the rectangular
1362 S. Qamar, W. Ashraf / Computer Physics Communications 184 (2013) 1349–1363
Fig. 16. Problem 5: 1D results along x = 0.30 at t = 0.5, κ(T ) = 0.
Fig. 17. Problem 6: 2D results on 128 × 128 mesh cells at t = 0.5, κ(T ) = 0.
domain [0, 2]×[0, 1], there is a 25 times denser cylindrical bubbleat (0.3, 0.5) whose radius is r = 0.15 and the number of grids are256×128. 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 introducedthrough the left boundary and ar = 1. The outflow boundaryconditions are applied at the right, lower and upper boundariesof the domain. We computed the solution without and with dif-fusion limit, where, κ(T ) = 10−3(1 + 10T 3) [3]. It is observedthat the incoming shock produces shock in the bubble and a re-
fracted shock. The contours for ρ, T and u at t = 0.6 are shownin Figs. 10–14. The comparisons of results in Figs. 15 and 16 showgood agreements between the proposed schemes and the resultsavailable in the literature. It can be observed that KFVS and centralupwind scheme produce comparable solutions and the NT schemeis diffusive. However, The KFVS scheme produces a more resolvedsolution (see Figs. 15 and 16).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 the
S. Qamar, W. Ashraf / Computer Physics Communications 184 (2013) 1349–1363 1363
Fig. 18. Problem 6: 1D results along y = 1.0 at t = 0.5, κ(T ) = 0.
box. The values for (ρ, T , u, v) are (1, 0.9, 0, 0) inside the circleand (25, 0.9, 0, 0) outside the circle. The simulation results areobtained at t = 0.5 on 128× 128 grid points. Figs. 17 and 18 showthe contour plots and comparison among the central-upwind, NT
central and KFVS schemes. Once again, a good performance of thecentral-upwind scheme canbe seen.However, KFVS scheme showsbetter performance in all three schemes.
5. Conclusions
We focused on the numerical solution of one- and two-dimensional radiation hydrodynamical equations (RHEs). Twodifferent types of central finite volume schemes were appliedto solve these equations, the central-upwind and the staggeredcentral schemes. The proposed numerical schemes preservesmonotonicity due to using MUSCL-type reconstruction and alsoavoid the detailed knowledge of a complicated exact/approximateRiemann solver. For validation, the KFVS scheme was also appliedto solve this model. A number of case studies were carried outand the accuracies of the schemes were analyzed quantitativelyand qualitatively. It was found that central-upwind schemes havebetter resolved discontinuous profiles as compared to the NTschemes. Further, it was concluded that the proposed numericalscheme gives comparable results with the KFVS scheme at lowCPU time. The suggestedmethod gives comparable accuracy to thescheme that uses Riemann solvers, e.g. Sekora and Stone [2], andis computationally less expensive due to its Riemann-solver freealgorithm.
Acknowledgment
A partial support by the Higher Education Commission (HEC) ofPakistan is gratefully acknowledged.
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