Contents. . . • Welcome, Bem-vindo! • SHARK-FV week • Tentative schedule p.2 • Themes p.2 • Participants p.2 • Abstracts p.5-42 • Hotel and restaurant p.44 • Organisation p.44 Need for help? Just ask! • Stéphane Clain • Gaspar Machado • Jorge Figueiredo • Rui Pereira • Raphaël Loubère [email protected]Make sure you can leave! Please take 1 minute during your stay to make sure that you are booked for a journey on a shuttle back to the airport to take your plane. Check with the locals. Hotel contact information: Hotel Parque do Rio, Caminho Padre Manuel de Sá Pereira, 4741-908 Fão, Portugal. +351 253 981 521 GPS: 8 47.097 W, 41 31.192 N Welcome, Bem-vindo! Welcome to the fourth experimental Sharing Higher-order Advanced Research Know-how on Finite Volume conference, SHARK-FV 2017 May 15th-19th. The main purpose of this conference is to strengthen the collaborations be- tween Finite Volume (FV) field actors and to share the burden of research and development of numerical codes. This workshop brings the opportunity for Researchers from International Universities and National Laboratories to dis- cuss the State-of-the-Art of high(er)-order Finite Volume methods for a large range of Physics and Engineering problems. The purposes of the workshops are threefold • to reinforce already existing collaborations; • to create new interactions between researchers in the same field; • to share detailed and technical experiences on specific issues and ex- change ideas, numerical codes, test cases... The area of Esposende between the ocean and Cavado river, the Hotel sur- rounded by luxurious calm gardens and pine-woods, the famous Portuguese cuisine, provide a perfect incubator for Exceptional Science! SHARK-FV week Your week will be organized as follows: Morning presentations are proposed followed by afternoon working sessions which are dedicated to intensive parallel workshops involving few participants. We expect a lot of time to be dedicated to discussions and small group working sessions, hence the few number of participants. We all should leave the SHARK workshop with new collaborators, new projects, advanced ones and a scientific momentum to attain the next workshop. 1
Please take 1 minute during yourstay to make sure that you arebooked for a journey on a shuttleback to the airport to take yourplane. Check with the locals.
Hotel contact information:Hotel Parque do Rio,Caminho Padre Manuel de SáPereira, 4741-908 Fão, Portugal.+351 253 981 521GPS: 8 47.097 W, 41 31.192 N
Welcome, Bem-vindo!
Welcome to the fourth experimental Sharing Higher-order Advanced ResearchKnow-how on Finite Volume conference, SHARK-FV 2017 May 15th-19th.The main purpose of this conference is to strengthen the collaborations be-tween Finite Volume (FV) field actors and to share the burden of research anddevelopment of numerical codes. This workshop brings the opportunity forResearchers from International Universities and National Laboratories to dis-cuss the State-of-the-Art of high(er)-order Finite Volume methods for a largerange of Physics and Engineering problems. The purposes of the workshopsare threefold
• to reinforce already existing collaborations;• to create new interactions between researchers in the same field;• to share detailed and technical experiences on specific issues and ex-
change ideas, numerical codes, test cases...
The area of Esposende between the ocean and Cavado river, the Hotel sur-rounded by luxurious calm gardens and pine-woods, the famous Portuguesecuisine, provide a perfect incubator for Exceptional Science!
SHARK-FV week
Your week will be organized as follows: Morningpresentations are proposed followed by afternoonworking sessions which are dedicated to intensiveparallel workshops involving few participants.We expect a lot of time to be dedicated to discussionsand small group working sessions, hence the fewnumber of participants.
We all should leave the SHARK workshop with new collaborators, new projects,advanced ones and a scientific momentum to attain the next workshop.
The talks are scheduled to last for 45 minutes with 5 minutes of questions/answers. Abstracts are proposed in pages5-42. After lunch, thematic discussion sessions are organized for interested people, the goal being to exchange andshare news, raise new problems and create a positive alchemy between the participants.Ideally such alchemy must generate bombastic discussions to feed the following work sessions, or free time.Afternoons are left free for each participant to construct his ideal and optimal schedule for the week. Be careful thatsome people may leave on Friday morning, so plan in advance your discussions and working sessions!
Common themes
Finite volume methods for Shallow Water equation
Reconstruction for finite volume methods
Discontinuous Galerkin schemes
Numerical Analysis of finite volume schemes
Finite volume and HPC (3D, NS, ...)
Finite volume for multi-physics and source terms
Finite volume on moving mesh
Low-Mach number and kinetic high-order methods
A posteriori MOOD stabilization
Modeling in geophysics
SPH particle scheme
AP schemes
Sponsors: The organizers acknowledge the financial supportof FEDER Funds through Programa Operacional Factores de
Competitividade — COMPETE and by Portuguese Funds throughFCT — Fundação para a Ciência e a Tecnologia, within the
Projects PTDC/MAT/121185/2010 andFCT-ANR/MAT-NAN/0122/2012 and ANR-12-IS01-0004
The next pages gather the abstracts of the talks given during SHARK-FV, here is a summary:
F. Vilar, page 5CELL-CENTERED LAGRANGIAN SCHEMES AND NODAL SOLVERS
G. Gassner, page 7ON THE CONSTRUCTION OF ENTROPY STABLE DG SCHEMES AND THEIR RELATION TO HIGH-ORDER FV
C. Escalante, page 8A NEW BILAYER SYSTEM WITH ENHANCED DISPERSIVE PROPERTIES.
I. Peshkov, page 11SYMMETRIC HYPERBOLIC MODELS AND ADER SCHEMES FOR MULTI- PHYSICS SIMULATIONS
F. Fambri, page 13SPECTRAL SEMI-IMPLICIT AND SPACE-TIME DISCONTINUOUS GALERKIN METHODS FOR THE INCOMPRESSIBLENAVIER-STOKES EQUATIONS ON STAGGERED ADAPTIVE CARTESIAN GRIDS
C. Fiorini, page 15HIGH-ORDER NUMERICAL SCHEME FOR SENSITIVITY ANALYSIS FOR EULER EQUATIONS
C.-D. Munz, page 17TBA
J.C. González-Aguire, page 19NUMERICAL SIMULATION OF BED LOAD AND SUSPEND LOAD TRANSPORT
T. Rey, page 23PROJECTIVE INTEGRATION FOR THE NONLINEAR BGK AND BOLTZ- MANN EQUATIONS
C. Berthon, page 25WELL-BALANCED SCHEMES FOR SHALLOW-WATER MODELS WITH STRONGLY NON-LINEAR SOURCE TERMS
X. Nogueira, page 27NEW APPLICATIONS OF THE SPH-ALE-MLS-MOOD METHOD
G.J. Machado, page 29ONE DIMENSIONAL STEADY-STATE EULER SYSTEM
M. Stauffert, page 31ON ALL-REGIME, HIGH-ORDER AND WELL-BALANCED LAGRANGE- PROJECTION TYPE SCHEMES FOR THESHALLOW WATER EQUATIONS
M.E. Vázquez-Cendón, page 33FINITE VOLUME METHODS FOR MULTI-COMPONENT EULER EQUA- TIONS WITH SOURCE TERMS IN NETWORKS
S. Busto, page 35A HIGH ORDER FV/FE PROJECTION METHOD FOR COMPRESSIBLE LOW-MACH NUMBER FLOWS
V. Michel-Dansac, page 37ASYMPTOTICALLY ACCURATE HIGH-ORDER SPACE AND TIME SCHEMES FOR THE EULER SYSTEM IN THE LOWMACH REGIME
W. Boscheri, page 39CWENO SCHEMES FOR CONSERVATION LAWS ON UNSTRUCTURED MESHES
A. Duran, page 41TBA
4
Purple SHARK-FV — May 15-19 2017 — Ofir, Portugal.
Sharing Higher-order Advanced Research Know-how on Finite Volumes
CELL-CENTERED LAGRANGIAN SCHEMES AND NODAL SOLVERS
François VilarInstitut Montpelliérain Alexander Grothendieck, Université de MontpellierPlace Eugène Bataillon, 34090 Montpellier, [email protected]
ABSTRACT
This talk aims at presenting the general framework of finite volumes cell-centered discretization throughthe design of multidimensional approximate Riemann solvers. Numerical results will be presented todepict the very high performance of the numerical schemes developed these past years, refer for instanceto [1, 2, 3, 4], and to Figure 1 and Figure 2.
KEYWORDS: Lagrangian gas dynamics, Multidimensional nodal solvers, Moving grids
This presentation is dedicated to the introduction of cell-centered Lagrangian schemes. It is well known thatfluid dynamics, and more generally continuum mechanics, relies on two kinematics descriptions: the Eulerianor spatial description and the Lagrangian or material description. In the former, the conservation laws arewritten using a fixed reference frame whereas in the latter they are written through the use of a time dependentreference frame that follows the fluid motion. Consequently, the referential will move and get deformed asthe fluid flows. Such representation is particularly well adapted to describe the time evolution of fluid flowscontained in regions undergoing large shape changes due to strong compressions or expansions. Further, in thisapproach, there is no mass flux across the boundary surface of a control volume moving with the fluid velocity.This representation thus provides a natural framework to track accurately material interfaces in multi-materialcompressible flows. Moreover, it avoids the appearance of numerical diffusion resulting from the discretizationof convection terms present in the Eulerian framework.Naturally, to work in such formalism one has to move the mesh, but in respect with some properties as thegeometric conservation law for instance. In other words, the question is how to define the velocity of gridnodes. This is the cornerstone of any Lagrangian scheme. And because this question is far from being obvious,the staggered approach has been preferred for 50 years. In this approach, the kinematic variables (vertexposition, velocity) are located at the nodes whereas the thermodynamic variables (density, pressure, internalenergy) are defined at the cell centers. The staggered grid schemes employed in most hydro-codes have beenremarkably successful over the past decades in solving complex multi-dimensional compressible fluid flows.However, they clearly have some theoretical and practical deficiencies such as mesh imprinting and symmetrybreaking. In addition, the fact that all variables are not conserved over the same space can make these schemesdifficult to handle when one wants to assess analytical properties of the numerical solution.However, remarkable progress has been done in the direction of cell-centered discretizations lately, whichhave made them very popular now. In this approach, a cell-centered placement of all hydrodynamic variablesis employed, which make them easier to manipulate and to analyze. Nonetheless, the tricky part lies in thedefinition of the nodal solvers.This talk aims at presenting the general framework of finite volumes cell-centered discretization through thedesign of multidimensional approximate Riemann solvers. Numerical results will be presented to depict thevery high performance of the numerical schemes developed these past years, refer for instance to [1, 2, 3, 4],and to Figure 1 and Figure 2.
5
0 0.2 0.4 0.6 0.8 1 1.20
0.2
0.4
0.6
0.8
1
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
(a) Pressure field
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1 1.2 1.4
solution
2nd order
(b) Density profiles
FIGURE 1: Second-order numerical results for a Sedov problem on an initial 30×30 Cartesian grid.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(a) Third-order numerical scheme
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(b) Exact solution
FIGURE 2: Final grids for a Taylor-Green vortex problem on an initial 10×10 Cartesian grid.
REFERENCES
[1] F. VILAR, C.-W. SHU AND P.-H. MAIRE, Positivity-preserving cell-centered Lagrangian schemes formulti-material compressible flows: Form first-order to high-orders. Part II: The 2D case. Journal of Com-putational Physics, 312:416-442, 2016.
[2] F. VILAR, C.-W. SHU AND P.-H. MAIRE, Positivity-preserving cell-centered Lagrangian schemes formulti-material compressible flows: Form first-order to high-orders. Part I: The 1D case. Journal of Com-putational Physics, 312:385-415, 2016.
[3] F. VILAR, P.-H. MAIRE AND R. ABGRALL, A discontinuous Galerkin discretization for solving the two-dimensional gas dynamics equations written under total Lagrangian formulation on general unstructuredgrids. Journal of Computational Physics, 276:188-234, 2014.
[4] F. VILAR, Cell-centered discontinuous Galerkin discretization for two-dimensional Lagrangian hydrody-namics. Computers and Fluids, 64:64-73, 2012.
6
Purple SHARK-FV — May 15-19 2017 — Ofir, Portugal.
Sharing Higher-order Advanced Research Know-how on Finite Volumes
ON THE CONSTRUCTION OF ENTROPY STABLE DG SCHEMES ANDTHEIR RELATION TO HIGH-ORDER FV
Gregor Gassner ∗
Mathematical Institute, University of Cologne, Weyertal 86-90, 50354 Cologne, Germany
ABSTRACT
In this talk, recent developments on the construction of high order DG schemes that are discretelyconsistent with the second law of thermodynamics are presented. Discrete consistency with the secondlaw of thermodynamics translates to entropy stability. Entropy stability was mainly introduced andconsidered early on by Tadmor and many developments for many different numerical methods exist.There is also some work on entropy stability in the context of DG, however this research relies on an exactevaluation of the inner products. With this assumption, the assumption of continuity is implied for the dis-cretisation. Consequently, these approaches assume stability to show stability. That is why we emphasisthe discrete consistency to the second law of thermodynamics, as we do not rely on an exact evalua-tion of the inner products. In fact, we rely on a strong relationship of a certain DG variant to high order FV.
The considered special variant of the DG methodology is the collocation spectral element methodwith Legendre-Gauss-Lobatto nodes. Extension to three spatial dimensions and curvilinear domains ispossible, however only when tensor product approximations are considered.
The entropy stable DG scheme implies numerical robustness of the high order method and fullcontrol over the dissipation of the scheme. These properties are desirable and offer great potential. On thedownside, entropy stability increases the computational costs of the method. It is still ongoing researchabout the real benefits of this novel approach.
We will focus in this talk on the following questions: Why does this approach only work for onespecific DG variant? What is the relation to high order Finite Volume schemes? What are the mainbuilding blocks? Why do we care about entropy conservation? What about hanging nodes? Can we nowcompute shocks with the entropy stable DG? What about hyperbolic problems with non-conservativesource terms?
Great White SHARK-FV — May 15-19 2017 — Ofir, Portugal.
Sharing Higher-order Advanced Research Know-how on Finite Volumes
A NEW BILAYER SYSTEM WITH ENHANCED DISPERSIVE PROPERTIES.
M.J. Castro a, C. Escalante a∗, E.D. Fernández-Nieto c, T. Morales de Luna b, G. Narbona-Reina c
a Departamento de A.M., E. e I.O. y Matemática Aplicada, Universidad de Málaga, Campus de Teatinos - 29080 Málaga,España.b Departamento de Matemáticas, Universidad de Córdoba, España.c Departamento de Matemática Aplicada I, ETS Arquitectura, Universidad de Sevilla, España.
ABSTRACT
In this work we consider a new layer-averaged non-hydrostatic system with enhanced dispersive proper-ties, e.g. phase and group velocity, as well as linear shoaling compared with exact linear theory.An efficient formally second-order well-balanced hybrid finite volume/difference numerical scheme isproposed. The scheme consists in a two-step algorithm. First, the hyperbolic part of the system is dis-cretized using a PVM path-conservative finite-volume method. Second, the dispersive terms are solved bymeans of compact finite differences. The numerical scheme proposed adapts well to GPU-architectures.The method has been applied to idealized and challenging experimental test cases over complex bathyme-tries which shows the efficiency and accuracy of the method.
INTRODUCTION
When modelling and simulating geophysical flows, the Nonlinear Shallow-Water equations, hereinafterSWE, is often a good choice as an approximation of the Navier-Stokes equations. Nevertheless, SWE do nottake into account effects associated with dispersive waves. In recent years, effort has been done in the derivationof relatively simple mathematical models for shallow water flows that include long nonlinear water waves. Seefor instance the works in [6, 7, 9, 10] among others.
The challenge is to improve nonlinear dispersive properties of the model by including information on thevertical structure of the flow while designing fast and efficient algorithms for its simulation. Here we shalluse one of the approaches introduced by Fernandez-Nieto et al in [4] with a slightly correction in the non-hydrostatic pressure, improving dispersive properties without an increase of the computational effort. Themodel will be solved numerically using a two step algorithm: on a first step we solve the two-layer shallow-water system in conservative form and on the second step we include the non-hydrostatic effects. Numericaltests and comparison with experimental data show the accuracy and efficiency of the approach.
DESCRIPTION OF THE MODEL
We consider one of the class of non-hydrostatic models introduced in [4] for two levels with a slightlycorrection in the non-hydrostatic pressure. As usual in shallow water models, the equations are obtained by aprocess of depth averaging on the vertical direction z. Total pressure is decomposed into a sum of hydrostaticand non-hydrostatic pressures. In this process, vertical velocity is assumed to have linear vertical profile at eachlevel, and non-hydrostatic pressure is assumed to have a discontinuity at the interface.
FIGURE 1: Sketch of the domain for the fluid problem
∂th+∂x (hu) = 0,
∂t (hu1)+∂x
(hu2
1 +12
gh2)−gh∂xH−G1(u) =−h(∂x p1 +σ1∂z p1)−
12
λλ −2
h(∂x pb +σ1 pb) ,
∂t (hw1)+∂x (hu1w1)−G1(w) =−∂z p1−1l1
λλ −2
pb,
∂t (hu2)+∂x
(hu2
2 +12
gh2)−gh∂xH−G2(u) =−h(∂x p2 +σ2∂z p2) ,
∂t (hw2)+∂x (hu2w2)−G2(w) =−∂z p2,
∂xuα−1/2 +σα−1/2∂zuα−1/2 +1h
∂zwα−1/2 = 0, α ∈ {1,2}.
(1)
being
p1 =12(pb + pI) , p2 =
12
pI, ∂z p1 =1l1(pI− pb) , ∂z p2 =−
1l2
pI,
uα−1/2 =12(uα+1 +uα) , ∂zuα−1/2 =
1lα
(uα+1−uα) ,
wα−1/2 =12(wα+1 +wα) , ∂zwα−1/2 =
1lα
(wα+1−wα) ,
hσα = ∂x (H− (α−1/2)lαh) , l1 + l2 = 1,
u0 = 0, w0 =−∂tH,
where t is time and g is gravitational acceleration. u1, u2 are depth averaged velocities in the x direction ateach level. w1, w2 are depth averaged velocities in the z direction at each level. pI, pb are the non-hydrostaticpressures at the interface and bottom respectively. The flow depth is h = η +H where η is the surface elevationmeasured from the still-water level and H is the still water depth.
NUMERICAL SCHEME
The model is solved numerically using a two-step algorithm: first the hyperbolic problem (SWE) is solved,then, in a second step, non-hydrostatic terms will be taken into account.
Following [3], we have developed an efficient second-order well-balanced numerical method, which com-bines finite-volume and finite-difference schemes: the hyperbolic part of the system is discretized using a PVMpath-conservative finite-volume method [1, 2, 8], and the non-hydrostatic terms with compact finite differencesusing one common arrangement of the variables, known as the Arakawa C-grid. The resulting ODE system isdiscretized using a TVD Runge-Kutta method [5]. Although the scheme implies solving a big linear system,we present an efficient way to solve it and easy to parallelize.
NUMERICAL TESTS
Some numerical tests including comparisons with laboratory data will be presented (see Figure 2).
9
FIGURE 2: Computed free surface. Small-scale bathymetry of the town of Seaside, Oregon.
ACKNOWLEDGMENT
This research has been supported by the Spanish Government through the Research projects MTM2015-70490-C2-1-R, MTM2015-70490-C2-2-R.
REFERENCES
[1] M.J. Castro, E. Fernández-Nieto, A class of computationally fast first order finite volume solvers: PVMmethods, SIAM Journal on Scientific Computing 34 (4) (2012) 173–196.
[2] M.J. Castro, J. Gallardo, C. Parés, High order finite volume schemes based on reconstruction of statesfor solving hyperbolic systems with nonconservative products. applications to shallow water systems,Mathematics of Computation 75 (2006) 1103–1134.
[3] C. Escalante, T. Morales De Luna, M.J. Castro. Non-hydrostatic pressure shallow flows: GPU implemen-tation using finite-volume and finite-difference scheme. 2017. Submitted to Applied Mathematics andComputation.
[4] E.D. Fernandez-Nieto, M. Parisot, Y. Penel, J. Sainte-Marie. Layer-averaged approximation of Euler equa-tions for free surface flows with a non-hydrostatic pressure. 2016. <hal-01324012>.
[5] S. Gottlieb, C.-W. Shu, Total variation diminishing runge-kutta schemes, Mathematics of Computation67 (221) (1998) 73–85.
[6] M.-O. Bristeau, A. Mangeney, J. Sainte-Marie, N. Seguin, An energy-consistent depth-averaged eulersystem: Derivation and properties, Discrete and Continuous Dynamical Systems Series B 20 (4) (2015)961–988.
[7] P. Madsen, O. Sørensen, A new form of the boussinesq equations with improved linear dispersion charac-teristics. part 2: A slowing varying bathymetry, Coastal Engineering 18 (1992) 183–204.
[8] C. Parés, Numerical methods for nonconservative hyperbolic systems: a theoretical framework, SIAM J.Numer. Anal. 1 (2006) 300–321.
[9] D. Peregrine, Long waves on a beach, Fluid Mechanics 27 (4) (1967) 815–827.
[10] Y. Yamazaki, Z. Kowalik, K. Cheung, Depth-integrated, non-hydrostatic model for wave breaking andrun-up, Numerical Methods in Fluids 61 (2008) 473–497.
10
Purple SHARK-FV — May 15-19 2017 — Ofir, Portugal.
Sharing Higher-order Advanced Research Know-how on Finite Volumes
SYMMETRIC HYPERBOLIC MODELS AND ADER SCHEMES FOR MULTI-PHYSICS SIMULATIONS
Ilya Peshkov a∗, Michael Dumbser b, Evgeniy Romenski c, Olindo Zanotti b.a Institut de Mathématiques de Toulouse, Université Toulouse III, Toulouse, France.b University of Trento, Trento, Italy.c Sobolev Institute of Mathematics, Novosibirsk, Russia.
ABSTRACT
We develop a symmetric hyperbolic thermodynamically compatible (SHTC) formalism for continuummodeling of nonlinear dynamic phenomena in fluids and solids. Our formalism allows to formulate math-ematical models in a consistent way for modeling of dissipative phenomena such as viscous momentum,mass, heat and electric charge transfer. A typical model within such the SHTC formalism is representedby a first order system of hyperbolic partial differential equations (PDEs) with algebraic source termswhich can be stiff. For the numerical treatment of the models, we chose the family of high-order one-stepADER-FV and ADER-DG methods.
INTRODUCTION
We develop a quite general family of first order symmetric hyperbolic systems of PDEs for modeling ofdynamic phenomena in fluids and solids. In particular, we are interested in describing dissipative phenomenasuch as viscous momentum, heat, mass and electric charge transfer typically modeled within the classicalsecond order parabolic theory. In particular, we have shown recently that our approach allows to describeviscous fluids and elasto-plastic solids in a one system of hyperbolic PDEs [1]. This model was then extendedto deal with motion of matter interacting with electro-magnetic fields [2]. We currently work on applyingthe model to flows with non-Newtonian rheology (viscoplastic fluids, dense granular flows), non-Fourier heatconduction. The multi-phase character of flow can be also taken into account as well. Extension of the modelto the relativistic case is of a particular interest because our hyperbolic approach allows only finite speeds forperturbation propagation even in the diffusion regime. Elasto-plastic deformation in solids are also naturallyaccounted for in the model.
The main challenges for numerical treatment of our models is the presence of stiff relaxation source termsand involution constraints like
∂∂ t
(rot(AAA)−aaa) = 0,∂∂ t
(div(BBB)−bbb) = 0, (1)
where AAA, BBB, aaa and bbb are some vector or tensor fields. The non-conservative terms are also presented but theirstructure is fully conditioned by the involution constraints (1).
The source terms in our models are of relaxation nature with a relaxation time τ usually playing the role ofa characteristic dissipation time scale. Such a source term becomes stiff if τ � T , where T is the characteristicmacroscopic time scale of the problem. For example, this situation arises when we deal with diffusion phe-nomena like Newtonian flows which classically modeled with the famous parabolic Navier-Stokes equations.As it is well known, stiff relaxation imposes severe time step restrictions for a numerical scheme if the standard
Purple SHARK-FV — May 15-19 2017 — Ofir, Portugal.
Sharing Higher-order Advanced Research Know-how on Finite Volumes
SPECTRAL SEMI-IMPLICIT AND SPACE-TIME DISCONTINUOUSGALERKIN METHODS FOR THE INCOMPRESSIBLE NAVIER-STOKESEQUATIONS ON STAGGERED ADAPTIVE CARTESIAN GRIDS
Francesco Fambri a∗, Michael Dumbser a
a DICAM, Department of Civil, Environmental and Mechanical Engineering, University of Trento, Via Mesiano, 77 -38123 Trento, Italy
ABSTRACT
We present two new families of arbitrary high order accurate spectral discontinuous Galerkin (DG) finiteelement method for the incompressible Navier-Stokes equations in two and three space dimensions on stag-gered space-time adaptive Cartesian meshes (AMR). The main advantage of making use of a semi-implicitdiscretization is that the numerical stability can be obtained for large time-steps without leading to an excessivecomputational demand [3]. While the pressure is defined on the control volumes of the main grid, the velocitycomponents are defined on face-based dual control volumes, leading to a spatially staggered mesh. In the firstfamily [1], high order of accuracy is achieved only in space, while a simple semi-implicit time discretization isderived by adopting an implicit discretization of the pressure gradient in the momentum equation and of the di-vergence of the velocity field in the continuity equation. In order to avoid a quadratic stability condition for theparabolic terms given by the viscous stress tensor, an implicit discretization is also used for the diffusive termsin the momentum equation. The second family of staggered DG schemes proposed in this work achieves highorder of accuracy also in time by expressing the numerical solution in terms of piecewise space-time polyno-mials [1]. In order to circumvent the low order of accuracy of the adopted fractional stepping, a simple iterativePicard procedure is introduced, which leads to a space-time pressure-correction algorithm. Moreover, in thiswork the presented semi-implicit approach is also implemented for the first time within a space-time adaptiveAMR framework in two and three space dimensions [2]. The real advantages of the staggered grid arise in thesolution of the Schur complement associated with the saddle point problem of the discretized incompressibleNavier-Stokes equations, i.e. after substituting the discrete momentum equations into the discrete continuityequation. This leads to a linear system for only one unknown, the scalar pressure. Indeed, the resulting linearpressure system is shown to be symmetric and positive-definite. Both linear systems for pressure and veloc-ity are very efficiently solved by means of a classical matrix-free conjugate gradient method, for which fastconvergence is observed without use of any preconditioner. Due to the explicit discretization of the nonlinearconvective terms, the final algorithm is stable within the classical CFL-type time step restriction for explicitDG methods. Moreover, by using a time-accurate local time stepping, each element can use its optimal timestep. The new spectral semi-implicit and space-time staggered DG schemes have been thoroughly verified forpolynomial degrees up to N = 11 for a large set of non-trivial test problems in two and three space dimensions,for which analytical, numerical or experimental reference solutions exist. To the knowledge of the authors, thisis the first staggered semi-implicit DG scheme for the incompressible Navier-Stokes equations on space-timeadaptive meshes in two and three space dimensions.
[1] F. Fambri and M. Dumbser. Spectral semi-implicit and space-time discontinuous Galerkin methods for theincompressible Navier-Stokes equations on staggered Cartesian grids. Applied Numerical Mathematics,110:41–74, (2016).
[2] F. Fambri and M. Dumbser. Semi-implicit discontinuous Galerkin methods for the incompressible Navier-Stokes equations on adaptive staggered Cartesian grids. Submitted to, ISSN arXiv:1612.09558.
[3] F. Fambri, M. Dumbser, and V. Casulli. An efficient semi-implicit method for three-dimensional non-hydrostatic flows in compliant arterial vessels. International Journal for Numerical Methods in BiomedicalEngineering, 30(11):1170âAS 1198, (2014).
[4] M. Dumbser and V. Casulli. A staggered semi-implicit spectral discontinuous galerkin scheme for theshallow water equations. Applied Mathematics and Computation, 219(15):8057 âAS 8077, (2013).
[5] M. Dumbser, F. Fambri, I. Furci, M. Mazza, M. Tavelli, and S. Serra- Capizzano. Staggered discontinuousGalerkin methods for the incompressible Navier-Stokes equations: spectral analysis and computationalresults. Submitted to, ISSN arXiv:1612.04529.
[6] M. Tavelli and M. Dumbser. A staggered space-time discontinuous Galerkin method for the three-dimensional incompressible Navier-Stokes equations on unstructured tetrahedral meshes. Journal of Com-putational Physics, 319: 294 âAS 323, (2016).
14
Purple SHARK-FV — May 15-19 2017 — Ofir, Portugal.
Sharing Higher-order Advanced Research Know-how on Finite Volumes
HIGH-ORDER NUMERICAL SCHEME FOR SENSITIVITY ANALYSIS FOR EULER EQUATIONS
C. Chalons a, R. Duvigneau b, C. Fiorini a∗
a Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 45, avenue des États-Unis, 78035Versailles, France.b Université Côte d’Azur, INRIA, CNRS, LJAD, INRIA Sophia-Antipolis Méditerranée Center, ACUMESProject-Team, 2004 route des Lucioles - B.P. 93, 06902 Sophia Antipolis, France.
ABSTRACT
Sensitivity analysis (SA) is the study of how changes in the inputs of a model affect the outputs. StandardSA techniques for PDEs, such as the continuous sensitivity equation method, call for the differentiationof the state variable. However, if the governing equations are hyperbolic PDEs, the state can be discon-tinuous and this generates Dirac delta functions in the sensitivity. The aim of this work is to define andapproximate numerically a system of sensitivity equations which is valid also when the state is discontin-uous: to do that, one can define a correction term to be added to the sensitivity equations starting from theRankine-Hugoniot conditions, which govern the state across a shock.
INTRODUCTION
Sensitivity analysis (SA) for partial differential equations (PDEs), concerns the quantification of changes ina PDEs solution due to perturbations in the model input. It has been a topic of active research for the last years,due to its many applications, for instance in uncertainty quantification, quick evaluation of close solutions [1],and optimization [2], to name but a few. Standard SA methods work only under certain hypotheses of regularityof the solution [3]. However, these assumptions are not verified in the case of hyperbolic systems of the generalform {
∂tU+∂xF(U) = 0, x ∈ R, t > 0,U(x,0) = U0(x),
(1)
due to possible discontinuities, which can occur even when the initial condition is smooth. If the state U isdiscontinuous, Dirac delta functions will appear in the sensitivity Ua = ∂aU. Here and throughout this work,a denotes the input parameter of the model which may vary and induce a non trivial sensitivity Ua of the statesolution U. By differentiating the system (1) with respect to the parameter of interest a and considering smoothsolutions of (1), we obtain the sensitivity equations:
{∂tUa +∂xF′(U)Ua = 0, x ∈ R, t > 0,Ua(x,0) = Ua,0(x).
(2)
At this stage, it is important to remark that the sensitivity system (2) admits the same real eigenvalues as thestate system (1), therefore the global system formed by (1)-(2) is only weakly hyperbolic. We recall that weakhyperbolicity means that the Jacobian matrix of the system admits real eigenvalues but is not R−diagonalizable.As a consequence and without any modification of (2), discontinuous weak solutions of the state variable Uwill generally induce Dirac delta functions in the sensitivity variable Ua, in addition to the usual discontinuity,so that the solutions of (2) have to be understood in the sense of measures.
The aim of this work is to define and approximate numerically a system of sensitivity equations which isvalid also when the state is discontinuous: to do that, one can define a correction term to be added to thesensitivity equations starting from the Rankine-Hugoniot conditions, which govern the state across a shock.We suggest the following term:
S =Ns
∑k=1
ρρρk(t)δ (x− xk,s(t)), (3)
where Ns is the number of discontinuities, ρρρk is the amplitude of the k−th correction (to be computed) and xs,k(t)the position of the k−th shock at time t. The amplitude ρρρk(t) is obtained by comparing an integral balance ona control volume for the equations (2) with the additional term (3) to the Rankine-Hugoniot conditions for thestate equations (1). One is thus led to set:
ρρρk(t) = ∂aσk(U+−U−), (4)
where the plus (respectively minus) indicates the value of the state to the right (respectively left) of the shock,and σk is the speed at which the shock travels.
We tested different numerical schemes: an exact Godunov-type method, for which we computed the ana-lytical solution of the Riemann problem for the state and for the sensitivity, a Roe-type approximate Riemannsolver, both first and second order, and a modified Godunov-type method without numerical diffusion, intro-duced in [4]. Concerning the discretisation of the source term, there are two main difficulties. First, we remarkthat from the definition of the amplitude (4), the source term is automatically zero in the regular zones where thestate is continuous, however this is not true in a numerical framework. Therefore, in order not to overcorrect, ashock detector needs to be introduced. Secondly, the expression (4) is valid only if the state U is locally con-stant to the left and to the right of the shock, which is true for first-order schemes, but not for higher-order ones.To overcome this problem in the case of the second-order scheme, instead of considering a piecewise affinefunction on the cells, we considered a piecewise constant function on the half of every cell, with an additionalRiemann problem in the cell itself.
REFERENCES
[1] R. Duvigneau, D. Pelletier. A sensitivity equation method for fast evaluation of nearby flows and uncer-tainty analysis for shape parameters. Int. J. of CFD, vol 20, 2006.
[2] J. Borggaard, J. Burns. A PDE Sensitivity Equation Method for Optimal Aerodynamic Design. Journal ofComputational Physics, vol 136, 1997.
[3] C. Bardos, O. Pironneau. A Formalism for the Differentiation of Conservation Laws. Compte rendu del’Académie des Sciences, vol 335, 2002.
[4] C. Chalons, P. Goatin. Godunov scheme and sampling technique for computing phase transitions in trafficflow modeling Interfaces and Free Boundaries, vol 10, 2008.
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Sharing Higher-order Advanced Research Know-how on Finite Volumes
INDUSTRIALIZATION OF HIGH ORDER SCHEMES
Claus-Dieter Munz a∗,a Institute of Aerodynamics and Gas Dynamics, University of Stuttgart, Germany
ABSTRACT
Despite the progress made in scientific computing, the simulation of many unsteady flow problems is stillimpractical for various important applications in terms of user time and computational resources. Besidethe increase of computer power a substantial improvement of the numerical methods is necessary, e.g., toperform high fidelity simulations of turbulent flows in industrial problems. High order schemes are suchcandidates for more efficient simulations of challenging problems. In this talk an overview is given aboutapplications of high order discontinuous Galerkin schemes in research topics of scientific engineering andindustrial problems that are considered at Stuttgart. The principal topic of the talk will be the simulationof multi-phase flow by sharp interface tracking.
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Sharing Higher-order Advanced Research Know-how on Finite Volumes
NUMERICAL SIMULATION OF BED LOAD AND SUSPEND LOAD TRANSPORT
J.C. González-Aguire a∗, M.E. Vázquez-Cendón b, J.A. González-Vázquez c, J. Alavez-Ramírez a,R.S. Casarín c
a División Académica de Ciencias Básicas, Universidad Juárez Autónoma de Tabasco. 86690 Tabasco, México.b Departamento de Matemática Aplicada, Universidade de Santiago de Compostela. 15782 Santiago de Compostela,España.c Instituto de Ingeniería, Universidad Nacional Autónoma de México. 04510 Ciudad de México, México.
ABSTRACT
In this work, one-dimensional bed load transport and suspend load transport simulations based on thedepth-averaged shallow water equations, mass conservation equation for sediment and the Exner equation,are presented. The sediment transport can be modeled through the system of partial differential equationswith source terms formed by the mass conservation and momentum conservation equations for the water-sediment mixture, the mass conservation equation for the sediment and the Exner equation for the bedconservation.Uncoupled numerical solution of the global system is considered. First, the hyperbolic system of partialdifferential equations formed by the conservation laws is solved using well-balanced Riemann solvers.Then, the Exner equation is solved.
INTRODUCTION
The floods are a problem which has increased during the last years due to climate change, it has detrimentimpact on the ecology and may cause potential risk on the human life and local economy [2]. Efficient esti-mations of impacts of floods with variable bed play an important role into establishing of a strategy of civilprotection. For these reasons, the development of numerical methods for prediction of floods dynamics anddispersion of sediment is a topic of interest. As a result of effort for build numerical methods to predict waterflow and its dynamics many codes were developed. Among them, Iber [3] was employed to simulate the floodswhich occurred in Villahermosa Mexico in 2007, giving good results as shown in [6], where a fixed bed wastaken into account.
The study of sediment transport is focused on the interrelationship between the moving water and the sed-iment materials [9]. It can be carried out through the study of the system of partial differential equations withsource terms made up by the mass and momentum conservation equations for water-sediment mixture, the massconservation equation for the sediment and the Exner equation for the bed conservation, namely,
• c(x, t) is the averaged-flux volumetric sediment concentration (ppm),
• z(x, t) is the depth of bed (m),
• ρ(x, t) is the density of water-sediment mixture (kg/m3),
• S f (x, t) is the friction slope which is computed using Manning formula (see [4, 13]),
• g is the gravitational acceleration (m/s2),
• p is the bed sediment porosity,
• E(x, t) is the erosion (m/s) (see [4, 13]),
• D(x, t) is the deposition (m/s) (see [4, 13]),
• ρw is the water density (kg/m3),
• ρs is the sediment density (kg/m3),
• ρ0 is the bed saturated density (kg/m3),
• qb(x, t) is the bed-load discharge (s2/m) (see [8, 9, 10]).
Equations (1), (2) and (3) constitute a hyperbolic system, which can be solved numerically using a range ofeffective schemes that can capture shocks and sharp fronts reasonably well. In this work, two different Riemannsolvers are used. The first scheme is based on the ideas put forward in [1], in which a general theory for sourceterm discretization was also introduced. The second scheme is based on the HLLC Riemann solver [7, 11].
The Exner equation for bed conservation (4) takes into account the bed load discharge, which can be cal-culated by deterministic laws or by probabilistic methods always supported by experimentation [10]. In thiswork, the Grass law [8] for computing the bed load discharge is used. It is the most basic law for the transportlaws. The solution of Exner equation is built following the work developed in [9] and by using the informationprovided by the Riemann solver.
The numerical results to be presented prove that the numerical schemes solve exactly the conservationproperty (rest at lake). Dam break flows are simulated, first the dam break cited by Cao in [4] is reproduced,and the numerical results obtained are contrasted with the numerical results displayed by Cao. Finally, theexperimental dam breaks cited in [5, 12, 13] are replicated and the numerical results obtained are comparedwith experimental data.
ACKNOWLEDGMENT
This research was partially supported by CONACYT through the national scholarship program, by SpanishMICINN project MTM2013-43745-R and by Xunta de Galicia and FEDER under research project GRC2013-014.
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REFERENCES
[1] A. Bermúdez, M. E. Vázquez-Cendón. Upwind methods for hyperbolic conservation laws with sourceterm. Computers & Fluids, vol 23(8), 1049-1071, 1994.
[2] F. Benkhaldoun, S. Sari, M. Seaid. A flux-limiter method for dam-break flow over erodible sediment beds.Applied Mathematical Modelling, vol 36, 4847-4861, 2012.
[3] E. Bladé, L. Cea, G. Corestein, E. Escolano, J. Puertas, E. Vázquez-Cendón, J. Dolz, A. Coll. Iber:herramienta de simulación numérica de flujo en ríos. Revista Internacional de Métodos Numéricos parael Cálculo y Diseño de Ingeniería, vol 30(1), 1-10, 2014.
[4] Z. Cao, G. Pender, S. Wallis, P. Carlling. Computational dam-break hydraulics over mobile sediment bed.Journal of Hydraulic Engineering, vol 130(7), 689-703, 2004.
[5] L. Francarrollo, L. Capart. Riemann wave description of erosional dam-break flows. Journal of FluidMechanic, vol 461, 183-238, 2002.
[6] J.C. González-Aguirre, M.E. Vázquez-Cendón, J. Alavez-Ramírez. Simulación numérica de inundacionesen Villahermosa México usando el código IBER. Ingeniería del agua, vol 20(4), 201-216, 2016.
[7] J.A. González-Vázquez. Desarrollo de un modelo de costa a largo plazo con transporte transversal. Tesisdoctoral, Universidad Nacional Autónoma de México, 2016.
[8] A. Grass. Sediment transport by waves and currents. SERC. London Cent. Mar. Technol, Report No. Fl;1981.
[9] C. Juez, J. Murillo, P. García-Navarro. A 2D weakly-coupled and efficient numerical model for transientshallow flow and movable bed. Advances in Water Resources, vol 71, 93-109, 2014.
[10] J. Murillo, P. García-Navarro. An Exner-based coupled model for two-dimensional transient flow overerodible bed. Journal of Computational Physics, vol 229, 8704-8732, 2010.
[11] J. Murillo, P. García-Navarro. Augmented version of the hll and hllc riemann solvers including sourceterm in one and two dimensions for shallow flow application. Journal of Computational Physics, 231(20),6861-6906, 2012.
[12] B. Spinewine, T. Zech. Small-scale laboratory dam break waves on movable beds. Journal of HydraulicResearch, vol 45(1), 73-86, 2007.
[13] W. Wu, S. S. Y. Wang. One-dimensional modeling of dam-break flow over movable beds. Journal ofHydraulic Research, vol 133(1), 48-58, 2007.
[14] D. Yung, L. Zuisen, Z. DeYu, K. YanPing. Coupling mechanism of mathematical model for sedimenttransport based on characteristic theory. China Science Technological Sciences, vol 59(11), 1696-1706,2016.
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PROJECTIVE INTEGRATION FOR THE NONLINEAR BGK AND BOLTZ-MANN EQUATIONS
W. Melis a, T. Rey b∗, G. Samaey a
a Department of Computer Science, K.U. Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium.b Laboratoire Paul Painlevé, Université de Lille, Cité Scientifique, 59655 Villeneuve d’Ascq, France.
ABSTRACT
We present high-order, fully explicit, asymptotic-preserving projective integration schemes for the non-linear BGK and Boltzmann equations. The methods first take a few small (inner) steps with a simple,explicit method (such as direct forward Euler) to damp out the stiff components of the solution. Then,the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. Based onthe spectrum of the collision operator, we deduce that, with an appropriate choice of inner step size, thetime step restriction on the outer time step as well as the number of inner time steps is independent ofthe stiffness of the source term. We illustrate the method with numerical results in one and two spatialdimensions.
LONG SUMMARY
The Boltzmann equation constitutes the cornerstone of kinetic theory. It describes the evolution of theone-particle mass distribution function f ε(x,v, t) ∈ R+ as:
∂t f ε + v ·∇x f ε =1εQ( f ε), (1)
where t ≥ 0 represents time, and (x,v)⊂RDx×Dv are the Dx-dimensional particle positions and Dv-dimensionalparticle velocities. In equation (1), the dimensionless constant ε > 0 determines the regime of the gas flow, forwhich we roughly identify the hydrodynamic regime (ε ≤ 10−4), the transitional regime (ε ∈ [10−4,10−1]),and the kinetic regime (ε ≥ 10−1). Furthermore, the left hand side of (1) corresponds to a linear transportoperator that comprises the convection of particles in space, whereas the right hand side contains the Boltzmanncollision operator that entails velocity changes due to particle collisions. However, due to its high-dimensionaland complicated structure, the Boltzmann collision operator is often replaced by simpler collision models thatcapture most essential features of the former, such as the so-called BGK operator.
In the limit ε → 0, the solution to equation (1) converges towards a gaussian distribution (the so-calledMaxwellian equilibrium) whose first moments in velocity (the density ρ , the mean velocity v and the tempera-ture T ) are solution to the compressible Euler system:
∂tρ +∇ · x(ρ v) = 0,
∂t(ρ v)+∇ · x(ρ v⊗ v + ρ T I) = 000,
∂tE +∇ · x(v(E +ρ T )) = 0,
(2)
in which E is the total energy.In this work, we construct a fully explicit, asymptotic-preserving, arbitrary order time integration method
for the stiff equation (1). The asymptotic-preserving property [1] implies that, in the limit when ε tends to zero,∗Correspondence to [email protected]
23
an ε-independent time step constraint, of the form ∆t = O(∆x), can be used, in agreement with the classicalhyperbolic CFL constraint for the limiting fluid equations (2). To achieve this, we will use a projective inte-gration method, which was introduced in [2] and first applied to kinetic equations in [3]. For a comprehensivereview of numerical schemes for collisional kinetic equations such as equation (1), we refer to [4]. Although itis known that an implicit treatment of (1) can be implemented explicitly, the order in time is usually restrictedto 2. Therefore, the main advantage of the proposed method is its arbitrary order in time.
Projective integration [2, 3] combines a few small time steps with a naive (inner) timestepping method (here,a direct forward Euler discretization) with a much larger (projective, outer) time step. The idea is sketched infigure 1. As we shall see in the talk, the parameters of these inner and outer integrators are directly linked tothe spectrum of the linearization of the collision operator.
timetn−1 tn tn+1
FIGURE 1: Sketch of projective integration. At each time, an explicit method is applied over a number ofsmall time steps (black dots) so as to stably integrate the fast modes. As soon as these modes are sufficientlydamped the solution is extrapolated using a much larger time step (dashed lines).
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 −10
1
1
1.5
2
2.5
3
3.5
4
4.5
xy
ρ(x
,0.8)
Density (t = 0.8)
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 −10
1
1
1.5
2
2.5
xy
T(x
,0.8)
Temperature (t = 0.8)
FIGURE 2: Example of a shock-bubble interaction, obtained in the fluid ε = 10−5 regime, with our projectiveRunge-Kutta 4 integration scheme. The transport part is solved using a WENO 3 method and the collision partwith a Fourier spectral scheme.
REFERENCES
[1] S. Jin Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations SIAM J. Sci.Comp., 1999, 21, 441-454.
[2] C. Gear & I. Kevrekidis Projective Methods for Stiff Differential Equations: Problems with Gaps in TheirEigenvalue Spectrum SIAM J. Sci. Comp., 2003, 24, 1091-1106.
[3] P. Lafitte & G. Samaey Asymptotic-preserving projective integration schemes for kinetic equations in thediffusion limit SIAM J. Sci. Comp., SIAM, 2012, 34, A579-A602.
[4] G. Dimarco & L. Pareschi Numerical methods for kinetic equations Acta Numerica, 2014, 23, 369-520.
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Sharing Higher-order Advanced Research Know-how on Finite Volumes
WELL-BALANCED SCHEMES FOR SHALLOW-WATER MODELS WITHSTRONGLY NON-LINEAR SOURCE TERMS
Christophe Berthon a∗,a Université de Nantes Laboratoire de Mathématiques Jean Leray 2 rue de la Houssinière - BP 92208 44322 NantesCedex 3
ABSTRACT
The present work is devoted to the derivation of accurate finite volume schemes to capture sophisticatedsteady states issuing from shallow-water models supplemented by strongly non-linear source terms. Two dis-tinct models are here considered. The first one is the Ripa model [1,5] to take into acount the temperaturepotential within the flow. The second model under consideration is the shallow-water model with Manningfriction. Both models involve very sophisticated steady states which are of prime importance for simulations.When compared to the well-known shallow-water model, the here considered steady states are not explicitlygiven. For instance, the steady states associated to the shallow-water model with Manning friction are solutionof a strongly non-linear algebraic relation. In the case of the Ripa model, the steady states are solutions of anon-solvable partial differential equation.
To approximate the solutions of these systems, we adopt finite volume methods deriving from Godunov-typeschemes [3]. Because of the source terms, the celebrate Harten, Lax and van Leer result [3], cannot be directlyapplied. However, after [2], the source term average can be easily introduced within an approximate Riemannsolver to get consistent schemes. Next, the main idea consists in electing a suitable source term average inorder to recover the required well-balanced property [6]; namely to exactly capture the steady states of interest.Considering the shallow-water model with Manning friction, we obtain an approximate Riemann solver whichexactly capture the solution of the non-linear equation governing the steady states. Let us underline that we geta well-balanced scheme witout solving the steady states.
Concerning the Ripa model, the situation turns out to be distinct since the steady states of interest aregoverned by a partial differential equation. Here, we exhibit three classes of solutions to be exactly capturedby the scheme (in [1] just two classes are considered). The required well-balance property is, once again,obtained by adopting a relevant source term average within the derived approximate Riemann solver. Theresulting scheme is thus proved to exactly capture the exhibited classes of steady states. Moreover, the schemeapproximate, in a sense to be specified, partial differential equation governing the steady states. It is worthnoticing that the derived approximate solvers preserves additional crucial properties (robustness and stability).
[1] A. Chertock, A. Kurganov, Y. Liu, Central-upwind schemes for the system of shallow water equationswith horizontal temperature gradients, Numerische Mathematik (2013), 1-45.
[2] G. Gallice, Positive and entropy stable Godunov-type schemes for gas dynamics and MHD equations inLagrangian or Eulerian coordinates, Numer. Math. (2003) 94, 673-713.
[3] A. Harten, P. Lax, B. Van Leer, On upstream differencing and Godunov-type schemes for hyperbolicconservation laws, SIAM review (1983) 25, 35-61.
[5] P. Ripa, On improving a one-layer ocean model with thermodynamics, Journal of Fluid Mechanics (1995)303, 169-202.
[6] C. Berthon, C. Chalons, A fully weel-balanced, positive and entropy satisfying Godunov-type methodfor the shallow-water equations, Math. of Comput., 85 (2016), pp. 1281-1307.
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NEW APPLICATIONS OF THE SPH-ALE-MLS-MOOD METHOD
L. Ramírez a, A. Piñeiro a, A. Eirís b, X. Nogueira a∗, S. Clain c, R. Loubère d, F. Navarrina a
a Group of Numerical Methods in Engineering, Universidade da Coruña, Campus de Elviña, 15071, A Coruña, Spainb Centro Universitario de la Defensa, Escuela Naval Militar, Plaza de España s/n, Marín, Spainc Centre of Mathematics, University of Minho, Campus de Azurém, 4080-058, Guimarães, PortugaldInstitut de Mathématiques de Toulouse et CNRS, Université de Toulouse, France
ABSTRACT
In this work we present our latest developments on the SPH-ALE method [1, 2]. In our approach [3],we use Moving Least Squares approximations for the higher-order computation of the left/rigth Riemannstates at the midpoint between pairs of particles. The stability of the scheme is achieved by the a posterioriMulti-dimensional Optimal Order Detection (MOOD) paradigm [4, 5, 6] instead of classical artificialviscosity approach. Here, we show the results of the application of this method to the shallow waterequations and to weakly compressible flows. We also present a modification of the formulation, that leadsto a more accurate integration. It consists on the substitution of the standard kernel integration by using aGalerkin approach based on MLS [7]. This modification allows the method to naturally fulfill the partitionof unity property, and it also improves other conservation properties. The resulting scheme is applied tothe resolution of the Linearized Euler Equations for aeroacoustics.
REFERENCES
[1] J.P. Vila, On Particle Weighted Methods and Smooth Particle Hydrodynamics, Mathematical Models andMethods in Applied Sciences, 9(2), 161–209, 1999.
[2] B. Ben Moussa, On the convergence of SPH method for scalar conservation laws with boundary condi-tions, Methods and Applications of Analysis, 13, 29–62, 2006.
[3] X. Nogueira, L. Ramírez, S. Clain,R. Loubère, L. Cueto-Felgueroso,I. Colominas, High-accurate SPHmethod with Multidimensional Optimal Order Detection limiting, Computer Methods in Applied Me-chanics and Engineering, 310:134–155, 2016.
[4] S. Clain, S. Diot, R. Loubère, A high-order finite volume method for systems of conservation laws- Mul-tidimensional Optimal Order Detection (MOOD), Journal of Computational Physics, 230:4028–4050,2011.
[5] S. Diot, S. Clain, R. Loubère, Improved detection criteria for the Multidimensional Optimal Order Detec-tion (MOOD) on unstructured meshes with very high-order polynomials, Computers and Fluids, 64:43–63,2012.
[6] R. Loubère, M. Dumbser and Steven Diot, A New Family of High Order Unstructured MOOD and ADERFinite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws, Communicationin Computational Physics, 16:718–763, 2014.
[7] E. Gaburov and K. Nitadori, Astrophysical Weighted Particle Magnetohydrodynamics, Monthly Noticesof the Royal Astronomical Society, 414, 129–154, 2011.
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ONE DIMENSIONAL STEADY-STATE EULER SYSTEM
S. Clain a, R. Loubère b, G.J. Machado a∗
a Centro de Matemática, Universidade do Minho, Campus de Gualtar - 4710-057 Braga, Portugal.b CNRS and Institut de Mathématiques de Bordeaux (IMB), Université de Bordeaux, Talence, France.
ABSTRACT
We propose a finite volume numerical scheme devoted to solve the one-dimensional steady-state Eulersystem. High-accuracy (up to the sixth-order presently) is achieved thanks to polynomial reconstructionswhile stability is provided with an a posteriori MOOD method which control the cell polynomial degreefor eliminating non-physical oscillations in the vicinity of discontinuities. Such a procedure demands thedetermination of a chain detector to discriminate between troubled and valid cells, a cascade of polynomialdegrees to be successively tested when oscillations are detected, and a parachute scheme correspondingto the last, viscous, and robust scheme of the cascade. The obtained results demonstrate that the schememanages to retrieve smooth solutions with optimal order of accuracy but also irregular solutions withoutspurious oscillations.
ACKNOWLEDGEMENTS
This research was financed by Portuguese Funds through FCT — Fundação para a Ciência e a Tecnologia,within the Project UID/MAT/00013/2013.
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ON ALL-REGIME, HIGH-ORDER AND WELL-BALANCED LAGRANGE-PROJECTION TYPE SCHEMES FOR THE SHALLOW WATEREQUATIONS
C. Chalons a, P. Kestener b, S. Kokh b,c, M. Stauffert a,b∗
a Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 78035 Versailles, France.b Maison de la Simulation USR 3441, Digiteo Labs, bât. 565, PC 190, CEA Saclay, 91191 Gif-sur-Yvette, France.c CEA/DEN/DANS/DM2S/STMF, CEA Saclay, 91191 Gif-sur-Yvette, France.
ABSTRACT
The purpose of this work is to design a high order scheme for the Shallow Water Equations (SWE) thanksto a Lagrange-Projection type approach.
We propose to extend the recent implicit-explicit schemes developed in [1, 2] in the framework of com-pressible single or two-phase flows. These methods enjoy several good features: they provide an accurateapproximation independently of the Mach regime. They also enable the use of time steps that are nolonger constrained by the sound velocity thanks to an implicit treatment of the acoustic waves and anexplicit treatment of the material waves.
We have for now studied two different extensions of the schemes: one towards the SWE in the Finite-Volume framework in [3], and one towards the Discontinuous Galerkin (DG) discretization for thebarotropic Euler equations [4]. In the present case, we propose a combination of both, that is to say aLagrange-Projection type DG scheme for the SWE system. We particularly focus on the discretization ofthe non-conservative terms, and more specifically on the well-balanced property of the method, which isnot trivial for high order schemes.
References
[1] M. Girardin. Méthodes numériques tout-régime et préservant l’asymptotique de type Lagrange-Projection : application aux écoulements diphasiques en régime bas Mach. PhD Thesis, University Pierreet Marie Curie Paris 6, 2014.
[2] C. Chalons, M. Girardin, S. Kokh. An all-regime Lagrange-Projection like scheme for the gas dynamicsequations on unstructured meshes. Communications in Computational Physics, 2016.
[3] C. Chalons, P. Kestener, S. Kokh, M. Stauffert. A large time-step and well-balanced Lagrange-Projectiontype scheme for the shallow-water equations. Communications in Mathematical Sciences, 2017.
[4] C. Chalons, M. Stauffert. A high-order Discontinuous Galerkin Lagrange-Projection scheme for thebarotropic Euler equations. To appear in FVCA8 conference proceedings, 2017.
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Sharing Higher-order Advanced Research Know-how on Finite Volumes
FINITE VOLUME METHODS FOR MULTI-COMPONENT EULER EQUA-TIONS WITH SOURCE TERMS IN NETWORKS
A. Bermúdez a,b, X. López a,b, M.E. Vázquez-Cendón a,b∗
a Departamento de Matemática Aplicada, Universidade de Santiago de Compostela (USC), Santiago de Compostela,Spain.b Instituto Tecnológico de Matemática Industrial (ITMATI), Santiago de Compostela, Spain.
ABSTRACT
With numerical simulation of gas transportation networks in view, a first-order well-balanced finitevolume scheme for the solution of a model for the flow of a multi-component gas in a pipe on non-flattopography is introduced.
The mathematical model consists of Euler equations with source terms arising from heat exchange andgravity and viscosity forces, coupled with the mass conservation equations of species.
We propose a segregated scheme in which the Euler and species equations are solved separately. Thismethodology leads to a flux vector in the Euler equations depending not only on the conservativevariables but also on time and space variables through the gas composition. This fact makes necessaryto add some artificial viscosity to the numerical flux which is done by introducing an additional sourceterm. Besides, in order to preserve the positivity of the species concentrations we discretize the flux in themass conservation equations for species in accordance with the upwind discretization of the total massconservation equation in the Euler system, [6]. As proposed in a previous reference by the authors, [4], thediscretization of the flux and source terms is made so as to ensure that the full scheme is well-balanced [8].
In this presentation the network is also considered. This requieres to deal with the junctions where theone-dimensional gas dynamic equations for pipes converging at the junction are coupled. The proposedtechnique consist in using one-dimensional models for the pipes with two-dimensional models for thejunctions, [5]. Numerical tests including both academic and real gas network problems are solved showingthe performance of the proposed methodology.
ACKNOWLEDGMENT
This work was supported by the Reganosa company, by FEDER and the Spanish Ministry of Science and In-novation under research projects ENE2013-47867-C2-1-R and MTM2013-43745-R, and by FEDER and Xuntade Galicia under research project GRC2013/014.
REFERENCES
[1] American Gas Association. AGA Report 8: Compressibility Factors of Natural Gas and Other RelatedHydrocarbon Gases (1992).
[2] M. K. Banda, M. Herty, A. Klar, Coupling conditions for gas networks governed by the isothermal Eulerequations, Networks and Heterogeneous Media 1 (2006) 295–314.
[3] A. Bermúdez, J. González-Díaz, F. J. González-Diéguez, A. M. González-Rueda, M. P. Fernández deCórdoba, Simulation and optimization models of steady-state gas transmission networks, Energy Procedia64 (2015) 130–139.
[4] A. Bermúdez, X. López, M.E. Vázquez-Cendón, Numerical solution of non-isothermal non-adiabatic flow of real gases in pipelines, Journal of Computational Physics, 323 (2016) 126–148.http://dx.doi.org/10.1016/j.jcp.2016.07.020
[5] A. Bermúdez, X. López, M.E. Vázquez-Cendón, Treating network junctions in finite volume solution oftransient gas flow models, submitted to Journal of Computational Physics (2016).
[6] A. Bermúdez, X. López, M.E. Vázquez-Cendón, Finite volume methods for multi-component Euler equa-tions with source terms, submitted to Computers and Fluids (2017).
[7] A. Bermúdez, M.E. Vázquez-Cendón, Upwind methods for hyperbolic conservation laws with sourceterms, Computers and Fluids, 23 (1994) 1049–1071.
[8] F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws: And well-balanced schemes for sources. Springer Science & Business Media (2004).
[9] P. Cargo, A.Y. LeRoux, A well balanced scheme for a model of atmosphere with gravity. Comptes Rendusde L Académie des Sciences Serie I-Mathematique, 318, (1994) 73–76.
[10] M.J. Castro, E. D. Fernández-Nieto, T. Morales de Luna, G. Narbona-Reina and C. Parés, A HLLCscheme for nonconservative hyperbolic problems. Application to turbidity currents with sediment trans-port, ESAIM: Mathematical Modelling and Numerical Analysis 47 1, (2013) 1–32.
[11] C. Chalons, F. Coquel, E. Godlewski, P. A. Raviart, N. Seguin, Godunov-type schemes for hyperbolicsystems with parameter-dependent source: the case of Euler system with friction. Mathematical Modelsand Methods in Applied Sciences, 20, (2010), no 11, 2109–2166.
[12] P. Chandrashekar, C. Klingenberg, A second order well-balanced finite volume scheme for Euler equationswith gravity, SIAM J. Sci. Comput., 37(3), (2015)
[13] D. Chargy, R. Abgrall, L. Fézoui, B. Larrouturou, Conservative numerical schemes for multicomponentinviscid flows. Rech. Aérospat. (English Edition) 1992, no. 2, 61–80.
[14] V. Desveaux, M. Zenk, C. Berthon, C. Klingenberg, A well-balanced scheme to capture non-explicitsteady states in the Euler equations with gravity, Int. J. Numer. Meth. Fluids 001 (2010) 1–23.
[15] P. García-Navarro, M.E. Vázquez-Cendón, On numerical treatment of the source terms in the shallowwater equations. Computers and Fluids 29 (2000), 951–979
[16] R. Käppeli, S. Mishra, Well-balanced schemes for the Euler equations with gravitation. Journal of Com-putational Physics, 259, (2014) 199–219.
[17] F. Ismail, P.L. Roe, Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks,J. Comput. Phys., 228 (2009) 5410–5436.
[18] B. Larrouturou, How to preserve the mass fractions positivity when computing compressible multi-component flows, J. Comput. Phys. 95 (1991), no. 1, 59–84.
[19] J. Luo, K. Xu,N. Liu, A well-balanced symplecticity-preserving gas-kinetic scheme for hydrodynamicequations under gravitational field. SIAM J. Sci. Comput. 33, (2011) 2356–2381.
[20] E. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer,3rd edition (2009).
[21] K. Xu, J. Luo, S. Chen, A Well-Balanced Kinetic Scheme for Gas Dynamic Equations under GravitationalField, Adv. Appl. Math. Mech., Vol. 2, No. 2, (2010) 200-210.
34
Purple SHARK-FV — May 15-19 2017 — Ofir, Portugal.
Sharing Higher-order Advanced Research Know-how on Finite Volumes
A HIGH ORDER FV/FE PROJECTION METHOD FOR COMPRESSIBLELOW-MACH NUMBER FLOWS
A. Bermúdez a, S. Busto a∗, J.L. Ferrín a, E.F. Toro b, M.E. Vázquez-Cendón a
a Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, ES-15782 Santiago de Compostela,Spain.b Laboratory of Applied Mathematics, DICAM, University of Trento, IT-38100 Trento, Italy. .
ABSTRACT
The purpose of this work is to introduce a finite volume/finite element projection method for incompress-ible and compressible low-Mach number flows. Mass conservation, momentum conservation, energyconservation and transport of species laws are coupled using the equation of state.
Starting with a 3D tetrahedral finite element mesh of the computational domain, the momentum equationis discretized by a finite volume method associated with a dual finite volume mesh where the nodes of thevolumes are the barycenters of the faces of the initial tetrahedra (see [4]).
Three different schemes are considered to solve the explicit transport-diffusion stage: a first orderscheme, a Kolgan-type scheme (see [5]) and a Local-ADER scheme (see [2], [3], [6]). To avoid spuriousoscillations in the Kolgan-type scheme a flux limiter is used. Meanwhile, the Local-ADER scheme isconstructed on the basis of ENO interpolations.
Concerning compressible flows, once the conservative variables related to the transport of species andthe energy conservation are computed, the equation of state is used in order to approximate the timederivative of the density (see [1]). Then, the pressure correction is computed by a piecewise linear finiteelement method associated with the initial tetrahedral mesh.
Eventually, several academic problems and classical test from fluid mechanics are presented.
ACKNOWLEDGMENT
This research was partially supported by Spanish MICINN projects MTM2008-02483, CGL2011-28499-C03-01 and MTM2013-43745-R; by the Spanish MECD under the grant FPU13/00279; by the Xunta de Galiciaunder grant Axudas de apoio á etapa predoutoral do Plan I2C; by Xunta de Galicia and FEDER under researchproject GRC2013-014 and by Fundación Barrié under grant Becas de posgrado en el extranjero.
REFERENCES
[1] A. Bermúdez. Continuum thermomechanics Birkhäuser, 2005.
[2] A. Bermúdez, S. Busto, M. Cobas, J.L. Ferrín, L. Saavedra and M.E. Vázquez-Cendón. Paths from math-ematical problem to technology transfer related with finite volume methods Proceedings of the XXIVCongress on Differential Equations and Applications/XIV Congress on Applied Mathematics, 43–54,Cádiz, Spain, 2015.
[3] A. Bermúdez, S. Busto, J.L. Ferrín, L. Saavedra, E.F. Toro, M.E. Vázquez-Cendón A projection hybridfinite volume-ADER/finite element method for turbulent Navier-Stokes Computational Mathematics, Nu-merical Analysis and Applications, SEMA-SIMAI Springer Series, vol 13, 2017.
[4] A. Bermúdez, J.L. Ferrín, L. Saavedra and M.E. Vázquez-Cendón. A projection hybrid finite vol-ume/element method for low-Mach number. J. Comp. Phys., vol 271, 360–378, Elsevier, 2014.
[5] L. Cea and M.E. Vázquez-Cendón. Analysis of a new Kolgan-type scheme motivated by the shallow waterequations. Appl. Num. Math., vol 62, 489–506, Elsevier, 2012.
[6] S. Busto, E.F. Toro & M.E. Vázquez-Cendón. Design and analysis of ADER-type schemes for modeladvection-diffusion-reaction equations J. Comp. Phys. vol 327, 553–575, Elsevier, 2016.
36
Purple SHARK-FV — May 15-19 2017 — Ofir, Portugal.
Sharing Higher-order Advanced Research Know-how on Finite Volumes
ASYMPTOTICALLY ACCURATE HIGH-ORDER SPACE AND TIMESCHEMES FOR THE EULER SYSTEM IN THE LOW MACH REGIME.
Raphaël Loubère b Victor Michel-Dansac a∗, Marie-Hélène Vignal a,a Université de Toulouse, Institut de Mathématiques, Toulouse, France b Université de Bordeaux, Institut deMathématiques, Talence, France
ABSTRACT
This article deals with the discretization of the compressible Euler system for all Mach numbers regimes.For highly subsonic flows, since acoustic waves are very fast compared to the velocity of the fluid, the gascan be considered as incompressible. From the numerical point of view, when the Mach number tends tozero, the classical Godunov type schemes present two main drawbacks: they lose consistency and theysuffer of severe numerical constraints for stability to be guaranteed since the time step must follow theacoustic waves speed. In this work, we propose to construct high accurate AP numerical scheme both inspace and time following the work in [1], IMEX techniques [2] and MOOD paradigm [3].Numerical tests for isentropic Euler equations both in 1D and 2D will be provided to assess the goodbehavior of our approach.
[1] G. Dimarco, R. Loubère, and M.-H. Vignal. Study of a new asymptotic preserving scheme for the Eulersystem in the low Mach number limit. To appear in SIAM SISC (2017)
[2] S. Clain, S. Diot, and R. Loubére. A high-order finite volume method for systems of conservation laws -Multi-dimensional Optimal Order Detection (MOOD). J. Comput. Phys. , 230(10):4028-4050, 2011.
[3] L. Pareschi and G. Russo. Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systemswith relaxation. J. Sci. Comput. , 25(1-2):129-155, 2005.
Purple SHARK-FV — May 15-19 2017 — Ofir, Portugal.
Sharing Higher-order Advanced Research Know-how on Finite Volumes
CWENO SCHEMES FOR CONSERVATION LAWS ON UNSTRUCTURED MESHES
M. Dumbser a∗, W. Boscheri a, M. Semplice b, G. Russo c
a University of Trento, Laboratory of Applied Mathematics, Via Mesiano 77, I-38123 Trento, Italyb Dipartmento di Matematica, Università di Torino, Via C. Alberto 10, I-10123 Torino, Italyc Department of Mathematics and Informatics, University of Catania, Viale A. Doria 6, I-95125 Catania, Italy
ABSTRACT
We present a novel arbitrary high order accurate central WENO spatial reconstruction procedure(CWENO) for the solution of nonlinear systems of hyperbolic conservation laws on fixed and movingunstructured simplex meshes in two and three space dimensions. Starting from the given cell averagesof a function on a triangular or tetrahedral control volume and its neighbors, the nonlinear CWENOreconstruction yields a high order accurate and essentially non-oscillatory polynomial that is defined ev-erywhere in the cell. Compared to other WENO schemes on unstructured meshes, the total stencil sizeis the minimum possible one, as in classical point-wise WENO schemes of Jiang and Shu. However, thelinear weights can be chosen arbitrarily, which makes the practical implementation on general unstruc-tured meshes particularly simple. We make use of the piecewise polynomials generated by the CWENOreconstruction operator inside the framework of fully discrete and high order accurate one-step ADER fi-nite volume schemes on fixed Eulerian grids as well as on moving Arbitrary-Lagrangian-Eulerian (ALE)meshes. The computational efficiency of the high order finite volume schemes based on the new CWENOreconstruction is tested on several two- and three-dimensional benchmark problems for the compressibleEuler equations and is found to be more efficient in terms of memory consumption and computational effi-ciency with respect to classical WENO reconstruction schemes on unstructured meshes. We also provideevidence that the new algorithm is suitable for implementation on massively parallel distributed memorysupercomputers, showing numerical examples run with several billions of degrees of freedom in space onmore than ten thousand CPU cores.
REFERENCES
[1] M. Dumbser, W. Boscheri, M. Semplice, and G. Russo CWENO schemes for conservation laws on un-structured meshes. SIAM Journal on Scientific Computing, 2017, submitted to.
Organizing InstitutionsCentro de Matemática, Universi-dade do Minho, Braga, PT.Institut Jean Leray, Nantes, FR.Institut de Mathématique de Bor-deaux, FR.
Scientific CommitteeCHRISTOPHE BERTHON, Univer-sité de Nantes, France.STÉPHANE CLAIN, Universidadedo Minho, Braga, Portugal.CHRISTOPHE CHALONS, UniversitéVersailles, Paris, France.MICHAEL DUMBSER, Universitàdegli studi di Trento, Italy.RAPHAËL LOUBÈRE, Université deBordeaux, France.ELENA VÁZQUEZ-CENDÓN,Universidade de Santiago deCompostela, Spain.XESÚS NOGUEIRA, Universidadeda Coruña, Spain.
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