contents welcome, bem-vindo! · contents... welcome, bem-vindo! shark-fv week temptative schedule...

Contents. . .• Welcome, Bem-vindo!• SHARK-FV week• Temptative schedule p.2• Hotel and restaurant p.2• Participants p.2• Abstracts p.5-34• Organisation p.36

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Just ask!

• Carolina Ribeiro• Stéphane Clain• Gaspar Machado• Jorge Figueiredo• Rui Pereira• Raphaël Loubère

[email protected]

Make sure you can leave!

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 first experimental Sharing Higher-order Advanced Know-howon Finite Volume conference, SHARK-FV 2014 from April the 28th to May the2nd.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... any useful material.

The area of Esposende between Atlantic ocean andCavado river, the Hotel surrounded by luxurious pri-vate calm gardens and peaceful pine-woods, the fa-mous Portuguese cuisine, provide a perfect incubatorfor Exceptional science!

SHARK-FV week

The organisation of the workshop is as follows: Morn-ing presentations are proposed followed by afternoonsessions which are dedicated to intensive parallel work-shops involving few collaborators.We expect a lot of time to be dedicated to discussions andsmall group working sessions, hence the few number ofparticipants!

@math.uminho.pt

[email protected]

[email protected]

[email protected]

[email protected]

[email protected]

http://www.math.univ-toulouse.fr/SHARK-FV/index.html

http://www.math.univ-toulouse.fr/SHARK-FV/index.html

Temptative schedule

Monday Tuesday Wednesday Thursday Friday8h00-9h00 Breakfast9h00-9h50 TORO p.10 DIMARCO p.18 BLACHÈRE p.26 Closing

Talks9h50-10h40 DIOT p.12 NARSKI p.21 CLAIN p.29 —10h40-11h00 Opening (11h-) Coffee break11h00-11h50 LOUBÈRE p.5 COSTA p.14 RISPOLI p.23 GALLOUET p.31 —

Talks11h50-12h40 FIGUEIREDO p.7 MACHADO p.17 VIGNAL p.24 VÁSQUEZ p.33 —12h40-13h20 DUMBSER p.8 Lunch13h30-14h30 Lunch

14h30-16h30Low Mach and Elliptic Very high-order Multi-physics, — table

Kinetic HO meth. and FVs FV methods source terms —

16h30-19h30Work Work Work Work —

Freesession session session session —

20h00- Dinner BANQUET Dinner

The talks are scheduled to last for 45 minutes with 5 minutes of questions/answers. Abstracts are proposed in pages5-34. After lunch, thematic round tables are organized for interested people, the goal being to exchange and sharenews, raise new problematic and create a positive alchemy between the participants. Ideally such alchemy mustgenerate bombastic discussions to feed the following work sessions, or free time scheduled after 16h30.This free time is 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!

Round tables will feature the following topics:

Elliptic finite volumemethodsIn the context of ellipticand parabolic systems ofequations one wonders howdo prospective high-orderfinite volume methodsbehave? Discussions mayrevolve around this topicand such approach.

Very high-order finite vol-umes methodsThe discussion may concernthe needs of very high-orderschemes according to theconsidered physics. It mayalso deal with the tech-niques to be derived in orderto suitably improve existingapproaches.

Low Mach number and ki-netic high-order methodsSharing experiences withnew techniques andschemes for all-speed flowsand kinetic equations willbe the focus of this dicus-sion as well as prospectiveapproaches (HPC, efficiency,high-order, etc.).

Finite volume for multi-physics system and sourcetermsOpportunity to discuss theimprovements and the rel-evance in deriving numeri-cal schemes to simulate verysophisticated models issuingfrom strongly complex mul-tifluid physics.

Hotel and restaurantHotel Parque do Rio (Caminho Padre Manuel de Sá Pereira) is located40km North of Oporto, between the Atlantic Ofir Beach and the CavadoRiver. Surrounded by luxurious private gardens and pinewoods offeringunforgettable peace and calm.

• Several multi-purpose salons and restaurant. Large garden andsolarium, two swimming pools, snack-bar with outdoor seating.

• “Papa amoras” restaurant-bar and chef Rui Paula the acclaimed chefof the DOC and DOP douro region restaurants, welcome us forbreakfasts at 7h-9h (7AM-9AM), lunches at 13h (1PM) and dinners20h (8PM).

• Wednesday evening is scheduled the exceptional banquet at 20h(8PM), tick your bookmark!

Back to Contents

ParticipantsFlorian Blachère, univ. NantesStephane Clain, univ. do MinhoRicardo Costa, univ. do MinhoGiacomo Dimarco, univ. FerraraSteven Diot, Los Alamos NationalLaboratoryMickael Dumbser, univ. TrentoJorge Figueiredo, univ. do MinhoFrancoise Foucher, univ. NantesThierry Gallouët, univ. MarseilleRaphael Loubère, univ. ToulouseGaspar Machado, univ. do MinhoJacek Narski, univ. ToulouseRui Pereira, univ. do MinhoCarolina Ribeiro, univ. do MinhoVitorio Rispoli, univ. ToulouseKhaled Saleh, univ. MarseilleEleuterio Toro, univ. TrentoElena Vásquez Cendón, univ. San-tiago de CompostelaM-H Vignal, univ. Toulouse

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http://www.estalagemparquedorio.com/en/

List of abstracts

The next pages gather the abstracts of the talks given during SHARK-FV, here is a summary:

R. Loubère, M. Dumbser, S. Diot, page 5COUPLING THE MULTI-DIMENSIONAL OPTIMAL ORDER DETECTION (MOOD) METHOD AND THE ARBITRARYHIGH ORDER DERIVATIVES (ADER) APPROACHES

J. Figueiredo , S. Clain, C. Ribeiro, page 7A NON-CONSERVATIVE FLUX APPROACH ENSURING THE C-PROPERTY FOR THE LAKE AT REST IN THE FRAME-WORK OF A MOOD BASED 2D SHALLOW-WATER SIXTH-ORDER FINITE VOLUME SCHEME

M. Dumbser, page 8HIGH ORDER ONE-STEP AMR AND ALE FINITE VOLUME METHODS FOR HYPERBOLIC PDE

E. Toro, page 10RECENT DEVELOPMENTS ON HIGH-ORDER ADER FINITE VOLUME SCHEMES

S. Diot, M.M. Francois, E.D. Dendy, page 12A VERY-HIGH-ORDER SHARP INTERFACE METHOD TO SIMULATE MULTI-MATERIAL FLOWS BASED ON THEMOOD CONCEPTS

R. Costa, S. Clain, G. J. Machado, page 14FINITE VOLUME SCHEME BASED ON CELL-VERTEX RECONSTRUCTIONS FOR ANISOTROPIC DIFFUSION PROB-LEMS WITH DISCONTINUOUS COEFFICIENTS

S. Clain, G.J. Machado, page 17A HIGH-ORDER FINITE VOLUME SCHEME FOR THE TIME-DEPENDENT CONVECTION-DIFFUSION EQUATION

G. Dimarco, L. Pareschi, page 18ASYMPTOTIC PRESERVING IMPLICIT-EXPLICIT RUNGE-KUTTA METHODS FOR NON LINEAR KINETIC EQUATIONS

G. Dimarco, J. Narski, R. Loubère, page 21TOWARDS AN ULTRA EFFICIENT KINETIC SCHEME : HIGH-PERFORMANCE COMPUTING

G. Dimarco, R. Loubère, V. Rispoli, page 23A MULTI-SCALE FAST SEMI-LAGRANGIAN METHOD FOR RAREFIED GAS DYNAMICS

G. Dimarco, R. Loubère, M.-H. Vignal, page 24FINITE VOLUMES SCHEMES PRESERVING THE LOW MACH LIMIT FOR EULER SYSTEMS

F. Blachère, R.Turpault, page 26AN ASYMPTOTIC AND ADMISSIBILITY PRESERVING FINITE VOLUME SCHEME FOR SYSTEMS OF CONSERVATIONLAWS WITH SOURCE TERMS ON 2D UNSTRUCTURED MESHES

S. Clain, G.J. Machado, R.M.S. Pereira, A. Boularas, page 29VERY HIGH ORDER FINITE VOLUME SCHEME FOR THE 2D AND 3D LINEAR CONVECTION DIFFUSION EQUATION

T. Gallouet, page 31COMPACTNESS OF SEQUENCES OF APPROXIMATE SOLUTIONS OF PARABOLIC EQUATIONS

A. Bermúdez, S. Busto, M. Cobas, J.L. Ferrín, L. Saavedra, M.E. Vázquez-Cendón, page 33A HIGH ORDER FINITE VOLUME/FINITE ELEMENT PROJECTION METHOD FOR LOW-MACH NUMBER FLOWSWITH TRANSPORT OF SPECIES

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SHARK-FV — April 28 - May 2 2014 — Ofir, Portugal.Sharing Higher-order Advanced Research Know-how on Finite Volumes

COUPLING THE MULTI-DIMENSIONAL OPTIMAL ORDER DETECTION(MOOD) METHOD AND THE ARBITRARY HIGH ORDER DERIVATIVES(ADER) APPROACHES

R. Loubère b∗, M. Dumbser a, S. Diot c

a Laboratory of Applied Mathematics. Department of Civil, Environmental and Mechanical Engineering, University ofTrento, Via Mesiano 77, I-38123 Trento (TN), Italy.b Institut de Mathématiques de Toulouse, Université de Toulouse 31062 Toulouse, France.c Los Alamos National Laboratory, Los Alamos, NM, 87545

ABSTRACT

In this paper, we investigate the coupling of the Multi-dimensional Optimal Order Detection (MOOD)method and the Arbitrary high order DERivatives (ADER) approach in order to design a new high orderaccurate, robust and computationally efficient Finite Volume (FV) scheme dedicated to solve nonlinearsystems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two andthree space dimensions, respectively. The Multi-dimensional Optimal Order Detection (MOOD) methodfor 2D and 3D geometries has been introduced in a recent series of papers [1, 2, 3] for mixed unstruc-tured meshes. It is an arbitrary high-order accurate Finite Volume scheme in space, using polynomialreconstructions with a posteriori detection and polynomial degree decrementing processes to deal withshock waves and other discontinuities. In the following work, the time discretization is performed withan elegant and efficient one-step ADER procedure [4, 5]. Doing so, we retain the good properties of theMOOD scheme, that is to say the optimal high-order of accuracy is reached on smooth solutions, whilespurious oscillations near singularities are prevented. The ADER technique permits not only to reducethe cost of the overall scheme as shown on a set of numerical tests in 2D and 3D, but it also increases thestability of the overall scheme. A systematic comparison between classical unstructured ADER-WENOschemes and the new ADER-MOOD approach has been carried out for high-order schemes in space andtime in terms of cost, robustness, accuracy and efficiency. The main finding of this paper is that the com-bination of ADER with MOOD generally outperforms the one of ADER and WENO either because atgiven accuracy MOOD is less expensive (memory and/or CPU time), or because it is more accurate for agiven grid resolution. A large suite of classical numerical test problems has been solved on unstructuredmeshes for three challenging multi-dimensional systems of conservation laws: the Euler equations ofcompressible gas dynamics, the classical equations of ideal magneto-Hydrodynamics (MHD) and finallythe relativistic MHD equations (RMHD), which constitutes a particularly challenging nonlinear systemof hyperbolic partial differential equation. All tests are run on genuinely unstructured grids composed ofsimplex elements.

ACKNOWLEDGMENTS

M.D. has been financed by the European Research Council (ERC) under the European Union’s SeventhFramework Programme (FP7/2007-2013) with the research project STiMulUs, ERC Grant agreement no. 278267.R.L. has been partially funded by the ANR under the JCJC project “ALE INC(ubator) 3D”. This work hasbeen authorized for publication under the reference LA-UR-13-28795. The authors would like to acknowl-edge PRACE for awarding access to the SuperMUC supercomputer based in Munich, Germany at the LeibnizRechenzentrum (LRZ).

∗Correspondence to [email protected]

5

REFERENCES

[1] S. Clain, S. Diot, and R. Loubère. A high-order finite volume method for systems of conservation laws-multi-dimensional optimal order detection (MOOD). Journal of Computational Physics, 230(10):4028 –4050, 2011.

[2] S. Diot, S. Clain, and R. Loubère. Improved detection criteria for the multi-dimensional optimal orderdetection (MOOD) on unstructured meshes with very high-order polynomials. Computers and Fluids,64:43 – 63, 2012.

[3] S. Diot, R. Loubère, and S. Clain. The MOOD method in the three-dimensional case: Very-high-orderfinite volume method for hyperbolic systems. International Journal of Numerical Methods in Fluids,73:362–392, 2013.

[4] M. Dumbser. Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier–Stokes equations. Computers & Fluids, 39:60–76, 2010.

[5] M. Dumbser, M. Castro, C. Parés, and E.F. Toro. ADER schemes on unstructured meshes for non-conservative hyperbolic systems: Applications to geophysical flows. Computers and Fluids, 38:1731 –1748, 2009.

6


A NON-CONSERVATIVE FLUX APPROACH ENSURING THE C-PROPERTYFOR THE LAKE AT REST IN THE FRAMEWORK OF A MOOD BASED 2DSHALLOW-WATER SIXTH-ORDER FINITE VOLUME SCHEME

J. Figueiredo a∗, S. Clain a, C. Ribeiro a

a Departamento de Matemática e Aplicações and Centro de Matemática, Universidade do Minho, Campus de Gualtar -4710-057 Braga, Portugal.

ABSTRACT

We solve the 2D shallow-water problem with bathymetric source term using a finite volume schemethat combines a Polynomial Reconstruction Operator (PRO) with a Multi-dimensional Optimal Detection(MOOD). This numerical scheme allows to obtain high-order of accuracy (up to sixth-order) for smoothsolutions. This is the first use of the PRO-MOOD technique in a non-conservative problem. Still, arelatively basic property such as the C-property for the lake at rest is not guaranteed when there arebathymetry variations, unless a specific approach is used. This is a very important issue when simulating,for instance, Tsunami wave propagation with varying bathymetry, since one wants to avoid the appearanceand development of small oscillations in the sea regions as long as they remain unperturbed by true waves.We show that the lake at rest can be reproduced exactly using a non-conservative flux together with theclassic conservative one independently of the overall scheme order. Furthermore, we show that this fluxscheme can be applied as long as the conservative flux being used admits a so-called physical bathymetryrepresentative (a new concept introduced specifically for this purpose), which is the case, for instance, ofthe Rusanov, HLL, and HLLC fluxes, even under relatively stiff bathymetries.


7


HIGH ORDER ONE-STEP AMR AND ALE FINITE VOLUME METHODSFOR HYPERBOLIC PDE

M. Dumbser b∗,a Laboratory of Applied Mathematics. Department of Civil, Environmental and Mechanical Engineering, University ofTrento, Via Mesiano 77, I-38123 Trento (TN), Italy.

ABSTRACT

In this talk we present a unified family of high order accurate finite volume and discontinuous Galerkinfinite element schemes on moving unstructured and adaptive Cartesian meshes for the solution of conser-vative and non-conservative hyperbolic partial differential equations.The PNPM approach adopted here uses piecewise polynomials uh of degree N to represent the data ineach cell. For the computation of fluxes and source terms, another set of piecewise polynomials wh ofdegree M ≥ N is used, which is computed from the underlying polynomials uh using a reconstructionor recovery operator. The PNPM method contains classical high order finite volume schemes (N = 0)and high order discontinuous Galerkin (DG) finite element methods (N = M) as two special cases of amore general class of numerical schemes. The schemes are derived in general ALE form so that Eulerianschemes on fixed meshes and Lagrangian schemes on moving meshes can be recovered as special cases ofthe ALE formulation. Furthermore, the method can also be naturally implemented on space-time adaptiveCartesian grids (AMR), together with time-accurate local time stepping (LTS). To assure the robustnessof the method at discontinuities, a nonlinear WENO reconstruction is performed. The time integration iscarried out in one single step using a high order accurate local space-time Galerkin predictor that is alsoable to deal with stiff source terms.Applications are shown for the compressible Euler and Navier-Stokes equations, for the MHD equationsand for the Baer-Nunziato model of compressible multi-phase flows.

REFERENCES

[1] M. Dumbser, A. Uriuuntsetseg, O. Zanotti. On ALE-Type One-Step WENO Finite Volume Schemes forStiff Hyperbolic Balance Laws. Communications in Computational Physics, 14:301–327, 2013.

[2] W. Boscheri and M. Dumbser. Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemeson Unstructured Triangular Meshes. Communications in Computational Physics, 14:1174–1206, 2013.

[3] M. Dumbser and W. Boscheri. High-Order Unstructured Lagrangian One-Step WENO Finite VolumeSchemes for Non-Conservative Hyperbolic Systems: Applications to Compressible Multi-Phase Flows.Computers and Fluids, 86:405–432, 2013.

[4] M. Dumbser, O. Zanotti, A. Hidalgo and D.S. Balsara. ADER-WENO Finite Volume Schemes with Space-Time Adaptive Mesh Refinement. Journal of Computational Physics, 248:257–286, 2013.

[5] M. Dumbser, A. Hidalgo and O. Zanotti. High Order Space-Time Adaptive ADER–WENO Finite Vol-ume Schemes for Non-Conservative Hyperbolic Systems. Computer Methods in Applied Mechanics andEngineering, 268:359–387, 2014.


9


RECENT DEVELOPMENTS ON HIGH-ORDER ADER FINITE VOLUMESCHEMES

Eleuterio Toro a

a University of Trento, Trento , Italy

ABSTRACT

First I briefly review the ADER approach for solving hyperbolic balance laws to arbitrary order of accu-racy in both space and time. Then I present a new, locally implicit solver for the Generalized RiemannProblem, the building block of ADER methods. The resulting explicit ADER schemes are then capableof dealing with balance laws with stiff source terms. Finally I describe the application of the new versionof ADER to solve general advection-diffusion reaction equations, that may also include source terms,reformulated as hyperbolic systems of balance laws via a relaxation procedure. Numerical examples areshown.

11


A VERY-HIGH-ORDER SHARP INTERFACE METHOD TO SIMULATEMULTI-MATERIAL FLOWS BASED ON THE MOOD CONCEPTS

S. Diot a∗, M.M. François a, E.D. Dendy b

a Fluid Dynamics and Solid Mechanics (T-3), Los Alamos National Laboratory, NM 87545, USA.b Computational Physics and Methods (CCS-2), Los Alamos National Laboratory, NM 87545, USA.

ABSTRACT

Simulations of multi-material compressible flows are of crucial interest for industrial and fundamentalresearches as many physical complex phenomena are driven by such hydrodynamics. Let us cite theexplosion of a star, transport in gas carrier or Inertial Confinement Fusion (ICF) for instance. The devel-opment of more robust and efficient methods for these complex problems is challenging but necessary toshorten the computational time to get the solution and therefore improve the current prediction capabili-ties. Aiming at this objective, I will present an original direct Eulerian Volume-Of-Fluid (VOF) method onunstructured meshes that uses ideas that are analogous to the ones in [3] in order to get a better approxima-tion of the volumes that are fluxed through the faces. This scheme better approximates the compressibilityof the velocity field and therefore allows to limit the mass exchange occurring at the material interface.The approach is validated for a purely advective problem of several materials on unstructured meshes.Comparisons to exact solutions will show that the approach reduces the errors, in particular when the flowis incompressible. Then we will apply this method to a reduced model for compressible material-fluidflows with single velocity and instantaneous pressure equilibrium. We will emphasize the advantagesof this method and point out the gain in accuracy obtained on classical test cases. In addition, we willstudy the efficiency of the proposed method by monitoring the ratio between errors and computationaltimes. Finally, preliminary results of the coupling between this method and the recently developed Mul-tidimensional Optimal Order Detection (MOOD) method [1, 2] will be provided. This shall demonstratethat designing a higher-order method through the MOOD concepts allows to reach optimal order whileensuring the method robustness.

ACKNOWLEDGMENTThis work was performed under the auspices of the National Nuclear Security Administration of the US

Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396 andsupported by the DOE Advanced Simulation and Computing (ASC) program. Approved for public releaseunder LA-UR-13-29379.

REFERENCES[1] S. Diot, S. Clain, R. Loubère, Improved detection criteria for the Multi-dimensional Optimal Order De-

tection (MOOD) on unstructured meshes with very high-order polynomials, Comput. Fluids 64 (2012)43–63.

[2] S. Diot, R. Loubère, S. Clain, The Multidimensional Optimal Order Detection method in the three-dimensional case: very high-order finite volume method for hyperbolic systems, Int. J. Numer. Meth.Fluids 73 (2013) 362–392.

[3] J. López, H. Hernández, P. Gómez, F. Faura, A volume of fluid method based on multidimensional advec-tion and spline interface reconstruction, J. Comp. Phys. 195 (2004) 718–742.


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FINITE VOLUME SCHEME BASED ON CELL-VERTEXRECONSTRUCTIONS FOR ANISOTROPIC DIFFUSION PROBLEMS WITHDISCONTINUOUS COEFFICIENTS

R. Costa a∗, S. Clain a,b, G. J. Machado a

a Departamento de Matemática e Aplicações e Centro de Matemática, Campus de Gualtar - 4710-057 Braga, Portugal.b Institut de Mathématiques de Toulouse, Université de Toulouse 31062 Toulouse, France.

ABSTRACT

We propose a new second-order finite volume scheme for non-homogeneous and anisotropic diffusionproblems based on cell to vertex reconstructions involving minimization of functionals to provide thecoefficients of the cell to vertex mapping. The method handles complex situations such as large precondi-tioning number diffusion matrices and very distorted meshes. Numerical examples are provided to showthe effectiveness of the method.

INTRODUCTION

The design of second-order schemes is still a challenging and important question. In fact, very high-ordermethods are rather complicated and require an important implementation effort whereas second-order meth-ods are quite simple and easy to code. A popular class of second-order finite volume schemes is based onvertex reconstructions using point-wise approximations on cells associated to a specific point location (usuallythe centroid). Then combining cell and vertex values, gradient approximations are evaluated to compute thediffusive flux on the interfaces.

In [2] we consider the homogeneous and isotropic situation where we evaluate the vertex values ψn fromthe cell values φi based on a simple linear combination to compute the coefficients βni. In the present study weextend the previous method to non-homogeneous and anisotropic problems.

FORMULATION

Let Ω be an open bounded polygonal domain of R2 with boundary ∂Ω. We split Ω into two non-overlappingsubdomains Ω1 and Ω2 sharing a common interface Γ where the diffusion tensor K is discontinuous with K =K1in Ω1 and K = K2 in Ω2. We seek function φ = φ1 in Ω1 and φ = φ2 in Ω2, solution of the anisotropic steady-state diffusion equations

∇ · (−K1∇φ1) = f1, in Ω1, ∇ · (−K2∇φ2) = f2, in Ω2,

where the source terms f1 and f2 are regular functions on Ω1 and Ω2, respectively. We prescribe the continuityboth for the flux and the function, namely K1∇φ1 ·n = K2∇φ2 ·n and φ1 = φ2 on Γ, the Dirichlet condition on∂Ω = ΓD and the Neumann condition on ∂Ω = ΓN.

CELL-VERTEX MAPPING

The generic finite volume discretization for each cell ci, i = 1, . . . , I cast in the residual form writes

Gi = ∑j∈ν(i)

|ei j||ci|

Fi j− fi,


15

where Fi j is an approximation of the diffusive flux through the edge ei j and fi is an approximation of the meanvalue of f over the cell ci.

Let φi, i = 1, . . . , I, be an approximation of φ on the mass centre qi of cell ci, and let ψn, n = 1, . . . ,N, bean approximation of φ on vertex vn. The goal is the design of a procedure to compute ψn from the neighbourpoint-wise cell values. We define ψn as

ψn = ∑i∈µ(n)

βniφi,

where µ(n) is the set of the indices of the neighbour cells. Let gather in vector Bn = (βni)i∈µ(n) the coefficientsof the linear combination of the cell data. We seek Bn such that Λ1(Bn) = 1, Λ2(Bn) = 0, and Λ3(Bn) = 0 where

Λ1(Bn) = ∑i∈µ(n)

βni, Λ2(Bn) = ∑i∈µ(n)

βni(qix− vnx), Λ3(Bn) = ∑i∈µ(n)

βni(qiy− vny),

in order to preserver first degree polynomials. To deal with situations where the solution is not unique, whichhappens when there are more than three cells in µ(n), we introduce the quadratic functional

E(Bn) =12 ∑

i∈µ(n)ωni(βni−θni)

2,

where θni are the target values such that ∑i∈µ(n) θni = 1, and ωni are positive weights. We then solve the problemwith the classical minimization with Lagrange multipliers where Λ1, Λ2 and Λ3 are the constraints.

At last, we compute the flux approximation Fi j based on a polynomial which interpolates the vertex valuesof ei j and the cell values of ci and c j. We compute the source term approximation fi based on the cell value φi

and on the vertex values of ψn of each vn ∈ ci.

NUMERICAL RESULTS

We briefly present a simulation where we consider the domain Ω = ]0,1[2 and the discontinuous diffusiontensor with

K1 =

[1 00 1

], K2 =

[100 00 0.01

].

The source terms are given by f1(x,y) = 2π2 cos(πx)sin(πy) on Ω1 =]0,0.5[×]0,1[ and by f2(x,y) = (1+0.012)cos(πx)sin(πy) on Ω2 =]0.5,1[×]0,1[. We compute the numerical solution using a triangular Delaunaymesh (see an example of the mesh in Fig.1, left) and we represent the result in Fig. 1 (right).

x

0.25

0.50.5

0

0.750.75

1

1

x

1

0

y

0.25 0.50.50

0.750.25 1

y

0.25

00.75 0 0.25 0.5 0.75 1

y

0

0.25

0.5

0.75

1

x

0 0.25 0.5 0.75 1

y

0

0.25

0.5

0.75

1

x

-0.01 0.495 1-0.01 0.495 1

FIGURE 1: Delaunay Mesh (left) and numerical solution on a fine mesh (right).

REFERENCES

[1] Clain, S., Machado, G. J., Nóbrega, J. M., Pereira, R. M. S.: A sixth-order finite volume method for theconvection-diffusion problem with discontinuous coefficients, Computer Methods in Applied Mechanicsand Engineering 267, 43–64 (2013).

[2] Costa, R., Clain, S., Machado, G. J.: New cell-vertex reconstruction for finite volume scheme: applicationto the convection-diffusion-reaction equation, under review.

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A HIGH-ORDER FINITE VOLUME SCHEME FOR THE TIME-DEPENDENTCONVECTION-DIFFUSION EQUATION

S. Clain a,b, G.J. Machado a∗

a Centre of Mathematics, University of Minho, Campus de Azurém, Guimarães, Portugal.b Institut de Mathématiques de Toulouse,Université de Toulouse, 31062 Toulouse, France.

ABSTRACT

The time discretization of a very high-order finite volume method may give rise to new numerical dif-ficulties resulting into accuracy degradations. Indeed, for the simple one-dimensional time-dependentconvection-diffusion equation for instance, a conflicting situation between the source term time discretiza-tion and the boundary conditions may arise when using the standard Runge-Kutta method. We propose analternative procedure by extending the Butcher Tableau to overcome this specific difficulty and achievefourth-, sixth- or eighth-order of accuracy schemes in space and time. To this end, a new finite volumemethod is designed based on specific polynomial reconstructions for the space discretization, while weuse the Extended Butcher Tableau to perform the time discretization. A large set of numerical tests hasbeen carried out to validate the proposed method (see [1] and [2]).

REFERENCES

[1] S. Clain, G.J. Machado, J.M. Nobrega, R.M.S. Pereira. A sixth-order finite volume method for multidomainconvection-diffusion problem with discontinuous coefficients. Computer Methods in Applied Mechanicsand Engineering, 267, 43–64, 2013.

[2] S. Clain, G.J. Machado. A very high-order finite volume method for the time-dependent convection-diffusion problem with Butcher tableau extension (submitted).


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ASYMPTOTIC PRESERVING IMPLICIT-EXPLICIT RUNGE-KUTTAMETHODS FOR NON LINEAR KINETIC EQUATIONS.

G. Dimarco b∗, L. Pareschi a,b

a Department of Mathematics and Computer Science, University of Ferrara, Italy,b Department of Mathematics and Computer Science, University of Ferrara, Italy &Institut de Mathématiques de Toulouse, Université de Toulouse 31062 Toulouse, France.

ABSTRACT

In this work, we discuss Implicit-Explicit (IMEX) Runge Kutta methods which are particularly adapted tostiff kinetic equations of Boltzmann type. We consider both the case of easy invertible collision operatorsand the challenging case of Boltzmann collision operators. We give sufficient conditions in order thatsuch methods are asymptotic preserving and asymptotically accurate. Their monotonicity properties arealso studied. In the case of the Boltzmann operator, the methods are based on the introduction of apenalization technique for the collision integral. This reformulation of the collision operator permits toconstruct penalized IMEX schemes which work uniformly for a wide range of relaxation times avoidingthe expensive implicit resolution of the collision operator. Finally we show some numerical results whichconfirm the theoretical analysis.

INTRODUCTION

The numerical solution of Boltzmann-type equations close to fluid regimes represents a real challenge fornumerical methods. In these regimes, in fact, the intermolecular collision rate grows exponentially and thecollisional time becomes very small. On the other hand, the actual time scale for evolution is the fluid dynamictime scale, which can be much larger than the collisional time. A non dimensional measure of the importanceof collision is given by the Knudsen number which is large in the rarefied regions and small in the fluid ones.Standard computational approaches lose their efficiency due to the necessity of using very small time steps indeterministic schemes or, equivalently, a large number of collisions in probabilistic approaches. Unfortunatelythe use of implicit solvers originates a prohibitive computational cost due to the high dimensionality and thenonlinearity of the collision operator.

In this talk, we develop Implicit-Explicit Runge-Kutta methods [1, 2] which are particularly efficient for stiffnon linear kinetic equations. First we consider the case where the implicit inversion of the collision term doesnot represent a problem, like for example the case of simple BGK operators. Asymptotic preservation propertiesand monotonicity are carefully studied and analyzed. Subsequently we deal with the challenging case of thefull Boltzmann equation. To this aim, we introduce a penalization strategy based on a decomposition of thegain term of the collision operator into an equilibrium and a non equilibrium part. This permits to derive newpenalized IMEX schemes which keep the good asymptotic preservation properties of standard IMEX schemesby avoiding the costly inversion of the collision term.

The methods studied are based on the following decomposition

R(Y ) = N(Y )+L(Y ), (1)

where N(Y ) represents the non-dissipative non-linear part and L(Y ) is a linear term such that L(Y ) = 0 impliesY = E(y). For example L(Y ) = A(E(y)−Y ) where A > 0 is an estimate of the Jacobian of R evaluated at


19

equilibrium. Note that, at variance with standard linearization techniques which operate on the short time scale,the operator is linearized on the asymptotically large time scale. This decomposition permits to apply IMEXtechniques which are implicit in the linear part and explicit in the non-linear part. The use of such techniques,as we will see, permits to achieve unconditionally stable and asymptotic preserving methods at the cost of anexplicit scheme.

REFERENCES

[1] G. Dimarco and L. Pareschi. Asymptotic preserving implicit-explicit Runge-Kutta methods for non linearkinetic equations. SIAM Journal of Numerical Analysis, Vol. 51, pp. 1064-1087 (2013)

[2] L. Pareschi and G. Russo Implicit-Explicit Runge-Kutta methods and applications to hyperbolic systemswith relaxation. J. Sci. Comput., 25 (2005), 129–155.

20


TOWARDS AN ULTRA EFFICIENT KINETIC SCHEME :HIGH-PERFORMANCE COMPUTING

G. Dimarco a, J. Narski b∗, R. Loubère b

a Department of Mathematics and Computer Science. University of Ferrara, Italy.b Institut de Mathématiques de Toulouse, Université de Toulouse 31062 Toulouse, France.

ABSTRACT

In this paper we present the new 3D/3D fast kinetic scheme developed in [1, 2] in its parallel version onclassical architecture using OPEN-MP and on GPU architecture using CUDA. The goal is to prove thatthis new scheme is well adapted to any type of parallelisation, and that the gain in CPU time is substantialon nowadays affordable computer. We briefly present the sequential version of our kinetic scheme andfocus on important details for a parallel implementation. Numerical tests are shown for the full 3D/3Dsimulations. These assess the very interesting speed-up factor gain between the sequential code and theparallel version.

INTRODUCTION

In many applications, the correct physical solution of systems far from thermodynamic equilibrium (rarifiedgases, plasma) require resolution of kinetic equations. These simulations are typically very resource consumingdue to the large dimension of the problem. Indeed, the distribution function depends on seven independentvariables: three space coordinates, three coordinates in the velocity space and the time.

Dimarco and Loubère have recently proposed in [1, 2] a method based on a splitting technique called FastKinetic Solver (FKS). The method relies on a Lagrangian transport scheme and a discretization of the velocityspace in the framework of so-called Discrete Velocity Models (DVM), where the velocity is discretized into aset of fixed velocities. The original equation is replaced by a set of linear transport equations plus a couplinginteraction term corresponding to the collision operator. This allowed to drastically reduce the computationalcost of the transport part. The FKS scheme was shown to be capable of solving the full six dimensional problem(3D in space + 3D in velocity) on a single laptop on reasonable meshes (1003 in space, 123 in velocity).Although the simulations were possible, the computational time remained long. The purpose of this work is toexplore a possibility of parallelization of the FKS method in order to reduce the computational time on multicore shared memory system and/or GPU.

PROBLEM FORMULATION

In this work, the following Boltzman-BGK equation is considered

∂t f +VVV ·∇XXX f =1τ(M f − f ), (1)

describing the distribution f = f (XXX ,VVV , t) of particles at position XXX ∈ Ω ⊂ Rdx , at time t > 0 and which movewith velocity VVV ∈ Rdv . We consider the general case in which we have dx = dv = d = 3 dimensions in spaceand velocity. The interaction operator is the BGK operator: collisions are modeled by a relaxation towards the


21

FIGURE 1: Kelvin Helmholtz instability obtained with GPU simulations at time steps: 4000, 6000 and 8000.

local thermodynamical equilibrium defined by the Maxwellian distribution function M f

M f = M f [ρ,UUU ,T ] (VVV ) =ρ

(2πθ)d/2 exp(−‖UUU−VVV‖2

2RT

), (2)

where ρ ∈ R and UUU ∈ Rdv are the density and mean velocity, T is the temperature and R the gas constant.The BGK equation is first discretized in the velocity space. The discrete velocity BGK model consists then

of a set of N evolution equations in the velocity space for fk of the form

∂t fk +VVV k ·∇XXX fk =1τ(Ek[F ]− fk), (3)

where Ek[F ] is a suitable approximation of M f and Nv is the total number of velocity grid points.Next, the splitting technique is applied to every equation of (3). First the transport step is solved exactly,

providing initial data for the relaxation step:

Transport stage−→ ∂t fk +VVV k ·∇XXX fk = 0, (4)

Relaxation stage−→ ∂t fk =1τ(Ek[F ]− fk). (5)

The equations a re finally discretized in space on a uniform Cartesian grid consisting of Ns points.

Parallelization

The main computational difficulty is related to the resolution of the relaxation stage, as it involves evaluationof exponential functions. Fortunately, this stage can be easily decoupled into Ns×Nv separate equations, each ofthem solved separately in parallel. Indeed, (5) involves no interaction between fk and fi for k 6= i and the spacecells are independent. The density, velocity and temperature used for computation of Ek[F ] can be precalculatedfor every space cell prior to the relaxation stage. The parallelization can be done in a following way :

1. Transport of particles. Move in parallel Nv particles.

2. Relaxation step. Perform in parallel the relaxation step for Nv×Ns particles, parallelization is performedon the external loop over Nv particles.

3. Update primitive variables. Update the density, mean velocity and temperature for every space cell.

In the numerical experiments performed this method showed a speed-up close to perfect on shared memorysystems with OMP interface. GPU parallelization allowed to obtain Kelvin Helmholtz instability on a 2003

space and 103 velocity grids in less then 32 hours — compared to 200 DAYS on a sequential machine.

REFERENCES

[1] G. DIMARCO, R. LOUBÈRE, Towards an ultra efficient kinetic scheme. Part I: basics on the BGK equa-tion, J. Comput. Phys., Vol. 255, 2013, pp 680-698.

[2] G. DIMARCO, R. LOUBÈRE, Towards an ultra efficient kinetic scheme. Part II: the high order case, J.Comput. Phys., Vol. 255, 2013, pp 699-719.

22


A MULTI-SCALE FAST SEMI-LAGRANGIAN METHOD FOR RAREFIEDGAS DYNAMICS

G. Dimarco a, R. Loubère b, V. Rispoli b∗

a Department of Mathematics and Computer Science, University of Ferrara, 44100 Ferrara, Italy.b Institut de Mathématiques de Toulouse, Université de Toulouse, 31062 Toulouse, France.

ABSTRACT

In this talk we present an extension of the method developed in [1, 2] aiming at the numerical resolutionof multi-scale problems arising in rarefied gas dynamics. The scope of this work is to consider situationsin which the whole domain does not demand the use of a kinetic model everywhere. In many realisticapplications some regions of the domain require a microscopic description, given by kinetic equations,while the rest of the domain can be treated with a coarser model of fluid type. Our aim is to showthat the kinetic scheme developed in previously cited articles is perfectly suited for building domaindecomposition strategies. Exploiting the latter, the numerical scheme’s efficiency greatly increases andit becomes very attractive with respect to classical numerical techniques for kinetic equations and multi-scale realistic problems. Several numerical results in the two and three dimensional settings are presented,which were obtained saving computational resources (CPU, memory and time). The presented work isthe object of a recent in-review paper [3].

REFERENCES

[1] G. Dimarco and R. Loubère. Towards an ultra efficient kinetic scheme. Part I: basics on the BGK equation.Journal of Computational Physics, vol. 255, pp. 680–698 (2013)

[2] G. Dimarco and R. Loubère. Towards an ultra efficient kinetic scheme. Part II: the high order case. Journalof Computational Physics, Vol. 255, pp. 699-719 (2013)

[3] G. Dimarco, R. Loubère and V. Rispoli, A multiscale fast semi-Lagrangian method for rarefied gas dy-namics. Submitted to Journal of Computational Physics.


23


FINITE VOLUMES SCHEMES PRESERVING THE LOW MACH LIMIT FOREULER SYSTEMS

G. Dimarco a,b, R. Loubère c, M.-H. Vignal b

a Mathematics Institute of Toulouse, University Toulouse 3, France.b Mathematics & Computer Science Department, University of Ferrara, Italy.c Mathematics Institute of Toulouse, CNRS, France.

ABSTRACT

I am interested in the so-called Asymptotic preserving schemes. These schemes are well known to bewell adapted for the resolution of multiscale problems in which several regimes are present.I will present the particular case of the low-Mach limit for Euler systems used in gas or fluid dynamics.The square of the Mach number is the ratio of the kinetic and thermic energies of the fluid. When thisratio tends to zero, the pressure waves are very fast and this yields the fluid incompressible.When a standard explicit finite volume scheme is used, it is well known that its time step is constrainedby the C.F.L. (Courant-Friedrichs, Levy) condition. In the low-Mach regime, this leads to time stepsinversely proportional to the pressure waves velocity which is very large. Thus, explicit schemes sufferfrom a severe numerical constraint in low-Mach regimes. Furthermore, these schemes are not consistentin this regime. This means that they do not capture the incompressible limit.Then, it is necessary to develop new schemes for bypassing these limitations. These new schemes mustbe stable and consistent in all regimes: from low Mach numbers to order one Mach numbersI will show how to construct such a scheme for Euler systems and I will present numerical results showingthe good behavior of these schemes in all regimes.

25


AN ASYMPTOTIC AND ADMISSIBILITY PRESERVING FINITE VOLUMESCHEME FOR SYSTEMS OF CONSERVATION LAWS WITH SOURCETERMS ON 2D UNSTRUCTURED MESHES

F. Blachère a∗, R. Turpault a

aLaboratoire de Mathématiques Jean Leray 2, rue de la Houssinière - BP 92208 44322 Nantes, France.

ABSTRACT

We introduce a new finite volume technique to obtain numerical schemes which preserve both asymp-totic and the set of admissible states on 2D unstructured meshes for the class of hyperbolic systems ofconservation laws with source terms described in [1].

The objective of this work is to design a suitable finite volume scheme to approximate the solutions ofhyperbolic systems of conservation laws with source terms which could be written as:

∂tU+div F(U) =−γ(U)R(U), (1)

where U∈RN , F is a smooth function whose Jacobian has real eigenvalues, γ ≥ 0 and R is a smooth functionwhich fulfills the compatibility conditions stated in [1].

The main characteristic of such systems is to degenerate when γ t→∞ into smaller parabolic systems of theform:

∂tu−div(M (u)∇u

)= 0, (2)

where u ∈ Rn is related to U and M is a positive function or a symmetric, positive and definite matrix.Examples of such systems include Euler equations with friction, the M1-model for radiative transfer, and

the telegraph equations.From the numerical point of view, the main difficulty is to construct asymptotic preserving schemes, i.e.

schemes that degenerate into consistent schemes for the diffusion equation (2). This question has been a majorissue during the last decade and several efficient asymptotic preserving schemes, based on different techniques,exist in 1D. In 2D, however, the problem is much more difficult. One of the reasons is that the target scheme inthe diffusion limit is often the classical two-point flux (a.k.a. FV4), which is not consistent anymore on generalmeshes.

Up to now, only two examples of such suitable 2D schemes exist for non admissible meshes: the MPFA-based scheme developed in [2] and the one constructed from the diamond scheme in [4]. Nevertheless, in bothcases, it is not possible to ensure the preservation of the set of admissible states on general meshes. Obviously,this property is critical in several configurations.

To overcome these difficulties, we propose a numerical procedure based on the technique developed byDroniou and Le Potier in [3] to obtain positivity-preserving discretization of the gradient for parabolic orelliptic equations. This technique serves as a target for an extension of the asymptotic-preserving proceduregiven in [1]. Thanks to it we are able to establish the preservation of the asymptotic and the set of admissiblestates under a classical hyperbolic-type CFL condition.


27

Theorem 1. Let us consider the Rusanov-like following numerical flux for the i-th edge of the mesh:

Fi.ni = Fi.ni−biθi

2∇iU.ni,

where:

• ∇iU.ni is Droniou and Le Potier’s approximation of the normal gradient of U on the i-th edge,

• Fi.ni = ∑j

λi, jF(UK)+F(U j)

2 .n j is a consistent approximation of F(U).ni such that λi, j ≥ 0,

• bi is greater than all waves speeds on the i-th edge, and θi = θi(U)≥ 0 depends on the mesh.

Using this flux along with the technique introduced in [1] leads to a numerical scheme which:

• degenerates into Droniou and Le Potier scheme for (2) in the asymptotic regime,

• preserves the set of admissible states under the following CFL condition:

maxj,K

(η j

∆tδK

)≤ 1

2,

where δK is a characteristic length of the cell K and η j = η j(bi,θi,λi, j).

Finally, high-order extensions may be considered using classical techniques, for instance MOOD or MUSCL,due to the form of the scheme.

REFERENCES

[1] C. Berthon, P. Le Floch, and R. Turpault. Late-time/stiff relaxation asymptotic-preserving approximationsof hyperbolic equations. Math. Comp. 82, pp 831-860, 2013.

[2] C. Buet, B. Després and E. Franck. Design of asymptotic preserving finite volume schemes for the hyper-bolic heat equation on unstructured meshes. Numer. Math., 122(2) pp 227-278, 2012.

[3] J. Droniou and C. Le Potier. Construction and convergence study of schemes preserving the elliptic localmaximum principle. SIAM J. Numer. Anal., 49(2) pp 459-490, 2011.

[4] C. Sarazin-Desbois. Méthodes numériques pour des systèmes hyperboliques avec terme source provenantde physiques complexes autour du rayonnement. Thèse de doctorat de l’Université de Nantes, 2013.

28


VERY HIGH ORDER FINITE VOLUME SCHEME FOR THE 2D AND 3DLINEAR CONVECTION DIFFUSION EQUATION

S. Clain a,b∗, G.J. Machado a, R.M.S. Pereira a, A. Boularas c

a Departamento de Matemática e Aplicações e Centro de Matemática, Campus de Gualtar - 4710-057 Braga, Portugal.b Institut de Mathématiques de Toulouse, Université de Toulouse 31062 Toulouse, France. c Laplace Centre, PaulSabatier University, 31062 Toulouse, France

ABSTRACT

A sixth-order finite volume method is proposed to solve the bidimensional linear steady-state convection-diffusion equation. A new class of polynomial reconstructions is proposed to provide accurate fluxes forthe convective and the diffusive operators. The method is also designed to compute accurate approx-imations even with discontinuous diffusion coefficient or velocity and remains robust for large Pécletnumbers. Discontinuous solutions deriving from the linear heat transfer Newton law are also consideredwhere a decomposition domain technique is applied to maintain an effective sixth-order approximation.We also propose a new technique to accuratly take a curved boundary into account maintaining the sixth-order of convergence and the method is extended to the three-dimensional case. For more details see[1, 2]

FRAMEWORK

Let Ω be a bounded polygonal domain of R2 or R3 and ∂Ω its boundary. We consider situations wherethe diffusion and the convection functions may comprise discontinuities, so domain Ω is partitioned in twosubdomains Ω1 and Ω2 sharing a common interface Γ:

∇.(V1φ1− k1∇φ1) = f1, in Ω1, (1a)

∇.(V2φ2− k2∇φ2) = f2, in Ω2, (1b)

k1∇φ1.nΓ = k2∇φ2.nΓ, on Γ, (1c)

φ1 = φ2 on Γ, (1d)

φ = φD, on ΓD, (1e)

− k∇φ .n = gP, on ΓP, (1f)

V.nφ − k∇φ .n = gT , on ΓT , (1g)

with adequate boundary conditions.Applying the generic finite volume procedure and using the Gauss quadrature formula with R points on theedges the sixth-order approximation writes

Gi = ∑j∈ν(i)

|ei j||ci|

R

∑r=1

ζrFi j,r− fi,

where Fi j,r stands for the numerical flux approximation of the physical flux function[V (qi j,r).n(qi j,r)φ(qi j,r)−

k(qi j,r)∇φ(qi j,r).n(qi j,r)]

evaluated at the Gauss point qi j,r.


29

MAIN ISSUES

The sixth-order finite volume method we propose is based on the following ingredients.

• We introduce a new class of polynomial reconstructions associated to cells for the convective part oredges/faces for the diffusive contribution. Conservative and non conservative reconstructions will beproposed based on a fuctional minimization where a weighted strategy provides the positivity preservingproperty for pure diffusive problems.

• We include a new technique to prescribe boundary conditions for curved domains up to the sixth-orderdeeply based on the reconstruction procedure we have adopted.

• Another issue concerns the treatement of discontinuous coefficients and even discontinuous solutionsstill preserving a sixth-order accuracy.

• The method is matrix free and the linear system is solved using a preconditionning GMRES method.We present a new and simple preconditionning matrix based on the low-order "Patankar" discretizationand a pseudo inverse procedure which dramatically reduces the number of iterations and save o lot ofcomputational resources.

NUMERICAL RESULTS

Numerical tests to assess the scheme capacity to handle classical situations have been proposed such as lowand large Péclet number problems, pure convective problem, discontinuous situations both for the coefficientsand the solutions. We also propose a concrete simulation deriving from polymer flow simulation.

FIGURE 1: Numerical simulation of a cooler with discontinuous coefficients

REFERENCES

[1] S. Clain, G. Machado, J. M. Nóbrega, R. Pereira, A sixth-order finite volume method for multidomainconvection-diffusion problem with discontinuous coefficients. Computer Methods in Applied Mechanicsand Engineering, 267, (2013), 43–64.

[2] A. Boularas, S. Clain, F. Baudoin, A sixth-order finite volume method for diffusion problem with curvedboundaries. Under preparation.

30


COMPACTNESS OF SEQUENCES OF APPROXIMATE SOLUTIONS OFPARABOLIC EQUATIONS

T. Gallouët a∗,a I2M, Université d’Aix-Marseille, 13453 Marseille cedex 13, France.

ABSTRACT

We present some compactness results useful for proving the existence of solutions for some parabolicequations by passing to the limit on approximate solutions. A main example is the case where theseapproximate solutions are obtained with Finite Volume schemes.

COMPACTNESS RESULTS

The main compactness results for proving existence of a solution for some nonlinear parabolic PDE aredue to J. L. Lions (in an hilbertian framework), J. P. Aubin (in the case of Lp-spaces, 1 < p < +∞) and J.Simon (in the case of L1-spaces). These results are proven with the compactness theorem of Kolmogorov (forLp-functions taking values in an Hilbert or Banach space).

When the parabolic PDE is discretized by a numerical scheme (with a space-time discretization), in order toprove the convergence of the approximate solution to the exact solution (as the discretization parameters tend to0), a possibility is to use a discrete version of these compactness results, using a family of approximate spaces.For some problems (as, for instance, the diffusion equation with the p-laplacian) it is sufficient to use a discreteversion of the so-called Aubin-Simon theorem. For some other cases, such as the Stefan problem, we have touse a discrete version of the Kolmogorov theorem itself.

We first give a version of the Kolmogorov theorem using a sequence of subspaces of a Banach space B. LetB be a Banach space and (Xn)n∈N be a sequence of Banach spaces included in B. We say that the sequence(Xn)n∈N is compactly embedded in B if all sequence (un)n∈N such that un ∈ Xn (for all n ∈ N) and (‖un‖Xn)n∈Nbounded is relatively compact in B.

Time compactness with a sequence of subspaces. Let 1≤ p <+∞, T > 0. Let B be a Banach space and(Xn)n∈N be a sequence of Banach spaces compactly embedded in B. Let ( fn)n∈N be a sequence of Lp((0,T ),B)satifying the following conditions• The sequence ( fn)n∈N is bounded in Lp((0,T ),B).• The sequence (‖ fn‖L1((0,T ),Xn))n∈N is bounded.• There exists a nondecreasing function η from (0,T ) to R+ such that limh→0+ η(h) = 0 and, for all h ∈

(0,T ) and n ∈ N,∫ T−h

0 ‖ fn(t +h)− fn(t)‖pBdt ≤ η(h).

Then, the sequence ( fn)n∈N is relatively compact in Lp((0,T ),B).

We now give a version of the Aubin-Simon theorem using a sequence of subspaces of B. Let B be a Banachspace, (Xn)n∈N be a sequence of Banach spaces included in B and (Yn)n∈N be a sequence of Banach spaces. Wesay that the sequence (Xn,Yn)n∈N is compact-continuous in B if the following conditions are satified• The sequence (Xn)n∈N is compactly embedded in B.• Xn ⊂ Yn (for all n ∈ N) and if the sequence (un)n∈N is such that un ∈ Xn (for all n ∈ N), (‖un‖Xn)n∈N

bounded and ‖un‖Yn → 0 (as n→+∞), then any subsequence converging in B converge (in B) to 0.


31

Aubin-Simon Theorem with a sequence of subspaces and a discrete derivative. Let 1 ≤ p < +∞. LetB be a Banach space, (Xn)n∈N be a sequence of Banach spaces included in B and (Yn)n∈N be a sequence ofBanach spaces. We assume that the sequence (Xn,Yn)n∈N is compact-continuous in B.

Let T > 0 and (un)n∈N be a sequence of Lp((0,T ),B) satifying the following conditions• For all n ∈ N, there exists N ∈N? and k1, . . . ,kN in R?

+ such that ∑Ni=1 ki = T and un(t) = vi for t ∈ (ti−1, ti),

i ∈ 1, . . . ,N, t0 = 0, ti = ti−1 + ki, vi ∈ Xn. (Of course, the values N, ki and vi are depending on n.)Furthermore, we define a.e. the function ðtun by setting ðtun(t) =

vi−vi−1ki

for t ∈ (ti−1, ti).• The sequence (un)n∈N is bounded in Lp((0,T ),B).• The sequence (‖un‖L1((0,T ),Xn))n∈N is bounded.• The sequence (‖ðtun‖Lp((0,T ),Yn))n∈N is bounded.Then there exists u ∈ Lp((0,T ),B) such that, up to a subsequence, un→ u in Lp((0,T ),B).

With the hypotheses of the previous theorems, another interesting question is to prove an additional regu-larity for u, namely that u ∈ Lp((0,T ),X) where X is some space closely related to the Xn (and included in B).We now precise the meaning of the sentence “X closely related to the Xn” and we give a regularity result.

Let B be a Banach space, (Xn)n∈N be a sequence of Banach spaces included in B and X be a Banach spaceincluded in B. We say that the sequence (Xn)n∈N is B-limit-included in X if there exist C ∈R such that if u is thelimit in B of a subsequence of a sequence (un)n∈N verifying un ∈ Xn and ‖un‖Xn ≤ 1, then u ∈ X and ‖u‖X ≤C.

Regularity of the limit. Let 1 ≤ p < +∞ and T > 0. Let B be a Banach space, (Xn)n∈N be a sequence ofBanach spaces included in B and B-limit-included in X (where X is a Banach space included in B).

Let T > 0 and, for n ∈ N, let un ∈ Lp((0,T ),Xn). We assume that the sequence (‖un‖Lp((0,T ),Xn))n∈N isbounded and that un→ u in Lp((0,T ),B) as n→+∞. Then u ∈ Lp((0,T ),X).

APPLICATION TO NUMERICAL SCHEMES

We consider here the case of a Finite Volume method with one unknown per cell and per time. With thenotations of the previous section, the B space is the space Lp(Ω) with (for instance) p = 1 or 2. We have asequence of spatial meshes, (Tn)n∈N. For a given n, the approximate space, Hn, is the set of functions constanton each cell. We set Xn = Yn = Hn. If u ∈ Hn, the Xn-norm is (for some q) a discrete W 1,q

0 -norm which reads,using some quite natural notations, for 1≤ q <+∞,

‖u‖q1,q,Tn

= ∑σ∈Eint ,σ=K|L

mσ dσ |uK−uL

dσ|q + ∑

σ∈Eext ,σ∈EK

mσ dσ |uK

dσ|q

and, for q = ∞, ‖u‖q1,∞,Tn

= maxMi,Me,M with

Mi = max|uK−uL|dσ

, σ ∈ Eint ,σ = K|L, Me = max|uK |dσ

, σ ∈ Eext ,σ ∈ EK, M = max|uK |, K ∈Tn.

The Yn-norm is a dual norm. For r ∈ [1,∞], ‖·‖−1,r,Tn is the dual norm of the norm ‖·‖1,q,Tn with q = r/(r−1).That is, for u ∈ Hn, ‖u‖−1,r,Tn = max∫Ω uvdx, v ∈ Hn,‖v‖1,q,Tn ≤ 1.

REFERENCES

[1] E. Chénier, R. Eymard, T. Gallouët and R. Herbin. An extension of the MAC scheme to locally refinedmeshes : convergence analysis for the full tensor time-dependent Navier-Stokes equations. Accepted forpublication in Calcolo, 2014.

[2] R. Eymard, P. Féron, T. Gallouët, C. Guichard and R. Herbin Gradient schemes for the Stefan problem.Submitted 2013.

[3] T. Gallouët and J. C. Latché. Compactness of discrete approximate solutions to parabolic PDEs - Appli-cation to a turbulence model. Commun. Pure Appl. Anal. 11 (2012), no. 6, 2371-2391.

32


A HIGH ORDER FINITE VOLUME/FINITE ELEMENT PROJECTIONMETHOD FOR LOW-MACH NUMBER FLOWS WITH TRANSPORT OFSPECIES

A. Bermúdez a, S. Busto a, M. Cobas a, J.L. Ferrín a, L. Saavedra c, M.E. Vázquez-Cendón a∗

a Departamento de Matemática Aplicada, Universidad de Santiago de Compostela 15782 Santiago de Compostela, Spain.b Departamento Fundamentos Matemáticos, Universidad Politécnica de Madrid E.T.S.I. Aeronáuticos. 28040 Madrid,Spain.

ABSTRACT

The purpose of the work is to present a finite volume/finite element projection method for low-Machnumber flows in both viscous and inviscid cases.

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 ofthe volumes are the barycenters of the faces of the initial tetrahedra. These volumes have been alreadyused for the 2D shallow water equation (see [1]) and allow for an easy implementation of flux boundaryconditions. The transport-diffusion stage is explicit. Upwinding of convective terms is done by classicalRiemann solvers as the Q-scheme of van Leer or the Rusanov scheme (see, for instance, [7]).

Concerning the projection stage, the pressure correction is computed by a piecewise linear finite elementmethod associated with the initial tetrahedral mesh (see [4] and [5]). Passing the information from onestage to the other is carefully made in order to get a stable global scheme (see [2]).

High order methods studied in [3] are analyzed and implemented to solve the transport-diffusion stagefor the convective and the diffusive terms. The numerical results are compared with those obtained withprofessional software.

In the present work it is also included the transport equations of species and the results obtained using aresolution coupled and uncoupled with the model of flows at low Mach number are analyzed. Moreover,this analysis will allow us to implement in a efficient way a k− ε model.

We present some academic problems in order to analyze the order of convergence as well as severalclassical test problems from fluid mechanics (see [6]). Numerical results are shown aiming to evaluatethe order of convergence of the method.

ACKNOWLEDGMENT

This project was co-financed with FEDER and Xunta de Galicia funds under the grant reference GRC2013-014.


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Local committeeSTÉPHANE CLAIN, Universidadedo Minho, Braga, Portugal.

JORGE FIGUEIREDO, Universidadedo Minho, Braga, Portugal.

GASPAR MACHADO, Universidadedo Minho, Braga, Portugal.

RUI PEREIRA, Universidade doMinho, Braga, Portugal.

CAROLINA RIBEIRO, Universidadedo Minho, Braga, Portugal.

Organizing InstitutionsCentro de Matematica, Universi-dade do Minho, Braga, Portugal.

Institut Jean Leray, Université deNantes, France.

Institut de Mathématique deToulouse, Université de Toulouse,France.

Organizing/Scientific Committee

CHRISTOPHE BERTHON, Université de Nantes, France.STÉPHANE CLAIN, Universidade do Minho, Braga, Portugal.FRÉDÉRIC COQUEL, Université Pierre et Marie Curie, Paris, France.STEVEN DIOT, Los Alamos National Laboratory, USA.MICHAEL DUMBSER, Università degli studi di Trento, Italy.ENRIQUE FERNÁNDEZ-NIETO, Universidad de Sevilla, Spain.THIERRY GALLOUËT, Université de Marseille, France.RAPHAËL LOUBÈRE, Université de Toulouse, France.CARLOS PARÉS, Universidad de Málaga, Spain.ELENA VÁZQUEZ CENDÓN, Universidade de Santiago de Compostela, Spain.

Sponsors: The organizers acknowledge the financial support of the Mathematical Institute of Toulouse (IMT),University Toulouse III, l’Agence National pour la Recherche (ANR project “ALE INC(ubator) 3D”), and by FEDERFunds 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 and FCT-ANR/MAT-NAN/0122/2012 and ANR-12-IS01-0004 GeoNum

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