Top

Journal of Scientific Computing

Published in:

04-05-2019

Improvement of the Hydrostatic Reconstruction Scheme to Get Fully Discrete Entropy Inequalities

Authors: Christophe Berthon, Arnaud Duran, Françoise Foucher, Khaled Saleh, Jean De Dieu Zabsonré

Published in: Journal of Scientific Computing | Issue 2/2019

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

This work is devoted to the derivation of an energy estimate to be satisfied by numerical schemes when approximating the weak solutions of the shallow water model. More precisely, here we adopt the well-known hydrostatic reconstruction technique to enforce the adopted Finite-Volume scheme to be well-balanced; namely to exactly preserve the lake at rest stationary solution. Such a numerical approach is known to get a semi-discrete (continuous in time) entropy inequality. However, a semi-discrete energy estimation turns, in general, to be insufficient to claim the required stability. In the present work, we adopt the artificial numerical viscosity technique to increase the desired stability and then to recover a fully discrete energy estimate. Several numerical experiments illustrate the relevance of the designed viscous hydrostatic reconstruction scheme.

previous article New Analysis of Galerkin FEMs for Miscible Displacement in Porous Media

next article Convergence Analysis of a Petrov–Galerkin Method for Fractional Wave Problems with Nonsmooth Data

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R., Perthame, B.: A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25(6), 2050–2065 (2004)MathSciNetMATHCrossRef

Audusse, E., Bouchut, F., Bristeau, M.-O., Sainte-Marie, J.: Kinetic entropy inequality and hydrostatic reconstruction scheme for the saint-venant system. Math. Comput. 85(302), 2815–2837 (2016)MathSciNetMATHCrossRef

Azerad, P., Guermond, J.-L., Popov, B.: Well-balanced second-order approximation of the shallow water equation with continuous finite elements. SIAM J. Numer. Anal. 55(6), 3203–3224 (2017)MathSciNetMATHCrossRef

Bermudez, A., Vazquez, M.E.: Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23(8), 1049–1071 (1994)MathSciNetMATHCrossRef

Berthon, C., Chalons, C.: A fully well-balanced, positive and entropy-satisfying godunov-type method for the shallow-water equations. Math. Comput. 85(299), 1281–1307 (2016)MathSciNetMATHCrossRef

Berthon, C., Marche, F.: A positive preserving high order VFRoe scheme for shallow water equations: a class of relaxation schemes. SIAM J. Sci. Comput. 30(5), 2587–2612 (2008)MathSciNetMATHCrossRef

F. Bouchut.: Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2004)

Bouchut, F., de Luna, T.Morales: A subsonic-well-balanced reconstruction scheme for shallow water flows. SIAM J. Numer. Anal. 48(5), 1733–1758 (2010)MathSciNetMATHCrossRef

Bouchut, F., de Luna, T.Morales: An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment. M2AN Math. Model. Numer. Anal. 42(4), 683–698 (2008)MathSciNetMATHCrossRef

10.

Cargo, P., Le Roux, A.-Y.: Un schéma équilibre adapté au modèle d’atmosphère avec termes de gravité. Comptes rendus de l’Académie des sciences. Série 1, Mathématique 318(1), 73–76 (1994)

11.

Chen, G., Noelle, S.: A new hydrostatic reconstruction scheme based on subcell reconstructions. SIAM J. Numer. Anal. 55(2), 758–784 (2017)MathSciNetMATHCrossRef

12.

Coquel, F., Saleh, K., Seguin, N.: A robust and entropy-satisfying numerical scheme for fluid flows in discontinuous nozzles. Math. Models Methods Appl. Sci. 24(10), 2043–2083 (2014)MathSciNetMATHCrossRef

13.

Couderc, F., Duran, A., Vila, J.-P.: An explicit asymptotic preserving low Froude scheme for the multilayer shallow water model with density stratification. J. Comput. Phys. 343, 235–270 (2017)MathSciNetMATHCrossRef

14.

Dafermos, C.M.: Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, (2010)

15.

Delestre, O., Lagrée, P.-Y.: A ‘well-balanced’ finite volume scheme for blood flow simulation. Internat. J. Numer. Methods Fluids 72(2), 177–205 (2013)MathSciNetCrossRef

16.

Dubois, F., Mehlman, G.: A non-parameterized entropy correction for Roe’s approximate Riemann solver. Numer. Math. 73, 169–208 (1996)MathSciNetMATHCrossRef

17.

Fernández-Nieto, E.D., Garres-Díaz, J., Mangeney, A., Narbona-Reina, G.: 2D granular flows with the \(\mu (I)\) rheology and side walls friction: a well-balanced multilayer discretization. J. Comput. Phys. 356, 192–219 (2018)MathSciNetMATHCrossRef

18.

Gallouët, T., Hérard, J.-M., Seguin, N.: Some recent finite volume schemes to compute Euler equations using real gas EOS. Int. J. Numer. Methods Fluids 39(12), 1073–1138 (2002)MathSciNetMATHCrossRef

19.

Gallouët, T., Hérard, J.-M., Seguin, N.: Some approximate Godunov schemes to compute shallow water equations with topography. Comput. Fluids 32(4), 479–513 (2003)MathSciNetMATHCrossRef

20.

Godlewski, E., Raviart, P.-A.: Hyperbolic systems of conservation laws, volume 3/4 of Mathématiques & Applications (Paris) [Mathematics and Applications]. Ellipses, Paris, (1991)

21.

Godlewski, E., Raviart, P.-A.: Numerical Approximation of Hyperbolic Systems of Conservation laws, Volume 118 of Applied Mathematical Sciences. Springer, New York (1996)MATHCrossRef

22.

Godunov, S.K.: A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. (N.S.) 47(89), 271–306 (1959)MathSciNetMATH

23.

Gosse, L.: A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comput. Math. Appl. 39(9), 135–159 (2000)MathSciNetMATHCrossRef

24.

Goutal, N., Maurel, F.: Proceedings of the 2nd workshop on dam-break wave simulation. Electricité de France, Direction des études et recherches (1997)

25.

Goutal, N., Maurel, F.: Dam-break wave simulation. In: Proceedings of the First CADAM workshop, (1998)

26.

Greenberg, J.M., Leroux, A.-Y.: A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33(1), 1–16 (1996)MathSciNetCrossRef

27.

Greenberg, J.M., Leroux, A.Y., Baraille, R., Noussair, A.: Analysis and approximation of conservation laws with source terms. SIAM J. Numer. Anal. 34(5), 1980–2007 (1997)MathSciNetMATHCrossRef

28.

Grenier, N., Vila, J.-P., Villedieu, P.: An accurate low-Mach scheme for a compressible two-fluid model applied to free-surface flows. J. Comput. Phys. 252, 1–19 (2013)MathSciNetMATHCrossRef

29.

Guermond, J.-L., Popov, B.: Viscous regularization of the euler equations and entropy principles. SIAM J. Appl. Math. 74, 284–305 (2014)MathSciNetMATHCrossRef

30.

Guermond, J.-L., Popov, B.: Invariant domains and second-order continuous finite element approximation for scalar conservation equations. SIAM J. Numer. Anal. 55, 3120–3146 (2017)MathSciNetMATHCrossRef

31.

Harten, A., Lax, P.D., Van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25, 35–61 (1983)MathSciNetMATHCrossRef

32.

Harten, A., Hyman, J.M.: A self-adjusting grid for the computation of weak solutions of hyperbolic conservation laws. J. Comput. Phys. 50, 235–269 (1983)MathSciNetMATHCrossRef

33.

Helluy, P., Hérard, J.-M., Mathis, H., Müller, S.: A simple parameter-free entropy correction for approximate Riemann solvers. Comptes Rendus Mécanique 338(9), 493–498 (2010)MATHCrossRef

34.

Jin, S.: A steady-state capturing method for hyperbolic systems with geometrical source terms. ESAIM: Math. Model. Numer. Anal. 35(04), 631–645 (2001)MathSciNetMATHCrossRef

35.

Lax, P.D.: Shock waves and entropy. In: Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971), pages 603–634. Academic Press, New York (1971)

36.

Lax, P.D.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, vol. 11. SIAM, New Delhi (1973)MATHCrossRef

37.

Lax, P.D., Wendroff, B.: Systems of conservation laws. Comm. Pure Appl. Math. 13, 217–237 (1960)MathSciNetMATHCrossRef

38.

LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)CrossRef

39.

Li, G., Xing, Y.: Well-balanced discontinuous Galerkin methods with hydrostatic reconstruction for the Euler equations with gravitation. J. Comput. Phys. 352, 445–462 (2018)MathSciNetMATHCrossRef

40.

Liang, Q., Marche, F.: Numerical resolution of well-balanced shallow water equations with complex source terms. Adv. Water Resour. 32(6), 873–884 (2009)CrossRef

41.

Maire, P.H., Abgrall, R., Breil, J., Ovadia, J.: A cell-centered Lagrangian scheme for two-dimensional compressible flow problems. SIAM J. Sci. Comput. 29, 1781–1824 (2007)MathSciNetMATHCrossRef

42.

Masella, J.-M., Faille, I., Gallouët, T.: On a rough godunov scheme. Int. J. for Comput. Fluid Dyn. 12(2), 133–150 (1999)MATHCrossRef

43.

Michel-Dansac, V., Berthon, C., Clain, S., Foucher, F.: A well-balanced scheme for the shallow-water equations with topography. Comput. Math. Appl. 72(3), 568–593 (2016)MathSciNetMATHCrossRef

44.

Michel-Dansac, V., Berthon, C., Clain, S., Foucher, F.: A well-balanced scheme for the shallow-water equations with topography or manning friction. J. Comput. Phys. 335, 115–154 (2017)MathSciNetMATHCrossRef

45.

Morales, T., Castro Díaz, M.J., Parés, C.: Reliability of first order numerical schemes for solving shallow water system over abrupt topography. Appl. Math. Comput. 219(17), 9012–9032 (2013)MathSciNetMATH

46.

von Neumann, J., Richtmyer, R.D.: A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys. 21, 232–237 (1950)MathSciNetMATHCrossRef

47.

Noh, W.F.: Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux. J. Comput. Phys. 72, 78–120 (1987)MATHCrossRef

48.

Perthame, B., Simeoni, C.: A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38(4), 201–231 (2001)MathSciNetMATHCrossRef

49.

Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43(2), 357–372 (1981)MathSciNetMATHCrossRef

50.

Serre, D.: Systems of conservation laws. 1. Cambridge University Press, Cambridge, (1999). Hyperbolicity, entropies, shock waves, Translated from the 1996 French original by I. N. Sneddon

51.

Tadmor, E.: Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comput. 43(168), 369–381 (1984)MathSciNetMATHCrossRef

52.

Tadmor, E.: The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Math. Comput. 49(179), 91–103 (1987)MathSciNetMATHCrossRef

53.

Tadmor, E.: Entropy stable schemes. Handbook of Numerical Methods for Hyperbolic Problems: Basic and Fundamental Issues, edited by R. Abgrall and C.-W. Shu (North-Holland, Elsevier, Amsterdam, 2017), 17:467–493 (2016)

54.

Toro, E.F.: Riemann solvers and numerical methods for fluid dynamics, 3rd edn. Springer, Berlin (2009). A practical introductionMATHCrossRef

55.

Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4(1), 25–34 (1994)MATHCrossRef

56.

Xu, K., Martinelli, L., Jameson, A.: Gas-kinetic finite volume methods, flux-vector splitting and artificial diffusion. J. Comput. Phys. 120, 48–65 (1995)MathSciNetMATHCrossRef

Title: Improvement of the Hydrostatic Reconstruction Scheme to Get Fully Discrete Entropy Inequalities
Authors: Christophe Berthon
Arnaud Duran
Françoise Foucher
Khaled Saleh
Jean De Dieu Zabsonré
Publication date: 04-05-2019
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 2/2019
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-019-00961-y

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 2/2019

Analysis on an HDG Method for the p-Laplacian Equations

A Viscosity-Independent Error Estimate of a Pressure-Stabilized Lagrange–Galerkin Scheme for the Oseen Problem

A Stabilized Semi-Implicit Euler Gauge-Invariant Method for the Time-Dependent Ginzburg–Landau Equations

Algorithm for Hamilton–Jacobi Equations in Density Space Via a Generalized Hopf Formula

Asymptotic Solutions for High Frequency Helmholtz Equations in Anisotropic Media with Hankel Functions

Performance Comparison of HPX Versus Traditional Parallelization Strategies for the Discontinuous Galerkin Method

Premium Partner