We present an alternative framework for designing efficient numerical schemes for non-conservative hyperbolic systems. This approach is based on the design of entropy conservative discretizations and suitable numerical diffusion operators that mimic the effect of underlying viscous mechanisms. This approach is illustrated by considering two model non-conservative systems: Lagrangian gas dynamics in non-conservative form and a form of isothermal Euler equations. Numerical experiments demonstrating the robustness of this approach are presented.
Mots-clés : non-conservative products, numerical schemes
@article{M2AN_2012__46_1_187_0,
author = {Fjordholm, Ulrik Skre and Mishra, Siddhartha},
title = {Accurate numerical discretizations of non-conservative hyperbolic systems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {187--206},
publisher = {EDP-Sciences},
volume = {46},
number = {1},
year = {2012},
doi = {10.1051/m2an/2011044},
mrnumber = {2846371},
zbl = {1272.65064},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2011044/}
}
TY - JOUR AU - Fjordholm, Ulrik Skre AU - Mishra, Siddhartha TI - Accurate numerical discretizations of non-conservative hyperbolic systems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 187 EP - 206 VL - 46 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2011044/ DO - 10.1051/m2an/2011044 LA - en ID - M2AN_2012__46_1_187_0 ER -
%0 Journal Article %A Fjordholm, Ulrik Skre %A Mishra, Siddhartha %T Accurate numerical discretizations of non-conservative hyperbolic systems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 187-206 %V 46 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2011044/ %R 10.1051/m2an/2011044 %G en %F M2AN_2012__46_1_187_0
Fjordholm, Ulrik Skre; Mishra, Siddhartha. Accurate numerical discretizations of non-conservative hyperbolic systems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 1, pp. 187-206. doi : 10.1051/m2an/2011044. http://archive.numdam.org/articles/10.1051/m2an/2011044/
[1] and , Two-layer shallow water system: a relaxation approach. SIAM. J. Sci. Comput. 31 (2009) 1603-1627. | MR | Zbl
[2] and , A comment on the computation of non-conservative products. J. Comput. Phys. 229 (2010) 2759-2763. | MR | Zbl
[3] , , , and , A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM. J. Sci. Comput. 25 (2004) 2050-2065. | MR | Zbl
[4] and , Finite volume solvers for multi-layer Saint-Venant system. Int. J. Appl. Math. Comput. Sci. 17 (2007) 311-319. | MR | Zbl
[5] , , and , Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes. J. Comput. Phys. 227 (2008) 8107-8129. | MR | Zbl
[6] , and , Continuous dependence of entropy solutions to the euler equations on the adiabatic exponent and mach number. Arch. Ration. Mech. Anal. 189 (2008) 97-130. | MR | Zbl
[7] , and , Definition and weak stability of nonconservative products. J. Math. Pures. Appl. 74 (1995) 483-548. | MR | Zbl
[8] , and , Energy preserving and energy stable schemes for the shallow water equations,Foundations of Computational Mathematics, Proc. FoCM held in Hong Kong 2008, London Math. Soc. Lecture Notes Ser. 363, edited by F. Cucker, A. Pinkus and M. Todd (2009) 93-139. | MR
[9] , and , Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography. J. Comput. Phys. 230 (2011) 5587-5609. | MR
[10] and , Hyperbolic systems of conservation laws. Ellipses (1991). | MR | Zbl
[11] , and , High order time discretization methods with the strong stability property, SIAM Rev. 43 (2001) 89-112. | MR | Zbl
[12] , Viscous shock profiles and primitive formulations. SIAM J. Numer. Anal. 29 (1992) 1592-1609. | MR | Zbl
[13] , Entropy weak solutions to nonlinear hyperbolic systems in nonconservative form. Comm. Partial Differential Equations 13 (1988) 669-727. | MR | Zbl
[14] and , Why nonconservative schemes converge to wrong solutions. Error analysis. Math. Comput. 62 (1994) 497-530. | MR | Zbl
[15] and , The Riemann problem for fluid flows in a nozzle with discontinuous cross-section. Commun. Math. Sci. 1 (2003) 763-797. | MR | Zbl
[16] , and , Fully discrete entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal. 40 (2002) 1968-1992. | MR | Zbl
[17] , Finite volume methods for hyperbolic problems.Cambridge university press, Cambridge (2002). | MR | Zbl
[18] , Shock waves for compressible Navier-Stokes equations are stable. Comm. Pure Appl. Math. 39 (1986) 565-594. | MR | Zbl
[19] and , Godunov method for non-conservative hyperbolic systems. Math. Model. Num. Anal. 41 (2007) 169-185. | Numdam | MR | Zbl
[20] and , On the well-balance property of Roe's method for nonconservative hyperbolic systems. Applications to shallow water equations. Math. Model. Num. Anal. 38 (2004) 821-852. | Numdam | MR | Zbl
[21] , Numerical methods for non-conservative hyperbolic systems: a theoretical framework. SIAM. J. Num. Anal. 44 (2006) 300-321. | MR | Zbl
[22] , The numerical viscosity of entropy stable schemes for systems of conservation laws, I. Math. Comp. 49 (1987) 91-103. | MR | Zbl
[23] , Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12 (2003) 451-512. | MR | Zbl
[24] and , Entropy stable approximations of Navier-Stokes equations with no artificial numerical viscosity. J. Hyperbolic Differ. Equ. 3 (2006) 529-559. | MR
[25] and , Energy preserving and stable approximations for the two-dimensional shallow water equations,in Mathematics and computation: A contemporary view, Proc. of the third Abel symposium. Ålesund, Norway, Springer (2008) 67-94. | MR
[26] , and , Conservative models and numerical methods for compressible two-phase flow. J. Sci. Comput. 42 (2010) 68-95. | MR | Zbl
Cité par Sources :
