# Accurate numerical discretizations of non-conservative hyperbolic systems

Ulrik Skre Fjordholm; Siddhartha Mishra

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

- Volume: 46, Issue: 1, page 187-206
- ISSN: 0764-583X

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topFjordholm, Ulrik Skre, and Mishra, Siddhartha. "Accurate numerical discretizations of non-conservative hyperbolic systems." ESAIM: Mathematical Modelling and Numerical Analysis 46.1 (2011): 187-206. <http://eudml.org/doc/222164>.

@article{Fjordholm2011,

abstract = {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.},

author = {Fjordholm, Ulrik Skre, Mishra, Siddhartha},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Non-conservative products; numerical schemes; non-conservative hyperbolic systems; entropy conservative discretizations; Lagrangian gas dynamics; isothermal Euler equations; numerical experiments},

language = {eng},

month = {10},

number = {1},

pages = {187-206},

publisher = {EDP Sciences},

title = {Accurate numerical discretizations of non-conservative hyperbolic systems},

url = {http://eudml.org/doc/222164},

volume = {46},

year = {2011},

}

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

DA - 2011/10//

PB - EDP Sciences

VL - 46

IS - 1

SP - 187

EP - 206

AB - 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.

LA - eng

KW - Non-conservative products; numerical schemes; non-conservative hyperbolic systems; entropy conservative discretizations; Lagrangian gas dynamics; isothermal Euler equations; numerical experiments

UR - http://eudml.org/doc/222164

ER -

## References

top- R. Abgrall and S. Karni, Two-layer shallow water system: a relaxation approach. SIAM. J. Sci. Comput.31 (2009) 1603–1627.
- R. Abgrall and S. Karni, A comment on the computation of non-conservative products. J. Comput. Phys.229 (2010) 2759–2763.
- E. Audusse, F. Bouchut, M.O. Bristeau, R. Klien and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM. J. Sci. Comput.25 (2004) 2050–2065.
- E. Audusse and M.O. Bristeau, Finite volume solvers for multi-layer Saint-Venant system. Int. J. Appl. Math. Comput. Sci.17 (2007) 311–319.
- M.J. Castro, P. LeFloch, M.L. Munoz Ruiz and C. Pares, Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes. J. Comput. Phys.227 (2008) 8107–8129.
- G.-Q. Chen, C. Christoforou and Y. Zhang, Continuous dependence of entropy solutions to the euler equations on the adiabatic exponent and mach number. Arch. Ration. Mech. Anal.189 (2008) 97–130.
- G. Dal Maso, P. LeFloch and F. Murat, Definition and weak stability of nonconservative products. J. Math. Pures. Appl.74 (1995) 483–548.
- U.S. Fjordholm, S. Mishra and E. Tadmor, 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.
- U.S. Fjordholm, S. Mishra and E. Tadmor, Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography. J. Comput. Phys.230 (2011) 5587–5609.
- E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws. Ellipses (1991).
- S. Gottlieb, C.-W. Shu and E. Tadmor, High order time discretization methods with the strong stability property, SIAM Rev.43 (2001) 89–112.
- S. Karni, Viscous shock profiles and primitive formulations. SIAM J. Numer. Anal.29 (1992) 1592–1609.
- P.G. LeFloch, Entropy weak solutions to nonlinear hyperbolic systems in nonconservative form. Comm. Partial Differential Equations13 (1988) 669–727.
- T.Y. Hou and P.G. LeFloch, Why nonconservative schemes converge to wrong solutions. Error analysis. Math. Comput.62 (1994) 497–530.
- P.G. LeFloch and M.D. Thanh, The Riemann problem for fluid flows in a nozzle with discontinuous cross-section. Commun. Math. Sci.1 (2003) 763–797.
- P.G. LeFloch, J.M. Mercier and C. Rohde, Fully discrete entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal.40 (2002) 1968–1992.
- R.J. LeVeque, Finite volume methods for hyperbolic problems.Cambridge university press, Cambridge (2002).
- T.P. Liu, Shock waves for compressible Navier–Stokes equations are stable. Comm. Pure Appl. Math.39 (1986) 565–594.
- M.L. Munoz Ruiz and C. Pares, Godunov method for non-conservative hyperbolic systems. Math. Model. Num. Anal.41 (2007) 169–185.
- C. Pares and M.J. Castro, 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.
- C. Pares, Numerical methods for non-conservative hyperbolic systems: a theoretical framework. SIAM. J. Num. Anal.44 (2006) 300–321.
- E. Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws, I. Math. Comp.49 (1987) 91–103.
- E. Tadmor, Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer.12 (2003) 451–512.
- E. Tadmor and W. Zhong, Entropy stable approximations of Navier–Stokes equations with no artificial numerical viscosity. J. Hyperbolic Differ. Equ.3 (2006) 529–559.
- E. Tadmor and W. Zhong, 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.
- E. Romenski, D. Drikakis and E. Toro, Conservative models and numerical methods for compressible two-phase flow. J. Sci. Comput.42 (2010) 68–95.

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