Godunov method for nonconservative hyperbolic systems

Muñoz-Ruiz, Marı́a Luz; Parés, Carlos

doi:10.1051/m2an:2007011

Cited by 64 publications

(20 citation statements)

References 21 publications

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“…Our conclusions, therefore, justify to search for robust and efficient high-order schemes for the approximation of nonconservative systems. In particular, Berthon and Coquel [1,2] and Chalons and Coquel [11], have introduced various numerical strategies for models of complex fluid flows including turbulence models, while Parés [30] and Muñoz-Ruiz and Parés [28] have developed many important applications.…”

Section: Introductionmentioning

confidence: 99%

Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes

Castro

LeFloch

Muñoz-Ruiz

et al. 2008

Journal of Computational Physics

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173

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We are interested in nonlinear hyperbolic systems in nonconservative form arising in fluid dynamics, and, for solutions containing shock waves, we investigate the convergence of finite difference schemes applied to such systems. According to Dal Maso, LeFloch, and Murat's theory, a shock wave theory for a given nonconservative system requires prescribing a priori a family of paths in the phase space. In the present paper, we consider schemes that are formally consistent with a given family of paths, and we investigate their limiting behavior as the mesh is refined. we first generalize to systems a property established earlier by Hou and LeFloch for scalar conservation laws, and we prove that nonconservative schemes generate, at the level of the limiting hyperbolic system, an convergence error source-term which, provided the total variation of the approximations remains uniformly bounded, is a locally bounded measure. This convergence error measure is supported on the shock trajectories and, as we demonstrate here, is usually "small". In the special case that the scheme converges in the sense of graphs -a rather strong convergence property often violated in practice-then this measure source-term vanishes. We also discuss the role of the equivalent equation associated with a difference scheme; here, the distinction between scalar equations and systems appears most clearly since, for systems, the equivalent equation of a scheme that is formally path-consistent depends upon the prescribed family of paths. The core of this paper is devoted to investigate numerically the approximation of several (simplified or full) hyperbolic models arising in fluid dynamics. This leads us to the conclusion that for systems having nonconservative products associated with linearly degenerate characteristic fields, the convergence error vanishes. For more general models, this measure is evaluated very accurately, especially by plotting the shock curves associated with each scheme under consideration; as we demonstrate, plotting the shock curves provide a convenient approach for evaluating the range of validity of a given scheme.

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Section: Introductionmentioning

confidence: 99%

Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes

Castro

LeFloch

Muñoz-Ruiz

et al. 2008

Journal of Computational Physics

Self Cite

221

173

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“…In this paper we develop a Godunov-type path-conservative method for the five-equation model of two-phase flow of Saurel and Abgrall [2] which is similar to that recently reported in [12]. The theoretical framework of Godunov-type path-conservative methods in [6,1] is adopted. Here we construct a specific scheme out of the family proposed in [1], in which the Godunov interface state computed from our HLLC-type solver is also used to compute a composite path.…”

Section: Introductionmentioning

confidence: 93%

“…For non-conservative systems there is a theory [1,7,23] in which generalized Rankine-Hugoniot conditions are established, involving a family of paths in phase space. Based on this theory, it has been possible to propose new numerical methodologies to solve non-conservative systems [1,6,18,23]. So far most applications of the path-conservative approach deal with shallow water type equations, with few applications to multiphase flows.…”

Section: Introductionmentioning

confidence: 99%

A path-conservative method for a five-equation model of two-phase flow with an HLLC-type Riemann solver

2011

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“…Fluctuations of the Godunov, Roe and other approximate Riemann solver types have been proposed in [21] and in [19,20].…”

Section: Numerical Schemesmentioning

confidence: 99%

Accurate numerical discretizations of non-conservative hyperbolic systems

Fjordholm

Mishra

2011

ESAIM: M2AN

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Abstract. We present an alternative framework for designing efficient numerical schemes for nonconservative 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.Mathematics Subject Classification. 65M06, 35L65.

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Godunov method for nonconservative hyperbolic systems

Cited by 64 publications

References 21 publications

Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes

Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes

A path-conservative method for a five-equation model of two-phase flow with an HLLC-type Riemann solver

Accurate numerical discretizations of non-conservative hyperbolic systems

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