Why nonconservative schemes converge to wrong solutions: error analysis
Mathematics of Computation
Nonclassical Shocks and Kinetic Relations: Finite Difference Schemes
SIAM Journal on Numerical Analysis
Numerical simulation of two-layer shallow water flows through channels with irregular geometry
Journal of Computational Physics
Numerical methods for nonconservative hyperbolic systems: a theoretical framework.
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Journal of Computational Physics
High Order Extensions of Roe Schemes for Two-Dimensional Nonconservative Hyperbolic Systems
Journal of Scientific Computing
Short Note: A comment on the computation of non-conservative products
Journal of Computational Physics
Journal of Computational Physics
Journal of Scientific Computing
A Duality Method for Sediment Transport Based on a Modified Meyer-Peter & Müller Model
Journal of Scientific Computing
Journal of Scientific Computing
A Simple Extension of the Osher Riemann Solver to Non-conservative Hyperbolic Systems
Journal of Scientific Computing
A Well-Balanced Path-Integral f-Wave Method for Hyperbolic Problems with Source Terms
Journal of Scientific Computing
Numerical Treatment of the Loss of Hyperbolicity of the Two-Layer Shallow-Water System
Journal of Scientific Computing
Advances in Engineering Software
Original Article: High order well-balanced scheme for river flow modeling
Mathematics and Computers in Simulation
Journal of Computational Physics
Journal of Computational Physics
A diffuse interface model with immiscibility preservation
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
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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.