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Numerical methods for nonconservative hyperbolic systems: A theoretical framework. (English) Zbl 1130.65089
The author considers the one-dimensional quasi-linear strictly hyperbolic system \(\partial W/\partial t+{\mathcal A}(W)\partial W/\partial x=0\) for \(x\in{\mathbb R}\) and \(t>0\). One assumes the definition of nonconservative products as Borel measures given by G. Dal Maso, P. G. LeFloch and F. Murat [J. Math. Pures Appl. 74, 483–548 (1995; Zbl 0853.35068)]. This definition, which depends on the choice of a family of paths in the phase space, allows one to give a rigorous definition of weak solutions to this equation. With this definition, a notion of entropy is chosen as the usual Lax’s concept or one related to an entropy pair. The classical theory of simple waves of hyperbolic systems of conservation laws and the results concerning the solutions of Riemann problems can be extended to the above system.
Once the difficult choice of a family of path is done, this article provides a theoretical framework for the numerical approximation of the corresponding weak solutions of the above system whose characteristic fields are genuinely nonlinear or linearly degenerate. The concept of path-conservative numerical schemes is introduced as a generalization of that of conservative schemes for systems of conservation laws, that is, it preserves the Borel measure related to the nonconservative products. Well-balance property, approximate Riemann solvers and high order methods based on reconstruction techniques are the object of the last three sections.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L60 First-order nonlinear hyperbolic equations
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
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