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Convergence of the upwind interfacesource method for hyperbolic conservation laws. (English) Zbl 1064.65098
Hou, Thomas Y. (ed.) et al., Hyperbolic problems: Theory, numerics, applications. Proceedings of the ninth international conference on hyperbolic problems, Pasadena, CA, USA, March 25–29, 2002. Berlin: Springer (ISBN 3-540-44333-9/hbk). 61-78 (2003).
Summary: This paper deals with typical questions arising in the analysis of numerical approximations for scalar conservation laws with a source term. The authors focus their attention on semi-discrete finite volume schemes, in the general case of a nonuniform spatial mesh.
To define appropriate discretizations of the source term, they introduce the formalism peculiar to the upwind interface source method and establish conditions on the numerical functions so that the discrete solver preserves the steady state solutions. Then a rigorous definition of consistency is formulated, adapted to the class of well-balanced schemes, for which they are able to prove a Lax-Wendroff type convergence theorem.
Some examples of numerical methods are discussed, in order to validate the proposed arguments .
For the entire collection see [Zbl 1024.00068].

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
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