Residual Distribution Schemes for Conservation Laws via Adaptive Quadrature

Abstract

This paper considers a family of nonconservative numerical discretizations for conservation laws which retain the correct weak solution behavior in the limit of mesh refinement whenever sufficient-order numerical quadrature is used. Our analysis of 2-D discretizations in nonconservative form follows the 1-D analysis of Hou and Le Floch [Math. Comp., 62 (1994), pp. 497--530]. For a specific family of nonconservative discretizations, it is shown under mild assumptions that the error arising from nonconservation is strictly smaller than the discretization error in the scheme. In the limit of mesh refinement under the same assumptions, solutions are shown to satisfy a global entropy inequality. Using results from this analysis, a variant of the "N" (Narrow) residual distribution scheme of van der Weide and Deconinck [Computational Fluid Dynamics '96, Wiley, New York, 1996, pp. 747--753] is developed for first-order systems of conservation laws. The modified form of the N-scheme supplants the usual exact single-state mean-value linearization of flux divergence, typically used for the Euler equations of gasdynamics, by an equivalent integral form on simplex interiors. This integral form is then numerically approximated using an adaptive quadrature procedure. This quadrature renders the scheme nonconservative in the sense described earlier so that correct weak solutions are still obtained in the limit of mesh refinement. Consequently, we then show that the modified form of the N-scheme can be easily applied to general (nonsimplicial) element shapes and general systems of first-order conservation laws equipped with an entropy inequality, where exact mean-value linearization of the flux divergence is notreadily obtained, e.g., magnetohydrodynamics, the Euler equations with certain forms of chemistry, etc. Numerical examples of subsonic, transonic, and supersonic flows containing discontinuities together with multilevel mesh refinement are provided to verify the analysis.