Summary: RESIDUAL DISTRIBUTION SCHEMES FOR CONSERVATION
LAWS VIA ADAPTIVE QUADRATURE
R´EMI ABGRALL AND TIMOTHY BARTH
SIAM J. SCI. COMPUT. c 2002 Society for Industrial and Applied Mathematics
Vol. 24, No. 3, pp. 732­769
Abstract. This paper considers a family of nonconservative numerical discretizations for conser-
vation 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. 497­530]. 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 en-
tropy 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. 747­753] 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

Source: Abgrall, Rémi - Institut de Mathematiques de Bordeaux, Université Bordeaux

Collections: Mathematics