 
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
sufficientorder numerical quadrature is used. Our analysis of 2D discretizations in nonconservative
form follows the 1D 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 firstorder systems of conservation laws. The modified form of the N
scheme supplants the usual exact singlestate meanvalue 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
