Quasimonotone Schemes for Scalar Conservation Laws. Part III

Abstract

In this paper the definition and analysis of the quasimonotone numerical schemes is extended to the general case $d > 1$, where d is the number of space variables. These schemes are constructed using the simple but very important class of monotone schemes that are defined by two-point monotone fluxes. To enforce the compactness in $L^\infty (L_{{\text{loc}}}^1 )$ of the sequence of approximate solutions, the case of meshes that are a Cartesian product of one-dimensional partitions is addressed. It is proved that the main stability and convergence results for one-dimensional quasimonotone schemes (of the first type) also hold in the general case. As a by-product of this theory, the theory of relaxed, quasimonotone schemes is developed. These schemes are $L^\infty $-stable, and they can be more accurate than the quasimonotone schemes; unfortunately, the compactness in $L^\infty (L_{{\text{loc}}}^1 )$ is lost. Nevertheless, if they converge, they do so to the entropy solution.