RESUMEN

In this paper, the technique of construction and analysis of quasimonotone finite-difference numerical schemes for scalar conservation laws in one space dimension, developed in Part I [SIAM J. Numer. Anal., 26 (1989), pp. 1325–1341], is extended to a wide class of Petrov–Galerkin finite-element methods. The resulting schemes are called the quasimonotone finite-element schemes. The approximate solution u_h is written as ū_h  + ŭ_h , where ū_h is a piecewise-constant function. The Petrov–Galerkin methods are then considered to be a set of equations that defines “the parameter” ŭ_h, plus a single equation, which is essentially a finite-difference scheme, that defines “the means” ŭ_h. All the results of the theory of quasimonotone finite-difference schemes can be coupling.

RESUMEN

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^∞ (L¹_loc) 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^∞-stable, and they can be more accurate than the quasimonotone schemes; unfortunately, the compactness in L^∞ (L¹_loc) is lost. Nevertheless, if they converge, they do so to the entropy solution.