Characteristic Galerkin Schemes for Scalar Conservation Laws in Two and Three Space Dimensions

Abstract

In this paper we are concerned with a finite element method for multidimensional scalar conservation laws: we describe a general formulation of the Euler characteristic Galerkin (ECG) scheme, motivated by key features of the one-dimensional ECG scheme. The method is defined by projecting the transport collapse operator onto a finite element space spanned by piecewise constant basis functions. The ECG scheme is {em TVD, monotone, maximum norm nonincreasing and unconditionally stable in the $L^{1}$-norm}. To the best of our knowledge, there is no other method which has all these properties. However, with piecewise constant basis functions, the ECG scheme is at most first-order accurate; greater accuracy can be obtained through a recovery procedure. Unlike conventional dimension-by-dimension methods, multidimensional ECG schemes contain important terms which describecorner effects. Nonuniform meshes and large time steps can be used with this class of schemes.