Numerical Analysis Group Research Report NA-92/1

P Lin, K W Morton, E Süli

In this paper we are concerned with a finite element method for multi-dimensional scalar conservation laws. We describe a general formulation of the Euler characteristic Galerkin (ECG) scheme, motivated by key features of the 1-D ECG scheme. This finite element method is defined through the projection onto piecewise constants of the transport collapse operator. Thus the scheme is TVD, monotone, maximum norm non-increasing and unconditionally stable in the L¹ norm. To the best of our knowledge, there is no other scheme having all these properties. However, it is at most first-order accurate; greater accuracy can be obtained through a recovery procedure. Being distinct from the usual dimension by dimension methods, the multi-dimensional ECG schemes contain important terms which describe the corner effects of the elements. Non-uniform meshes and large time steps can be used in these schemes.

Subject classifications:

AMS(MOS): 65M60, 65N30, 35L65

Key words and phrases:

Characteristic Galerkin method, Riemann-Stieltjes integral, transport-collapse operator, multi-dimensional conservation laws, corner effects.

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