Propagation and spurious reflection in finite-element approximations of hyperbolic equations

Abstract

There are a variety of spurious phenomena in the numerical simulation of wave propagation which manifest themselves by the appearance of short-wavelength oscillations near discontinuities and computational boundaries. The appropriate tool for the description of those phenomena is that given by Fourier analysis. In particular, introduction in the analysis of the concepts of energy propagation and group velocity (from mathematical physics) produces a set of particularly interesting results. The corresponding mathematics have been reasonably well described in the recent literature for usual finite-difference schemes. But several new aspects present themselves in the numerical schemes based on finite-element methods. We will, in this paper, examine wave propagation in the numerical approximation of hyperbolic equations obtained with the linear finite-element—Galerkin method, pointing out where the corresponding properties differ from those of finite differences. In particular, it will be shown that • The energy conserved in finite-element schemes is not the usual square of the l 2 norm (as it is with finite differences). • The expression of this energy is given by a modified form of Parseval's relation. • Group velocities of rightgoing and leftgoing solutions corresponding to the same frequency are not equal. Illustration of spurious reflection at downwind boundaries, upwind boundaries and at interfaces in mesh refinement are given together with their mathematics.