Stability analysis for Eulerian and semi-Lagrangian finite-element formulation of the advection-diffusion equation

Abstract

This paper analyzes the stability of the finite-element approximation to the linearized two-dimensional advection-diffusion equation. Bilinear basis functions on rectangular elements are considered. This is one of the two best schemes as was shown by Neta and Williams [1]. Time is discretized with the theta algorithms that yield the explicit (theta = 0), semi-implicit (theta = 1/2), and implicit (theta = 1) methods. This paper extends the results of Neta and Williams [1] for the advection equation. Giraldo and Neta [2] have numerically compared the Eulerian and semi-Lagrangian finite-element approximation for the advection-diffusion equation. This paper analyzes the finite element schemes used there. The stability analysis shows that the semi-Lagrangian method is unconditionally stable for all values of a while the Eulerian method is only unconditionally stable for 1/2 < theta < 1. This analysis also shows that the best methods are the semi-implicit ones (theta = 1/2). In essence this paper analytically compares a semi-implicit Eulerian method with a semi-implicit semi-Lagrangian method. It is concluded that (for small or no diffusion) the semi-implicit semi-Lagrangian method exhibits better amplitude, dispersion and group velocity errors than the semi-implicit Eulerian method thereby achieving better results. In the case the diffusion coefficient is large, the semi-Lagrangian loses its competitiveness. Published by Elsevier Science Ltd.