Two-dimensional dispersion analyses of finite element approximations to the shallow water equations

Abstract

Dispersion analysis of discrete solutions to the shallow water equations has been extensively used as a tool to define the relationships between frequency and wave number and to determine if an algorithm leads to a dual wave number response and near 2△x oscillations. In this paper, we explore the application of two-dimensional dispersion analysis to cluster based and Galerkin finite element-based discretizations of the primitive shallow water equations and the generalized wave continuity equation (GWCE) reformulation of the harmonic shallow water equations on a number of grid configurations. It is demonstrated that for various algorithms and grid configurations, contradictions exist between the results of one-dimensional and two-dimensional dispersion analysis as a result of subtle changes in the mass matrix. Numerical experiments indicate that the two-dimensional dispersion analysis correctly predicts the existence and onset of near 2△x noise in the solution. © 2004 John Wiley and Sons, Ltd.