Fourier stability analysis of two-dimensional finite element schemes for shallow water equations

Abstract

This article presents a stability-based analysis of amplification factors obtained using Fourier or von Neumann method for the finite element formulation of shallow water equations. A Galerkin finite element model is developed for two-dimensional shallow water equations to obtain a linearised form of the error equations. Fourier analysis is performed at the element as well as nodal levels to propose time-step criteria for consistent, upwind and lumped methods using explicit, semi-implicit and implicit schemes. The minimum and upper bound Courant numbers are computed for each finite element scheme using the coefficient method. We observed a close agreement between the amplification factors determined using integer and half-integer multiples of Courant numbers of scalar and matrix eigenvalue problems and that of minimum and upper bound Courant numbers of the coefficient method. We demonstrate that ignoring the amplification factor behaviour at the computational grid node may result in an inappropriate representation of numerical stability for certain schemes. The amplitude behaviour as a function of the Courant and wave numbers is analysed for square elements for two-dimensional field solution at each computational grid node. A relationship is derived between element error-squared norm and nodal error-squared norm.