# CAIMS Prize Award

Institute of Applied Mathematics, University of British Columbia

Supervisor: J.M. Varah

Thesis title: Dispersion analyses of finite element solutions of the shallow water equations

Abstract:

This thesis investigates the accuracy and stability of finite element solutions of the shallow water equations. The method of investigation is referred to as a dispersion analysis. It compares numerical phase velocities, group velocities, and wave amplification factors to their analytic counterparts.

Chapter 1 discusses the shallow water equations, finite element and finite difference methods, and reviews previous work. The advantages and disadvantages of a dispersion analysis are also discussed.

Chapters 2 and 3 are restricted to numerical solutions of the one dimensional linearized shallow water equations. The phase and group velocities of eight spatial discretizations are calculated and examined for their relative merits. The most accurate two-step time-stepping methods are found for three finite element spatial discretizations; the wave equation model of Gray and Lynch, the Galerkin method with linear basis functions, and the Galerkin method which combines quadratic basis functions for velocity with linear functions for elevation. It is shown that with an appropriate time-stepping method, lumping the wave equation model need not cause an accuracy loss.

Chapter 4 extends the analysis to the linearized two dimensional equations. Finite element solutions are computed for two configurations of triangular elements. Two finite element methods, Thacker's method and the lumped wave equation model, are shown to be cost competitive and as accurate as the Richardson-Sielecki explicit finite difference method. The analysis also suggests that finite element meshes comprised of equilateral triangles most accurately represent phase and group velocity.

Chapter 5 extends the one dimensional dispersion analysis to include boundary conditions. The stability and relative accuracy of several absorbing boundary conditions are examined. Accuracy is evaluated through the calculation of reflection coefficients. An unstable boundary condition of the type examined by Trefethen is also found.