Solutions of the shallow water equations on a sphere

Abstract

Various numerical solutions of the shallow water equations in two dimensions are studied in an effort to develop a computational technique applicable to hydrodynamics in spherical geometry. The equations are first cast in a form which allows periodic boundary conditions in both angular coordinates. Explicit numerical solutions using leap-frog centered differencing in time and either second, fourth, or compact fourth order centered spatial differencing are studied. The fourth order compact differencing is found to be easily adapted to spherical geometry and is superior to the second order technique. We also consider an alternating-direction implicit (ADI) scheme in an attempt to increase computational efficiency by taking larger time steps. Both analytically steady state and time dependent solutions are examined to investigate stability properties and discretization errors in time and space. Implicit methods require more computation per time step than explicit methods for solution of the shallow water equations. However, the total time for a simulation can be less with the implicit method. The ADI formalism also has advantages of importance for more physically complex problems.