An Efficient Semi-Lagrangian and Alternating Direction Implicit Method for Integrating the Shallow Water Equations

Abstract

A grid-point method for integrating the shallow water equations based on a split semi-Lagrangian treatment of advection and an Eulerian alternating direction implicit treatment of the adjustment terms is presented. The scheme is simpler than the semi-implicit scheme, involving only the solution of linear tridiagonal systems of equations rather than a Helmholz equation. The theoretical properties of the scheme are examined for the E-grid. It is unconditionally stable for advection and for simple Rossby waves and has a very lenient stability criterion for gravity-inertia waves. Though two-level in time, it is shown to give second-order accuracy for both types of wave solution. No splitting errors occur in either case. The scheme is used to carry out 24-hour integrations of a limited area barotropic model with real initial data. It is shown to be more efficient than a previous semi-Lagrangian scheme presented by Bates and McDonald.