The implementation of the semi-implicit scheme in cell-integrated semi-lagrangian models

Abstract

The cell-integrated semi-Lagrangian method, in which trajectories from the corner points of a grid cell define its extent at a previous time, may be applied to a set of equations in Lagrangian form derived from the complete set of primitive equations to construct a numerical model which conserves exactly the discrete forms of the global integral constraints of mass, momentum, entrophy and total energy. Due to the fulfilment of these integral constraints one should expect the numerical model to be absolutely stable. Experiments with a simple one-dimensional shallow water model show, however, that a time step dependent instability develops when a CFL criterion for gravity waves is exceeded. Analysis of experiments with one-dimensional shallow water models unveils the mechanism of the instability. Using again for simplicity one-dimensional models a main achievement is a successful implementation of the semi-implicit time stepping scheme in the cell-integrated models. © 1997 Taylor & Francis Group, LLC.