Compact Finite Difference Schemes for Ocean Models: 1. Ocean Waves

A general framework is derived which leads to generic expressions of discrete dispersion relationships for inertia–gravity and Rossby waves, valid for every finite difference scheme and every type of grid. These relationships are used to investigate the performance of fourth-order and sixth-order compact implicit schemes on Arakawa grids A–E. It is shown that the use of compact schemes leads to very clear improvement in approximating frequency and group velocity of inertia–gravity waves. On the other hand, increasing the order of the schemes does not necessarily improve the accuracy of the discrete dispersion relationship in the case of Rossby waves. Globally, the fourth-order family is found to be a good compromise, which improves significantly the quality of the approximation of the dispersion properties with regard to conventional second-order centered schemes.