THE OVER-RELAXATION FACTOR IN THE NUMERICAL SOLUTION OF THE OMEGA EQUATION¹

Abstract

Results of tests for the optimum over-relaxation coefficients in the numerical relaxation of the omega equation we presented. One case considers a strong upper-level development for tests on a fixed grid using one-, two-, and three-dimensional forms of the omega equation. In the other case of a "classical storm" the omega equation is relaxed in its three-dimensional form using several different horizontal grids.In the threedimensional tests, the observed over-relaxation coefficients are found to be less than the values given by the theory for all grid sizes considered. A sharp cut-off is found to occur shortly after the optimum over-relaxation value is reached regardless of the number of dimensions of the equation or the size of the grid.For the one-and two-dimensional tests, the relaxation scheme agreesv ery well with theory.

Some Practical Aspects of Overrelaxation in Diagnostic and Forecast Calculations

“…In a wide class of geostrophic, quasigeostrophic and stream function models, there is a jury problem in which the matrix A and the vector b are known functions of the state of the system (for instance, ,in the simplest barotropic models A is a fixed matrix and b is the time derivative of the relative vorticity), and the solution vector X is the time derivative of the geopotential or stream field. Some models require repeated solutions of the vertical velocity (omega) equation discussed by Stuart and O'Neill [9] and O'Brien [8].…”