30. Options for solving elliptic equations

The external mode elliptic equation is solved by conjugate gradients with two possible forms for the numerics of the coefficient matrix. One form of the coefficient matrix must be chosen. In previous versions of MOM, a solution by sucessive over relaxation was available but this has been eliminated in favor of the conjugate gradient technique.

The accuracy of the external mode solution for the stream function $\Delta\psi = \psi^{\tau+1} - \psi^{\tau-1}$ is determined by setting a tolerence (see Section 14.4.5). The preferred method of solution is by conjugate gradients which is an iterative technique. The iteration is ended and the equation is considered solved when the estimated sum of future corrections to $\Delta \psi$ (the truncated ones) is less than the specified tolerence. The estimated sum is approximated assuming a geometric decrease in the maximum correction to $\Delta \psi$ with time.

Typically, as a rule of thumb, the number of scans taken to solve the external mode equation should not be $<< max\;(imt,\;jmt)$ when the forcing (i.e. windstress) is time dependent. If the number of scans is much less than this, it may indicate that the specified tolerence is set too large. As an example, if the specified tolerence is 10⁸, then in 10,000 time steps the computed solution is guaranteed to be within 1 Sverdrup $(1 \mbox{x} 10^{12})$ of the true solution at every point. This is a worst case assuming systematic errors always of the same sign. If time steps are very small, then $\Delta \psi$ will be small and the tolerence should be decreased particularly if long running experiments are involved. The amount of error that is tolerable is dependent on the goal of the experiment.

In cases where the windstress is set constant or zero, the number of scans may eventually get very small as equilibrium is approached. The reason is that a guess at the solution is made from the previous solutions. If the solution is not evolving, then eventually, the guess is within the specified tolerence and zero scans result. It is left up to the researcher to determine the appropriate value of the tolerence. The number of times that the scans are $< max\;(imt,\;jmt)$ are counted and a message is printed at the end of the execution.