One of the most significant advances in numerical forecasting has been the introduction of the semi-implicit time differencing technique, which allows a computational saving of factors up to three over conventional explicit methods. The method does, however, involve casting the primitive equations in a more elaborate form and solution procedures of greater sophistication are required.
Along with this increased complexity have come several difficulties of formulation, including the derivation of the Laplacian operator for the Helmholtz equations; and the selection of an appropriate computational grid on which to solve these equations. In the past, semi-implicit schemes have encountered trouble from the above-mentioned sources and were only able to be successfully implemented after careful reworking of the methods (McGregor and Leslie (1977), McGregor et al. (1978)).
Because of the important economic value of the semi-implicit approach, it is timely to emphasize again what has been learned about the vagaries of the semi-implicit method, particularly when a scheme proposed recently by Kudoh (1978) appears to have overlooked the problems.