Semi-Implicit Time Differencing

The basic ideas of semi-implicit time differencing are reviewed, including analytical results not widely referenced in the meteorological literature. It is shown that solving the resulting implicit equations by simple fixed-point iteration requires a relatively small time step to guarantee convergence, thus defeating the purpose of the semiimplicit method. In contrast, classical iteration methods (e.g., Jacobi, Gauss-Seidel) converge regardless of the time step if the implicit problem is elliptic. These ideas are illustrated in detail for the shallow-water equations in one dimension and sketched for the two-dimensional case. A detailed analysis of absolute stability is combined with previously derived conditions on the order of accuracy to identify promising schemes for the time integration of meteorological models. All appropriate cases of combined linear multistep methods of orders up to three (explicit) and two (implicit) are considered. In particular, a stable and promising semi-implicit formulation of the third-order Adams-Bashforth method is proposed. Technical Report No. 2002-01 Department of Mathematics and Computer Science Clarkson University, Potsdam, NY 13699–5815 This work was supported by the U. S. Department of Energy Subcontract to DOE Cooperative Agreement DE-FC02-01ER63163