Splitting Methods for Three-Dimensional Transport Models with Interaction Terms

Abstract

We investigate the use of splitting methods for the numerical integration of three-dimensional transport-chemistry models. In particular, we investigate various possibilities for the time discretization that can take advantage of the parallelization and vectorization facilities offered by multi-processor vector computers. To suppress wiggles in the numerical solution, we use third-order, upwind-biased discretization of the advection terms, resulting in a five-point coupling in each direction. As an alternative to the usual splitting functions, such as co-ordinate splitting or operator splitting, we consider a splitting function that is based on a three-coloured hopscotch-type splitting in the horizontal direction, whereas full coupling is retained in the vertical direction. Advantages of this splitting function are the easy application of domain decomposition techniques and unconditional stability in the vertical, which is an important property for transport in shallow water. The splitting method is obtained by combining the hopscotch-type splitting function with various second-order splitting formulae from the literature. Although some of the resulting methods are highly accurate, their stability behaviour (due to horizontal advection) is quite poor. Therefore we also discuss several new splitting formulae with the aim to improve the stability characteristics. It turns out that this is possible indeed, but the price to pay is a reduction of the accuracy. Therefore, such methods are to be preferred if accuracy is less crucial than stability; such a situation is frequently encountered in solving transport problems. As part of the project TRUST (Transport and Reactions Unified by Splitting Techniques), preliminary versions of the schemes are implemented on the Cray C98 4256 computer and are available for benchmarking.