Splitting methods for three-dimensional bio-chemical transport

Splitting methods for the time integration of three-dimensional transport-chemistry models offer interesting prospects: second-order accuracy can be combined with sufficient stability, and the amount of implicitness can be reduced to a manageable level. Furthermore, exploiting the parallelization and vectorization features of the algorithm, a realistic simulation with many species over long time intervals becomes feasible. As an alternative to the usual splitting functions, such as co-ordinate splitting or operator splitting, we discuss in this paper a splitting function that is of hopscotch type. Both for a second-order, symmetric spatial discretization (resulting in a three-point coupling in each direction), and for a third-order, upwind discretization (giving rise to a five-point coupling, in general), we define a particular variant of this hopscotch splitting. These splitting functions will be combined with an appropriate splitting formula, resulting in second-order (in time) splitting methods. A common feature of both hopscotch splitting functions is that we have only coupling in the vertical direction, resulting in a stability behaviour that is independent of the vertical mesh size; this is an important property for transport in shallow water. Another characteristic of this hopscotch- type splitting is that it allows for an easy application of domain decomposition techniques in the horizontal directions. Two choices for the splitting formula will be presented. The resulting methods have been applied to a large-scale test problem and the numerical results will be discussed. Furthermore, we show performance results obtained on a Cray C98/4256. As part of the project TRUST (Transport and Reactions Unified by Splitting Techniques), preliminary versions of the schemes are available for benchmarking.