SPLITTING METHODS FOR PROBLEMS WITH DIFFERENT TIMESCALES

The time step for the leapfrog scheme for a symmetric hyperbolic syste m with multiple timescales is limited by the Courant-Friedrichs-Lewy c ondition based on the fastest speed present. However, in many physical cases, most of the energy is in the slowest wave, and for this wave t he use of the above time step implies that the time truncation error i s much smaller than the spatial truncation error. A number of methods have been proposed to overcome this imbalance-for example, the semi-im plicit method and the additive splitting technique originally proposed by Marchuk with variations attributable to Strang, and Klemp and Wilh elmson. An analysis of the Marchuk splitting method for multiple times cale systems shows that if a time step based on the slow speed is used , the accuracy of the method cannot be proved, and in practice the met hod is quite inaccurate. If a time step is chosen that is between the two extremes, then the Klemp and Wilhelmson method can be used, but on ly if an ad hoc stabilization mechanism is added. The additional compu tational burden required to maintain the accuracy and the stability of the split-explicit method leads to the conclusion that it is no more efficient than the leapfrog method trivially modified to handle comput ationally expensive smooth forcing terms. Using the mathematical analy sis developed in a previous manuscript, it is shown that splitting sch emes are not appropriate for badly skewed hyperbolic systems. In a num ber of atmospheric models, the semi-implicit method is used to treat t he badly skewed vertical sound wave terms. This leads to the excitatio n of the high-frequency waves in a nonphysical manner. It is also show n that this is equivalent to solving the primitive equations; that is, a model using this method for the large-scale case will be ill posed at the lateral boundaries. The multiscale system for meteorology was i ntroduced by Browning and Kreiss to overcome exactly these problems.