Analysis of parallel versus sequential splittings for time-stepping physical parameterizations

Various numerical issues concerning different approaches to the time stepping of physical parameterizations in numerical weather prediction (NWP) and climate models are examined. Parallel-split and sequential-split methods are explained and analyzed in the context of simple model equations. Terms arising from the use of splitting techniques produce erroneous solutions if the time step is large ( of the size typically used in semi-implicit semi-Lagrangian models). Errors in steady-state solutions are examined in particular, as these may lead to systematic biases and climate drift. Splitting methods are then applied to a multiple timescale problem. For large time steps, a useful scheme should produce an accurate discrete representation of the reduced system, which has fast modes removed. Parallel splitting may be of limited use because only explicit versions model reduced systems without splitting errors, but such versions cannot stably integrate fast modes with acceptably large time steps. In a numerical context, sequential-splitting schemes are more flexible. Second-order schemes can be more accurate than first-order ones if the time step is very large, provided careful data initialization is performed to prevent noisy solutions for stiff problems.