A Problem with the Robert–Asselin Time Filter for Three-Time-Level Semi-Implicit Semi-Lagrangian Discretizations

Abstract

Three-time-level schemes for first-order differential equations are affected by the existence of computational modes. The Robert?Asselin filter was developed in an Eulerian context to selectively damp such modes while leaving the physical solution virtually untouched. Here previous analysis of this time filter for simple equation sets is extended to three-time-level semi-implicit semi-Lagrangian discretizations of the hydrostatic primitive equations. For Courant numbers greater than one-half, a surprising loss of selectivity is found when compared to that previously observed in the fully Eulerian context. Not only can the physical solution be undesirably more damped than the computational one, it can be significantly more damped. This problem is particularly serious in polar regions of a global gridpoint model due to the convergence of the meridians, although its severity might be reduced by using horizontal diffusion or spatial filtering to make the flow field more isotropic. (It is, however, no more serious there than elsewhere for a triangularly truncated spectral model because of its isotropic representation.) The problem's source is attributed to the current practice of integrating the equations along the trajectories but time filtering at grid points. Although time filtering along a trajectory would mitigate the problem, it would be costly in practice because of the additional computations required.