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The one-parameter three-time-level family of O(Δt2)-accurate schemes, introduced in Rivest et al. to address the problem of the spurious resonant response of semi-implicit semi-Lagrangian schemes at large Courant number, has been generalized to a two-parameter family by introducing the possibility of evaluating total derivatives using an additional time level. The merits of different members of this family based on both theory and results are assessed. The additional degree of freedom might be expected a priori to permit a reduction of the time truncation errors while still maintaining stability and avoiding spurious resonance. Resonance, stability, and truncation error analyses for the proposed generalized family of schemes are given. The subfamily that is formally O(Δt3)-accurate is unfortunately unstable for gravity modes. Sample integrations for various members of the generalized family are shown. Results are consistent with theory, and stable nonresonant forecasts at large Courant number are possible for a range of values of the two free parameters. Of the two methods proposed in Rivest et al. for computing trajectories, the one using a piecewise-defined trajectory is to be preferred to that using a single great-circle arc since it is more accurate at a large time step for some members of the generalized family.
Côté et al. (Fri,) studied this question.