A numerical method based on leapfrog and a fourth-order implicit time filter

A time-stepping scheme is proposed. It is based on the leapfrog method and a fourth-order time filter. The scheme requires only one evaluation per time step and uses an implicit filter, but the effort needed to implement it in an explicit manner is trivial. Comparative tests demonstrate that the proposed scheme produces numerical approximations that are more stable and highly accurate compared to the standard Robert-Asselin (RA) and the Robert-Asselin-Williams (RAW) filtered leapfrog scheme, even though both methods use filter coefficients that are tuned such that the 2Δt modes are damped at the same rate. Formal stability analysis demonstrates that the proposed method generates amplitude errors of O[(Δt)⁴], implying third-order accuracy. This contrasts with the O[(Δt)²] errors produced by the standard RA and RAW filtered leapfrog. A second scheme that produces amplitude errors of O[(Δt)⁶] is also presented. The proposed scheme is found to do well at controlling numerical instabilities arising in the diffusion equation and in nonlinear computations using Lorenz's system and the global shallow-water spectral model. In addition to noticeably improving the resolution of the physical modes, the proposed method is simple to implement and has a wider region of stability compared to the existing time-filtered leapfrog schemes. This makes the proposed method a potential alternative for use in atmospheric, oceanic, and climate modeling.