Deriving the Error of Time Filtered Leapfrog Scheme via Modified Equations

The leapfrog (LF) scheme is a cornerstone of numerical weather prediction and large-scale atmospheric modeling due to its computational efficiency and ability to preserve the amplitude of pure oscillations during long integrations. However, the three-time-level nature of the LF method introduces a parasitic computational mode that can grow over time and contaminate physical solutions. Traditionally, the Robert-Asselin (RA) filter has been employed to suppress this mode, but it inadvertently damps the physical mode, reducing the LF scheme's formal accuracy from second to first order. This research provides a rigorous mathematical analysis of modern time filters—specifically the Robert Asselin (RA), Robert Asselin Williams (RAW), and higher-order Robert Asselin (hoRA) filters— using the method of modified equations to evaluate phase and amplitude errors. By solving the linear system for each filtered scheme, we derive equivalent linear multistep methods and their corresponding two-term modified equations. Our findings confirm that the RAW filter significantly mitigates the physical mode damping of the RA filter, recovering second-order accuracy when parameters are optimally tuned (e.g., █(α=0.53)). Furthermore, the hoRA filter demonstrates even higher performance, attaining second-order accuracy generally and third-order accuracy for the specific choice of █(β=0.4). Numerical tests on the oscillation equation validate these theoretical derivations, showing that the hoRA filter yields the lowest amplitude and phase error magnitudes compared to the RA and RAW alternatives.