The Third-Order Adams-Bashforth Method: An Attractive Alternative to Leapfrog Time Differencing

The third-order Adams–Bashforth method is compared with the leapfrog scheme. Like the leapfrog scheme, the third-order Adams–Bashforth method is an explicit technique that requires just one function evaluation per time step. Yet the third-order Adams–Bashforth method is not subject to time splitting instability and it is more accurate than the leapfrog scheme. In particular, the O[(Δt)4] amplitude error of the third-order Adams–Bashforth method can be a marked improvement over the O[(Δt)2] amplitude error generated by the Asselin-filtered leapfrog scheme—even when the filter factor is very small. The O[(Δt)4] phase-speed errors associated with third-order Adams–Bashforth time differencing can also be significantly less than the O[(Δt)2] errors produced by the leapfrog method. The third-order Adams–Bashforth method does use more storage than the leapfrog method, but its storage requirements are not particularly burdensome. Several numerical examples are provided illustrating the superiority of third-order Adams–Bashforth time differencing. Other higher-order alternatives to the Adams–Bashforth method are also surveyed. A discussion is presented describing the general relationship between the truncation error of an ordinary differential solver and the amplitude and phase-speed errors that develop when the scheme is used to integrate oscillatory systems.