An iterated, multipoint differential transform method for numerically evolving partial differential equation initial-value problems

Abstract

Traditional numerical techniques for solving time-dependent partial differential equation (PDE) initial-value problems (IVPs) store a truncated representation of the function values and a certain number of their time derivatives at each time step. Although redundant in the dx → 0 limit, what if spatial derivatives were also stored? This paper presents an iterated, multipoint differential transform method (IMDTM) for numerically evolving PDE IVPs. Using this scheme, we demonstrate that stored spatial derivatives can be propagated in an efficient and self-consistent manner and can effectively contribute to the evolution procedure in a way that can confer several advantages, including aiding in solution verification. Lastly, in order to efficiently implement the IMDTM scheme, a generalized finite-difference stencil formula is derived that can take advantage of multiple higher-order spatial derivatives when computing even-higher-order derivatives. As demonstrated here, the performance of these techniques compares favorably to other explicit evolution schemes in terms of speed, memory footprint and accuracy.