An Iterated, Multipoint Differential Transform Method for Numerically Evolving PDE IVPs

TL;DR

This paper introduces an iterated, multipoint differential transform method (IMDTM) that stores and propagates spatial derivatives for solving PDE initial-value problems, improving efficiency, accuracy, and solution verification.

Contribution

The paper presents a novel IMDTM scheme that efficiently propagates spatial derivatives and derives a generalized finite-difference stencil for higher-order derivatives, enhancing PDE evolution methods.

Findings

IMDTM improves speed and accuracy over traditional methods

Stored spatial derivatives aid in solution verification

The generalized stencil efficiently computes higher-order derivatives

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 some 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, it is demonstrated 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 which can confer several advantages, including aiding solution verification. Lastly, in order to efficiently implement the IMDTM scheme, a generalized finite-difference stencil formula is derived which can take advantage of multiple higher-order spatial derivatives when computing even-higher-order derivatives. As is demonstrated, the performance of these techniques compares favorably to other explicit evolution schemes in terms of speed, memory footprint and accuracy.