Method of lines transpose: High order L-stable O(N) schemes for parabolic equations using successive convolution

TL;DR

This paper introduces a high-order, L-stable, fast convolution-based solver for nonlinear parabolic equations, combining successive convolution techniques with dimensionally split methods for efficient multi-dimensional problems.

Contribution

It develops a novel high-order, L-stable solver using successive convolution and Green's functions, applicable to nonlinear parabolic equations in multiple dimensions.

Findings

Achieves high order accuracy in space and time.

Demonstrates stability and convergence through resolvent expansions.

Successfully applied to heat, Allen-Cahn, and Fitzhugh-Nagumo equations.

Abstract

We present a new solver for nonlinear parabolic problems that is L-stable and achieves high order accuracy in space and time. The solver is built by first constructing a single-dimensional heat equation solver that uses fast O(N) convolution. This fundamental solver has arbitrary order of accuracy in space, and is based on the use of the Green's function to invert a modified Helmholtz equation. Higher orders of accuracy in time are then constructed through a novel technique known as successive convolution (or resolvent expansions). These resolvent expansions facilitate our proofs of stability and convergence, and permit us to construct schemes that have provable stiff decay. The multi-dimensional solver is built by repeated application of dimensionally split independent fundamental solvers. Finally, we solve nonlinear parabolic problems by using the integrating factor method, where we apply the basic scheme to invert linear terms (that look like a heat equation), and make use of Hermite-Birkhoff interpolants to integrate the remaining nonlinear terms. Our solver is applied to several linear and nonlinear equations including heat, Allen-Cahn, and the Fitzhugh-Nagumo system of equations in one and two dimensions.