2015/08/13 by Matthew F. Causley, Hana Cho, Causley, Matthew F. +5 · 5 citations
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Applied mathematics #Computer science #Convolution (computer science) #Differential Equations and Numerical Methods #Heat equation #Hermite polynomials #Mathematical analysis #Mathematical optimization #Mathematics #Nonlinear system #Numerical methods for differential equations #Physics #Resolvent #Solver
paper · pdf · doi:10.48550/arxiv.1508.03105
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2015/08/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
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.