High-Order Multiderivative IMEX Schemes

TL;DR

This paper extends high-order multiderivative IMEX schemes based on Hermite interpolation for stiff differential equations, demonstrating their stability, convergence, and applicability to PDEs, with potential benefits in parallelism and memory efficiency.

Contribution

It generalizes 4th-order IMEX schemes to arbitrary high orders (6th to 12th), proves their convergence, and applies Hermite methods to PDEs for the first time.

Findings

Developed high-order Hermite IMEX schemes up to 12th order.

Proved stability and convergence of these schemes.

Applied methods to both ODEs and PDEs, including Burgers' equation.

Abstract

Recently, a 4th-order asymptotic preserving multiderivative implicit-explicit (IMEX) scheme was developed (Sch\"utz and Seal 2020, arXiv:2001.08268). This scheme is based on a 4th-order Hermite interpolation in time, and uses an approach based on operator splitting that converges to the underlying quadrature if iterated sufficiently. Hermite schemes have been used in astrophysics for decades, particularly for N-body calculations, but not in a form suitable for solving stiff equations. In this work, we extend the scheme presented in Sch\"utz and Seal 2020 to higher orders. Such high-order schemes offer advantages when one aims to find high-precision solutions to systems of differential equations containing stiff terms, which occur throughout the physical sciences. We begin by deriving Hermite schemes of arbitrary order and discussing the stability of these formulas. Afterwards, we demonstrate how the method of Sch\"utz and Seal 2020 generalises in a straightforward manner to any of these schemes, and prove convergence properties of the resulting IMEX schemes. We then present results for methods ranging from 6th to 12th order and explore a selection of test problems, including both linear and nonlinear ordinary differential equations and Burgers' equation. To our knowledge this is also the first time that Hermite time-stepping methods have been applied to partial differential equations. We then discuss some benefits of these schemes, such as their potential for parallelism and low memory usage, as well as limitations and potential drawbacks.