High-Order Schemes of Exponential Time Differencing for Stiff Systems with Nondiagonal Linear Part

TL;DR

This paper develops high-order exponential time differencing schemes for stiff systems with nondiagonal linear operators, using numerical calculation of exponential coefficients via auxiliary problems, and tests their accuracy and efficiency on complex physical models.

Contribution

It introduces a novel approach to implement exponential time differencing for nondiagonal systems by numerically computing exponential coefficients through auxiliary problems.

Findings

Methods show high accuracy in numerical experiments.

Schemes demonstrate improved computational performance.

Effective for complex physical models like Cahn--Hilliard and Fokker--Planck equations.

Abstract

Exponential time differencing methods is a power tool for high-performance numerical simulation of computationally challenging problems in condensed matter physics, fluid dynamics, chemical and biological physics, where mathematical models often possess fast oscillating or decaying modes -- in other words, are stiff systems. Practical implementation of these methods for the systems with nondiagonal linear part of equations is exacerbated by infeasibility of an analytical calculation of the exponential of a nondiagonal linear operator; in this case, the coefficients of the exponential time differencing scheme cannot be calculated analytically. We suggest an approach, where these coefficients are numerically calculated with auxiliary problems. We rewrite the high-order Runge--Kutta type schemes in terms of the solutions to these auxiliary problems and practically examine the accuracy and computational performance of these methods for a heterogeneous Cahn--Hilliard equation, a sixth-order spatial derivative equation governing pattern formation in the presence of an additional conservation law, and a Fokker--Planck equation governing macroscopic dynamics of a network of neurons.