A semi-generating function approach to the stability of implicit-explicit multistep methods for nonlinear parabolic equations

TL;DR

This paper introduces a semi-generating function approach for analyzing the stability of high-order implicit-explicit multistep methods applied to nonlinear parabolic equations, providing a unified framework and new methods.

Contribution

A novel semi-generating function technique is developed for stability analysis of IEMS methods, enabling the design of high-order methods with guaranteed stability for nonlinear parabolic equations.

Findings

Established unconditional stability criteria for IEMS methods.

Revisited and compared existing IEMS methods up to fifth order.

Proposed new eighth-order IEMS methods with high controllability intensity.

Abstract

The rigorous stability analysis of high-order implicit-explicit multistep (IEMS) methods for nonlinear parabolic equations by using discrete energy arguments is a long standing open issue due to their non-A-stable property. A novel semi-generating function approach combined with the global discrete energy analysis is suggested to the stability and convergence analysis of general IEMS methods for nonlinear parabolic equations. Inspired from the Grenander-Szeg\"{o} theorem for the Toeplitz matrix, the semi-generating function approach is used to handle the three groups of discrete coefficients via three complex rational polynomials on the unit circle. A unified theoretical framework is then presented to establish the unconditional stability of IEMS methods if the minimum eigenvalue of composite convolution kernels for the implicit part is properly large and the spectral norm bound of composite convolution kernels for the explicit part is properly small. An indicator, called implicit-explicit controllability intensity, is then introduced to evaluate the degree of controllability of implicit part over explicit part. Some of existing IEMS methods, up to the fifth-order time accuracy, are revisited and compared by computing the associated implicit-explicit controllability intensities such that one can choose certain IEMS method or proper parameter to maintain the unconditional stability for a specific nonlinear parabolic model. We also propose a new parameterized class of IEMS methods, up to the eighth-order time accuracy, which satisfy the priori settings of our theory and have a large value of the implicit-explicit controllability intensity by choosing proper parameter so that they would be well suited for a wide class of nonlinear parabolic problems.