A semi-generating function approach to the stability of implicit-explicit multistep methods for nonlinear parabolic equations
Abstract
The rigorous stability analysis of high-order implicit-explicit linear multistep (IELM) methods for nonlinear parabolic equations by using discrete energy arguments is a long standing open issue due to their non-A-stability property. A novel semi-generating function approach combined with a global discrete energy analysis is suggested for the stability and convergence of general IELM methods in solving nonlinear parabolic equations. Inspired from the Grenander-Szegő theorem for Toeplitz matrices, the semi-generating function approach is used to handle the three groups of discrete coefficients via three complex polynomials on the unit circle. A unified theoretical framework is then presented to establish the unconditional stability of IELM 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 the implicit part over the explicit part. Some of the existing IELM methods, up to fifth-order time accuracy, are revisited and compared by computing the associated implicit-explicit controllability intensities such that one can choose an IELM method or proper parameter to maintain the unconditional stability for a specific nonlinear parabolic model. We also propose a new parameterized class of IELM methods, up to the ninth-order time accuracy, which satisfy the a priori settings of our theory and have a large value of the implicit-explicit controllability intensity by choosing a proper parameter so that they would be well suited for a wide class of nonlinear parabolic problems.
Related
- Energy stability of explicit--implicit--null time discretizations for nonlinear degenerate parabolic equations with state-dependent coefficients
- Stability and Error Analysis of Multistep IMEX Local Discontinuous Galerkin Methods for One-Dimensional Nonlinear Fourth-Order Evolution Problems
- Harnack-type inequalities for nonlocal doubly nonlinear parabolic equations
- Liouville Theorems for Semilinear Parabolic Equations Involving the Nonlinear Gradient Term
- Characterization of intrinsic parabolic Harnack inequalities for doubly nonlinear equations of slow diffusion type
BibTeX
@misc{liao2026semigenerating,
title = {{A semi-generating function approach to the stability of implicit-explicit multistep methods for nonlinear parabolic equations}},
author = {Hong-lin Liao and Chaoyu Quan and Tao Tang and Tao Zhou},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2605.05619},
eprint = {2605.05619},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.05619},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A semi-generating function approach to the stability of implicit-explicit multistep methods for nonlinear parabolic equations
AU - Hong-lin Liao
AU - Chaoyu Quan
AU - Tao Tang
AU - Tao Zhou
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2605.05619
UR - https://arxiv.org/abs/2605.05619
AB - The rigorous stability analysis of high-order implicit-explicit linear multistep (IELM) methods for nonlinear parabolic equations by using discrete energy arguments is a long standing open issue due to their non-A-stability property. A novel semi-generating function approach combined with a global discrete energy analysis is suggested for the stability and convergence of general IELM methods in solving nonlinear parabolic equations. Inspired from the Grenander-Szegő theorem for Toeplitz matrices, the semi-generating function approach is used to handle the three groups of discrete coefficients via three complex polynomials on the unit circle. A unified theoretical framework is then presented to establish the unconditional stability of IELM 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 the implicit part over the explicit part. Some of the existing IELM methods, up to fifth-order time accuracy, are revisited and compared by computing the associated implicit-explicit controllability intensities such that one can choose an IELM method or proper parameter to maintain the unconditional stability for a specific nonlinear parabolic model. We also propose a new parameterized class of IELM methods, up to the ninth-order time accuracy, which satisfy the a priori settings of our theory and have a large value of the implicit-explicit controllability intensity by choosing a proper parameter so that they would be well suited for a wide class of nonlinear parabolic problems.
ER -