A unified framework for original energy dissipation of equilibrium-preserving exponential Runge--Kutta methods for L2 gradient flows
Abstract
This paper develops a unified framework for establishing original energy dissipation of equilibrium-preserving exponential time differencing Runge--Kutta (ETDRK) methods for L2 gradient flows. The analysis is based on an auxiliary continuous evolution representation of a general ETDRK method, which allows the energy variation to be controlled without imposing positive-definiteness conditions on the Butcher coefficients or introducing stabilization terms. For a broad class of equilibrium-preserving ETDRK schemes, we derive an explicitly computable time-step restriction. Explicit restrictions are obtained for representative schemes of orders two through five, and the framework also applies to arbitrarily high-order interpolation-based ETDRK methods. For the latter class, the resulting admissible step sizes are substantially larger than those obtained from the previous interpolation-specific analysis. Numerical experiments for the Allen--Cahn equation confirm the theoretical dissipation restrictions.
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BibTeX
@misc{quan2026unified,
title = {{A unified framework for original energy dissipation of equilibrium-preserving exponential Runge--Kutta methods for $L^2$ gradient flows}},
author = {Chaoyu Quan and Pinzhong Zheng and Zhi Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11532},
eprint = {2610.11532},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11532},
}
RIS
TY - UNPB
TI - A unified framework for original energy dissipation of equilibrium-preserving exponential Runge--Kutta methods for L^2 gradient flows
AU - Chaoyu Quan
AU - Pinzhong Zheng
AU - Zhi Zhou
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11532
UR - https://arxiv.org/abs/2610.11532
AB - This paper develops a unified framework for establishing original energy dissipation of equilibrium-preserving exponential time differencing Runge--Kutta (ETDRK) methods for L^2 gradient flows. The analysis is based on an auxiliary continuous evolution representation of a general ETDRK method, which allows the energy variation to be controlled without imposing positive-definiteness conditions on the Butcher coefficients or introducing stabilization terms. For a broad class of equilibrium-preserving ETDRK schemes, we derive an explicitly computable time-step restriction. Explicit restrictions are obtained for representative schemes of orders two through five, and the framework also applies to arbitrarily high-order interpolation-based ETDRK methods. For the latter class, the resulting admissible step sizes are substantially larger than those obtained from the previous interpolation-specific analysis. Numerical experiments for the Allen--Cahn equation confirm the theoretical dissipation restrictions.
ER -