arXiv subjects

Pinzhong Zheng

Publications and source records attributed to Pinzhong Zheng.

2 recordsLinked to original sources

arXiv2026

A unified framework for original energy dissipation of equilibrium-preserving exponential Runge--Kutta methods for $L^2$ gradient flows

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.

math.NA↗

arXiv2024

Maximum bound principle and original energy dissipation of arbitrarily high-order rescaled exponential time differencing Runge-Kutta schemes for Allen--Cahn equations

The energy dissipation law and the maximum bound principle are two critical physical properties of the Allen--Cahn equations. While many existing time-stepping methods are known to preserve the energy dissipation law, most apply to a modified form of energy. In this work, we demonstrate that, when the nonlinear term of the Allen--Cahn equation is Lipschitz continuous, a class of arbitrarily high-order exponential time differencing Runge--Kutta (ETDRK) schemes preserve the original energy dissipation property, under a mild step-size constraint. Additionally, we guarantee the Lipschitz condition on the nonlinear term by applying a rescaling post-processing technique, which ensures that the numerical solution unconditionally satisfies the maximum bound principle. Consequently, our proposed schemes maintain both the original energy dissipation law and the maximum bound principle and can achieve arbitrarily high-order accuracy. We also establish an optimal error estimate for the proposed schemes. Some numerical experiments are carried out to verify our theoretical results.

math.NA↗