Arbitrary High-Order Maximum Principle-Preserving and Energy Dissipating Schemes for Gradient Flows

TL;DR

This paper introduces a novel framework for high-order, structure-preserving numerical schemes for gradient flows, ensuring maximum principle and energy dissipation simultaneously with improved accuracy and stability.

Contribution

The authors develop a new predictor-corrector framework that guarantees energy stability and maximum principle preservation for high-order schemes applied to gradient flows.

Findings

The new schemes achieve high-order accuracy while preserving energy dissipation.

Numerical experiments demonstrate the schemes' efficiency and ability to avoid oscillations.

The framework is applicable to exponential time differencing Runge-Kutta methods.

Abstract

For gradient flows, the existing structure-preserving schemes are difficult to achieve arbitrary high-order accuracy in time while preserving maximum-principle (MBP) and energy dissipating simultaneously. In this paper, we develop a new framework for constructing structure-preserving schemes which shall preserve those nice properties. By introducing KKT-conditions for energy dissipating and bound-preserving, we rewrite the original gradient flow into an expanded and coupled system. We shall utilize a novel predictor-corrector-corrector framework, termed the PCC method, which consists of a prediction from any numerical scheme to the user's favor, followed by two correction steps designed to enforce energy stability and MBP, respectively. We take the exponential time differencing Runge-Kutta scheme (ETDRK) as an example and establish the unique solvability and robust error analysis for our new framework. Extensive numerical experiments are provided to validate the efficiency and accuracy of our new approach. Enough numerical comparisons with the existing popular schemes are shown that our structure-preserving schemes can avoid numerical oscillations and capture the exact evolution of energy.