Energy-dissipative spectral renormalization exponential integrator method for gradient flow problems

TL;DR

This paper introduces a high-order, energy-dissipative spectral renormalization exponential integrator for gradient flow problems, improving accuracy and efficiency through a novel TDSR factor and adaptive time-stepping.

Contribution

The paper develops a new spectral renormalization exponential integrator that enforces energy dissipation and relaxes time step restrictions for gradient flow simulations.

Findings

Achieves high-order temporal accuracy.

Demonstrates energy dissipation law preservation.

Shows improved efficiency with adaptive time-stepping.

Abstract

In this paper, we present a novel spectral renormalization exponential integrator method for solving gradient flow problems. Our method is specifically designed to simultaneously satisfy discrete analogues of the energy dissipation laws and achieve high-order accuracy in time. To accomplish this, our method first incorporates the energy dissipation law into the target gradient flow equation by introducing a time-dependent spectral renormalization (TDSR) factor. Then, the coupled equations are discretized using the spectral approximation in space and the exponential time differencing (ETD) in time. Finally, the resulting fully discrete nonlinear system is decoupled and solved using the Picard iteration at each time step. Furthermore, we introduce an extra enforcing term into the system for updating the TDSR factor, which greatly relaxes the time step size restriction of the proposed method and enhances its computational efficiency. Extensive numerical tests with various gradient flows are also presented to demonstrate the accuracy and effectiveness of our method as well as its high efficiency when combined with an adaptive time-stepping strategy for long-term simulations.