Arbitrarily High-order Unconditionally Energy Stable Schemes for Gradient Flow Models Using the Scalar Auxiliary Variable Approach

TL;DR

This paper introduces high-order, unconditionally energy stable numerical schemes for gradient flow models using the scalar auxiliary variable approach, enabling larger time steps and broad applicability across thermodynamically consistent models.

Contribution

The paper develops a general high-order scalar auxiliary variable (HSAV) method that achieves arbitrary order accuracy and unconditional energy stability for gradient flow models.

Findings

HSAV schemes reach high order accuracy in numerical experiments.

HSAV schemes allow larger time steps compared to standard SAV methods.

Numerical results confirm unconditional energy stability and efficiency.

Abstract

In this paper, we propose a novel family of high-order numerical schemes for the gradient flow models based on the scalar auxiliary variable (SAV) approach, which is named the high-order scalar auxiliary variable (HSAV) method. The newly proposed schemes could be shown to reach arbitrarily high order in time while preserving the energy dissipation law without any restriction on the time step size (i.e., unconditionally energy stable). The HSAV strategy is rather general that it does not depend on the specific expression of the effective free energy, such that it applies to a class of thermodynamically consistent gradient flow models arriving at semi-discrete high-order energy-stable schemes. We then employ the Fourier pseudospectral method for spatial discretization. The fully discrete schemes are also shown to be unconditionally energy stable. Furthermore, we present several numerical experiments on several widely-used gradient flow models, to demonstrate the accuracy, efficiency and unconditionally energy stability of the HSAV schemes. The numerical results verify that the HSAV schemes can reach the expected order of accuracy, and it allows a much larger time step size to reach the same accuracy than the standard SAV schemes.