A general class of linear unconditionally energy stable schemes for the gradient flows

TL;DR

This paper introduces a broad class of linear, high-order, unconditionally energy stable schemes for gradient flows, based on the SAV technique and general linear time discretization, applicable to various models.

Contribution

It develops a general framework for linear energy stable schemes using SAV and GLTD, encompassing existing and new schemes with high-order accuracy.

Findings

Schemes are unconditionally energy stable under algebraic stability of GLTD.

Convergence order depends on stage order and extrapolation points.

Numerical tests confirm stability, accuracy, and effectiveness.

Abstract

This paper studies a class of linear unconditionally energy stable schemes for the gradient flows. Such schemes are built on the SAV technique and the general linear time discretization (GLTD) as well as the linearization based on the extrapolation for the nonlinear term, and may be arbitrarily high-order accurate and very general, containing many existing SAV schemes and new SAV schemes. It is shown that the semi-discrete-in-time schemes are unconditionally energy stable when the GLTD is algebraically stable, and are convergent with the order of $\min\{\hat{q},\nu\}$ under the diagonal stability and some suitable regularity and accurate starting values, where $\hat{q}$ is the generalized stage order of the GLTD and $\nu$ denotes the number of the extrapolation points in time. The energy stability results can be easily extended to the fully discrete schemes, for example, if the Fourier spectral method is employed in space when the periodic boundary conditions are specified. Some numerical experiments on the Allen-Cahn, Cahn-Hilliard, and phase field crystal models are conducted to validate those theories as well as the effectiveness, the energy stability and the accuracy of our schemes.