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

TL;DR

This paper develops and analyzes a broad class of linear, unconditionally energy stable schemes for gradient flows, extending previous work with more general linear time discretizations and validating stability through numerical experiments.

Contribution

It introduces a new class of SAV-GL schemes with non-algebraically stable parameters and proves their unconditional energy stability using algebraic identities and inequalities.

Findings

Schemes are unconditionally energy stable for gradient flows.

Numerical experiments confirm stability and appropriate time step sizes.

Fourier pseudo-spectral method effectively eliminates aliasing errors.

Abstract

This paper continues to study linear and unconditionally modified-energy stable (abbreviated as SAV-GL) schemes for the gradient flows. The schemes are built on the SAV technique and the general linear time discretizations (GLTD) as well as the extrapolation for the nonlinear term. Different from [44], the GLTDs with three parameters discussed here are not necessarily algebraically stable. Some algebraic identities are derived by using the method of undetermined coefficients and further used to establish the modified-energy inequalities for the unconditional modified-energy stability of the semi-discrete-in-time SAV-GL schemes. It is worth emphasizing that those algebraic identities or energy inequalities are not necessarily unique for some choices of three parameters in the GLTDs. Numerical experiments on the Allen-Cahn, the Cahn-Hilliard and the phase field crystal models with the periodic boundary conditions are conducted to validate the unconditional modified-energy stability of the SAV-GL schemes, where the Fourier pseudo-spectral method is employed in space with the zero-padding to eliminate the aliasing error and the time stepsizes for ensuring the original-energy decay are estimated by using the stability regions of our SAV-GL schemes for the test equation. The resulting time stepsize constraints for the SAV-GL schemes are almost consistent with the numerical results on the above gradient flow models.