Improving the Accuracy and Consistency of the Scalar Auxiliary Variable (SAV) Method with Relaxation

TL;DR

This paper introduces the relaxed-SAV (RSAV) method, enhancing the original SAV approach by penalizing numerical errors to improve accuracy and consistency in solving thermodynamically consistent PDEs.

Contribution

The paper proposes a relaxation technique for the SAV method, called RSAV, which maintains its advantages while significantly improving accuracy and consistency.

Findings

RSAV improves the accuracy of auxiliary variables.

RSAV maintains energy stability and linearity.

Numerical examples demonstrate enhanced performance.

Abstract

The scalar auxiliary variable (SAV) method was introduced by Shen et al. and has been broadly used to solve thermodynamically consistent PDE problems. By utilizing scalar auxiliary variables, the original PDE problems are reformulated into equivalent PDE problems. The advantages of the SAV approach, such as linearity, unconditionally energy stability, and easy-to-implement, are prevalent. However, there is still an open issue unresolved, i.e., the numerical schemes resulted from the SAV method preserve a "modified" energy law according to the auxiliary variables instead of the original variables. Truncation errors are introduced during numerical calculations so that the numerical solutions of the auxiliary variables are no longer equivalent to their original continuous definitions. In other words, even though the SAV scheme satisfies a modified energy law, it does not necessarily satisfy the energy law of the original PDE models. This paper presents one essential relaxation technique to overcome this issue, which we named the relaxed-SAV (RSAV) method. Our RSAV method penalizes the numerical errors of the auxiliary variables by a relaxation technique. In general, the RSAV method keeps all the advantages of the baseline SAV method and improves its accuracy and consistency noticeably. Several examples have been presented to demonstrate the effectiveness of the RSAV approach.