Improving the accuracy and consistency of the scalar auxiliary variable (SAV) method with relaxation

The scalar auxiliary variable (SAV) method was introduced by Shen et al. in [36] 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 resulting 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 relaxedSAV (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

Automated summary

The scalar auxiliary variable (SAV) method is widely used in solving thermodynamically consistent PDE problems, but its numerical scheme raises the issue of preserving the energy law. This paper presents the relaxedSAV (RSAV) method, which overcomes this issue and improves accuracy and consistency.