  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.
