Efficient and accurate exponential SAV algorithms with relaxation for dissipative system

Abstract

In this paper, we construct two kinds of exponential SAV approach with relaxation (R-ESAV) for dissipative system. The constructed schemes are linear and unconditionally energy stable. They can guarantee the positive property of SAV without any assumption compared with R-SAV and R-GSAV approaches, preserve all the advantages of the ESAV approach and satisfy dissipation law with respect to a modified energy which is directly related to the original free energy. We also give the rigorous consistency estimates of the constructed schemes for the L2 gradient flows. Moreover, the second version of R-ESAV approach is easy to construct high-order BDFk schemes. Especially for Navier–Stokes equations, we construct two kinds of novel schemes based on the R-ESAV method. Finally, ample numerical examples are presented to exhibit that the proposed approaches are accurate and effective.