Step-by-step solving schemes based on scalar auxiliary variable and invariant energy quadratization approaches for gradient flows

Abstract

In this paper, we propose several novel numerical techniques to deal with nonlinear terms in gradient flows. These step-by-step solving schemes, termed 3S-SAV and 3S-IEQ schemes, are based on recently popular scalar auxiliary variable (SAV) and invariant energy quadratization (IEQ) approaches. By introducing a novel auxiliary variable η to replace the original one in the traditional SAV approach, we rewrite the equivalent gradient flow systems. Then, linear, decoupled, and unconditionally energy stable numerical schemes are constructed. More importantly, the phase function ϕ and auxiliary variable η can be calculated step-by-step which can save more CPU time in calculation. Similar procedure can also be used to modify the IEQ approach. Specially, the proposed 3S-IEQ approach only needs to solve linear equation with constant coefficients while the system with variable coefficients must be calculated for the traditional IEQ approach. Two comparative studies of traditional SAV/IEQ and 3S-SAV/3S-IEQ approaches are considered to show the accuracy and efficiency. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy.