A weighted scalar auxiliary variable method for solving gradient flows: bridging the nonlinear energy-based and Lagrange multiplier approaches

Abstract: Two primary scalar auxiliary variable (SAV) approaches are widely applied for simulating gradient flow systems, i.e., the nonlinear energy-based approach and the Lagrange multiplier approach. The former guarantees unconditional energy stability through a modified energy formulation, whereas the latter preserves original energy stability but requires small time steps for numerical solutions. In this paper, we introduce a novel weighted SAV method which integrates these two approaches for the first time. Our method leverages the advantages of both approaches: (i) it ensures the existence of numerical solutions for any time step size with a sufficiently large weight coefficient; (ii) by using a weight coefficient smaller than one, it achieves a discrete energy closer to the original, potentially ensuring stability under mild conditions; and (iii) it maintains consistency in computational cost by utilizing the same time/spatial discretization formulas. We present several theorems and numerical experiments to validate the accuracy, energy stability and superiority of our proposed method.