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

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 latt...

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Bibliographic Details
Published in:arXiv.org 2024-11
Main Authors: Huang, Qiong-Ao, Jiang, Wei, Jerry Zhijian Yang, Cheng, Yuan
Format: Article
Language:English
Subjects:
Energy
Gradient flow
Lagrange multiplier
Mathematical analysis
Stability
Online Access:Get full text
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Summary: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.
ISSN:2331-8422