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

Qiong-Ao Huang, Wei Jiang, Jerry Zhijian Yang, Cheng Yuan

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.

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.

Paper Structure

This paper contains 12 sections, 8 theorems, 59 equations, 11 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonlinear energy-based SAV approach
  4. Lagrange multiplier-based SAV approach
  5. Weighted SAV approach
  6. First-order weighted SAV scheme
  7. Second-order weighted SAV scheme
  8. Numerical results
  9. Convergence rate
  10. Dynamic evolution in 2D
  11. Dynamic evolution in 3D
  12. Conclusion

Key Result

Theorem 1

(cf. ShenJ18ShenJ19) The nonlinear energy-based SAV-BE scheme eqa4 is unconditionally energy stable in the sense that

Figures (11)

  • Figure 1: (a): The evolution of weighting coefficient $\lambda_{\min}$ of curve I simulated using the weighted SAV-BE scheme \ref{['eqc3']} with $\tau=10^{-3}$ and $\gamma=0$. (b)-(c): The curve of the left-hand side of nonlinear scalar equation \ref{['eqc7']} (or $h(r,\lambda)$ as defined in \ref{['thmh1']}) with respect to $r$ at $t=1$ and $t=10$.
  • Figure 2: (a): The evolution of weighting coefficient $\lambda_{\min}$ of curve I simulated using the weighted SAV-BE scheme \ref{['eqc3']} with $\tau=10^{-3}$ and $\gamma=2$. (b)-(c): The curve of the left-hand side of nonlinear scalar equation \ref{['eqc7']} (or $h(r,\lambda)$ as defined in \ref{['thmh1']}) with respect to $r$ at $t=1$ and $t=10$.
  • Figure 3: Several snapshots of curve I simulated using the weighted SAV-BE scheme \ref{['eqc3']} with different $\tau$, respectively, where $\gamma=2$.
  • Figure 4: The evolution of (a) weighting coefficient $\lambda_{\min}$, (b) normalized energy, and (c) relative mass deviation of curve I simulated using the weighted SAV-BE scheme \ref{['eqc3']} with different time step sizes, respectively, where $\gamma=2$.
  • Figure 5: Several snapshots of curve II simulated using the SAV-CN scheme (second row) and the weighted SAV-CN scheme \ref{['eqd1']} (third row), respectively, where $\tau=10^{-3}$ and $\gamma=2$.
  • ...and 6 more figures

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • remark 1
  • remark 2
  • ...and 3 more