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A Regularized Auxiliary Variable (RAV) Approach for Gradient Flows

Zhaoyang Wang, Ping Lin

Abstract

In this paper, we propose a regularized auxiliary variable (RAV) approach and construct accurate and robust time-discrete schemes for a large class of gradient flows. By introducing an auxiliary variable $r=0$ and constructing an auxiliary equation that naturally fits into the energy relation, the numerical solution $r^{n+1}$ of the auxiliary variable is corrected at each time step to preserve consistency with the original system. The developed RAV scheme satisfies unconditional energy stability with respect to the original variables, and in certain cases the original energy law can be directly recovered. Furthermore, we obtain a uniform bound on the norm of the numerical solution, which allows us to establish the optimal error estimate in $L^\infty(0,T;H^2)$ for the second-order scheme without any restriction on the time step. We present ample numerical results, including comparisons with the scalar auxiliary variable (SAV) approach, to demonstrate the accuracy and effectiveness of the proposed RAV approach.

A Regularized Auxiliary Variable (RAV) Approach for Gradient Flows

Abstract

In this paper, we propose a regularized auxiliary variable (RAV) approach and construct accurate and robust time-discrete schemes for a large class of gradient flows. By introducing an auxiliary variable and constructing an auxiliary equation that naturally fits into the energy relation, the numerical solution of the auxiliary variable is corrected at each time step to preserve consistency with the original system. The developed RAV scheme satisfies unconditional energy stability with respect to the original variables, and in certain cases the original energy law can be directly recovered. Furthermore, we obtain a uniform bound on the norm of the numerical solution, which allows us to establish the optimal error estimate in for the second-order scheme without any restriction on the time step. We present ample numerical results, including comparisons with the scalar auxiliary variable (SAV) approach, to demonstrate the accuracy and effectiveness of the proposed RAV approach.

Paper Structure

This paper contains 12 sections, 4 theorems, 80 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The regularized auxiliary variable (RAV) approach and its schemes
  3. Gradient flows of a single function
  4. Extended to higher-order schemes
  5. RAV approach for gradient flows of multiple functions
  6. Error analysis
  7. Numerical experiments
  8. Cahn-Hilliard model
  9. Phase-field crystal (PFC) model
  10. Phase-field vesicle model
  11. Phase-field surfactant model
  12. Conclusions and remarks

Key Result

Theorem 2.1

(Energy stability) For $Q^{n+1}\geq 0$, the second-order RAV scheme (the2-4) is unconditionally energy stable in the sense that the following discrete energy law holds: When $Q^{n+1}<0$, we have Moreover, if $Q^{n+1}-r^{n}\geq 0$, the scheme satisfies the original energy dissipation law $E[\phi^{n+1}]-E[\phi^{n}]\leq 0$. $\blacktriangleleft$$\blacktriangleleft$

Figures (10)

  • Figure 1: Snapshots of the phase variable $\phi$ computed by the SAV–CN scheme at $t=5$. The line graphs give the discrepancy between the auxiliary variable and the original variable.
  • Figure 2: Snapshots of the phase variable $\phi$ computed by the RAV scheme (\ref{['the2-3']}) at $t=5$. The line graphs give the discrepancy between the auxiliary variable and the original variable.
  • Figure 3: Evolution of total energy for the RAV scheme with different time steps.
  • Figure 4: Snapshots of the phase variable $\phi$ computed by the SAV–CN scheme at $t=40$. The line graphs give the discrepancy between the auxiliary variable and the original variable.
  • Figure 5: Snapshots of the phase variable $\phi$ computed by the RAV scheme (\ref{['the2-3']}) at $t=40$. The line graphs give the discrepancy between the auxiliary variable and the original variable.
  • ...and 5 more figures

Theorems & Definitions (10)

  • Remark 2.1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.3
  • Theorem 3.1
  • proof