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The Efficient Variable Time-stepping DLN Algorithms for the Allen-Cahn Model

YiMing Chen, Dianlun Luo, Wenlong Pei, Yulong Xing

Abstract

We consider a family of variable time-stepping Dahlquist-Liniger-Nevanlinna (DLN) schemes, which is unconditional non-linear stable and second order accurate, for the Allen-Cahn equation. The finite element methods are used for the spatial discretization. For the non-linear term, we combine the DLN scheme with two efficient temporal algorithms: partially implicit modified algorithm and scalar auxiliary variable algorithm. For both approaches, we prove the unconditional, long-term stability of the model energy under any arbitrary time step sequence. Moreover, we provide rigorous error analysis for the partially implicit modified algorithm with variable time-stepping. Efficient time adaptive algorithms based on these schemes are also proposed. Several one- and two-dimensional numerical tests are presented to verify the properties of the proposed time adaptive DLN methods.

The Efficient Variable Time-stepping DLN Algorithms for the Allen-Cahn Model

Abstract

We consider a family of variable time-stepping Dahlquist-Liniger-Nevanlinna (DLN) schemes, which is unconditional non-linear stable and second order accurate, for the Allen-Cahn equation. The finite element methods are used for the spatial discretization. For the non-linear term, we combine the DLN scheme with two efficient temporal algorithms: partially implicit modified algorithm and scalar auxiliary variable algorithm. For both approaches, we prove the unconditional, long-term stability of the model energy under any arbitrary time step sequence. Moreover, we provide rigorous error analysis for the partially implicit modified algorithm with variable time-stepping. Efficient time adaptive algorithms based on these schemes are also proposed. Several one- and two-dimensional numerical tests are presented to verify the properties of the proposed time adaptive DLN methods.
Paper Structure (23 sections, 7 theorems, 105 equations, 2 figures, 18 tables, 1 algorithm)

This paper contains 23 sections, 7 theorems, 105 equations, 2 figures, 18 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Notations
  3. The Modified DLN Algorithm
  4. Unconditional Stability in Model Energy
  5. Error Analysis
  6. The DLN-SAV algorithm
  7. The algorithm
  8. Numerical implementation
  9. Unconditional Stability in Model Energy
  10. Time Adaptivity
  11. Numerical Tests
  12. One-dimensional Traveling Wave Solution
  13. Constant Time Step
  14. Variable Time Step
  15. Adaptive Algorithms
  16. ...and 8 more sections

Key Result

Lemma 2.1

\newlabellemma:Gstab For any sequence $\{ y_{n} \}_{n=0}^{N}$ in $L^{2}(\Omega)$ over $\mathbb{R}$, $n \in \{ 1,2, \cdots, N-1 \}$ and $\theta \in [0,1]$, we have where the $G$-norm $\| \cdot \|_{G(\theta)}$ is defined by and the coefficients $\gamma_{\ell }^{(n)}$ are given by

Figures (2)

  • Figure 6.1: 2D Adaptive modified DLN algorithm with $\theta= 1$ converges to steady state with 5794 time steps.
  • Figure 6.2: 2D Adaptive DLN-SAV algorithm with $\theta= 1$ converges to steady state with 5896 time steps.

Theorems & Definitions (18)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 8 more