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Energy dissipation law and maximum bound principle-preserving linear BDF2 schemes with variable steps for the Allen-Cahn equation

Bingyin Zhang, Hongfei Fu, Rihui Lan, Shusen Xie

TL;DR

This work develops a linear, second-order, variable-step BDF2-sESAV scheme for the Allen–Cahn equation that preserves discrete energy dissipation and the MBP. Central to the construction is a novel auxiliary functional $V(ullet)$ that ensures the first-order SAV approximation does not compromise second-order accuracy, enabling stabilized MBP-preserving schemes with a kernel-recombination framework. The unbalanced BDF2-sESAV-I scheme achieves unconditional energy dissipation under $0<r_k<4.864-oldsymbol{ abla}$ and MBP under $0<r_k<1+ oot2$, with rigorous $H^1$ and $L^0$ error estimates and a proven relation between modified and original energies. Numerical tests corroborate the theory, showing robust MBP and energy decay on uniform and adaptive grids and substantial speedups from adaptive time stepping for long-time simulations, making the method practical for phase-field computations.

Abstract

In this paper, we propose and analyze a linear, structure-preserving scalar auxiliary variable (SAV) method for solving the Allen--Cahn equation based on the second-order backward differentiation formula (BDF2) with variable time steps. To this end, we first design a novel and essential auxiliary functional that serves twofold functions: (i) ensuring that a first-order approximation to the auxiliary variable, which is essentially important for deriving the unconditional energy dissipation law, does not affect the second-order temporal accuracy of the phase function $φ$; and (ii) allowing us to develop effective stabilization terms that are helpful to establish the MBP-preserving linear methods. Together with this novel functional and standard central difference stencil, we then propose a linear, second-order variable-step BDF2 type stabilized exponential SAV scheme, namely BDF2-sESAV-I, which is shown to preserve both the discrete modified energy dissipation law under the temporal stepsize ratio $ 0 < r_{k} := τ_{k}/τ_{k-1} < 4.864 - δ$ with a positive constant $δ$ and the MBP under $ 0 < r_{k} < 1 + \sqrt{2} $. Moreover, an analysis of the approximation to the original energy by the modified one is presented. With the help of the kernel recombination technique, optimal $ H^{1}$- and $ L^{\infty}$-norm error estimates of the variable-step BDF2-sESAV-I scheme are rigorously established. Numerical examples are carried out to verify the theoretical results and demonstrate the effectiveness and efficiency of the proposed scheme.

Energy dissipation law and maximum bound principle-preserving linear BDF2 schemes with variable steps for the Allen-Cahn equation

TL;DR

This work develops a linear, second-order, variable-step BDF2-sESAV scheme for the Allen–Cahn equation that preserves discrete energy dissipation and the MBP. Central to the construction is a novel auxiliary functional that ensures the first-order SAV approximation does not compromise second-order accuracy, enabling stabilized MBP-preserving schemes with a kernel-recombination framework. The unbalanced BDF2-sESAV-I scheme achieves unconditional energy dissipation under and MBP under , with rigorous and error estimates and a proven relation between modified and original energies. Numerical tests corroborate the theory, showing robust MBP and energy decay on uniform and adaptive grids and substantial speedups from adaptive time stepping for long-time simulations, making the method practical for phase-field computations.

Abstract

In this paper, we propose and analyze a linear, structure-preserving scalar auxiliary variable (SAV) method for solving the Allen--Cahn equation based on the second-order backward differentiation formula (BDF2) with variable time steps. To this end, we first design a novel and essential auxiliary functional that serves twofold functions: (i) ensuring that a first-order approximation to the auxiliary variable, which is essentially important for deriving the unconditional energy dissipation law, does not affect the second-order temporal accuracy of the phase function ; and (ii) allowing us to develop effective stabilization terms that are helpful to establish the MBP-preserving linear methods. Together with this novel functional and standard central difference stencil, we then propose a linear, second-order variable-step BDF2 type stabilized exponential SAV scheme, namely BDF2-sESAV-I, which is shown to preserve both the discrete modified energy dissipation law under the temporal stepsize ratio with a positive constant and the MBP under . Moreover, an analysis of the approximation to the original energy by the modified one is presented. With the help of the kernel recombination technique, optimal - and -norm error estimates of the variable-step BDF2-sESAV-I scheme are rigorously established. Numerical examples are carried out to verify the theoretical results and demonstrate the effectiveness and efficiency of the proposed scheme.

Paper Structure

This paper contains 18 sections, 13 theorems, 128 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. A novel linear second-order variable-step BDF2-sESAV scheme
  3. Preliminaries
  4. An essential auxiliary functional
  5. Variable-step BDF2-sESAV schemes
  6. Approximation error of $V ( g_h(\hat{\phi}^{n}, R^{n-1}) ) -1$
  7. Discrete energy dissipation law and MBP
  8. Discrete maximum bound principle
  9. Discrete energy dissipation law
  10. Error analysis
  11. Numerical experiments
  12. Temporal convergence
  13. Preservation of energy dissipation law and MBP
  14. Long-term simulation with adaptive time-stepping
  15. Concluding remarks
  16. ...and 3 more sections

Key Result

Lemma 2.1

Assume the stepsize ratio condition $0 < r_k < r_{\max} := 4.864 - \delta$ for $2 \leq k \leq n$ holds, where $0< \delta < 4.864$ is any given small constant. Then, the convolution kernels $\{ b^{(n)}_{n-k} \}_{k=1}^{n}$ are positive definite, that is, for any real sequence $\left\{w_k\right\}_{k=0 where $G[ w^n ] := \frac{r_{n+1} \sqrt{r_{\max }}}{2\left(1+r_{n+1}\right)} \frac{w_n^2}{\tau_n}$.

Figures (12)

  • Figure 1: Plot of $V(z)$ on the interval $[-1,3]$
  • Figure 2: Approximations of $g_h(\hat{\phi}, R)$ and $V ( g_h(\hat{\phi}, R) )$ to $1$
  • Figure 3: The $L^{\infty}$-norm error of $\phi$ (left) and the error of $g_h(\phi,R)$ (right) generated by the (variable-step) BDF2-sESAV-I scheme: the double-well potential
  • Figure 4: The $L^{\infty}$-norm error of $\phi$ (left) and the error of $g_h(\phi,R)$ (right) generated by the (variable-step) BDF2-sESAV-I scheme: the Flory--Huggins potential
  • Figure 5: Evolution of simulated solutions in maximum-norm computed by the BDF2-sESAV schemes with different stabilization terms: the double-well potential
  • ...and 7 more figures

Theorems & Definitions (29)

  • Lemma 2.1
  • proof
  • Lemma 2.2: SINUM_2020_Liao
  • Lemma 2.3: SIREV_Du_2021
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6: JCM_Tang_2016CMS_Shen_2016
  • Lemma 2.7
  • proof
  • Remark 2.8
  • ...and 19 more