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Convergence of long-time stable variable-step arbitrary order ETD-MS scheme for gradient flows with Lipschitz nonlinearity

Wenbin Chen, Zhaohui Fu, Shun Wang, Xiaoming Wang

TL;DR

This work develops variable-step, arbitrarily high-order exponential time differencing multistep (ETD-MS) schemes for gradient flows with Lipschitz nonlinearity, proving unconditional energy stability via a slightly perturbed (modified) energy. It establishes optimal convergence rates under mild conditions on step sizes and local step ratios. The authors adapt the framework to the no-slope-selection NSS thin-film epitaxial growth model, verifying assumptions and showing stability bounds and energy behavior. Numerical experiments, including adaptive time stepping, confirm the theoretical results and demonstrate efficient long-time integration with accurate coarsening dynamics.

Abstract

We analyze a variable-step extension of a family of arbitrarily high-order exponential time differencing multistep (ETD-MS) schemes recently developed by the authors. We prove that the schemes are unconditionally stable in the sense that a modified energy-representing a slight perturbation of the original energy-decreases monotonically over time, provided the nonlinearity is Lipschitz continuous in some appropriate sense. Moreover, we establish optimal-order convergence under mild conditions on the time-step size and local time-step ratio. Numerical experiments on the thin film epitaxial growth model without slope selection, employing a novel variable-step second-order scheme, validate the theoretical findings as well as its potential in developing highly efficient time-adaptive solution.

Convergence of long-time stable variable-step arbitrary order ETD-MS scheme for gradient flows with Lipschitz nonlinearity

TL;DR

This work develops variable-step, arbitrarily high-order exponential time differencing multistep (ETD-MS) schemes for gradient flows with Lipschitz nonlinearity, proving unconditional energy stability via a slightly perturbed (modified) energy. It establishes optimal convergence rates under mild conditions on step sizes and local step ratios. The authors adapt the framework to the no-slope-selection NSS thin-film epitaxial growth model, verifying assumptions and showing stability bounds and energy behavior. Numerical experiments, including adaptive time stepping, confirm the theoretical results and demonstrate efficient long-time integration with accurate coarsening dynamics.

Abstract

We analyze a variable-step extension of a family of arbitrarily high-order exponential time differencing multistep (ETD-MS) schemes recently developed by the authors. We prove that the schemes are unconditionally stable in the sense that a modified energy-representing a slight perturbation of the original energy-decreases monotonically over time, provided the nonlinearity is Lipschitz continuous in some appropriate sense. Moreover, we establish optimal-order convergence under mild conditions on the time-step size and local time-step ratio. Numerical experiments on the thin film epitaxial growth model without slope selection, employing a novel variable-step second-order scheme, validate the theoretical findings as well as its potential in developing highly efficient time-adaptive solution.

Paper Structure

This paper contains 12 sections, 4 theorems, 54 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. The continuous problem
  3. The temporal semi-discrete scheme and its long time stability
  4. The scheme
  5. Energy stability
  6. Convergence analysis
  7. Application to no-slope-selection (NSS) thin-film epitaxial growth model
  8. Numerical results
  9. Temporal convergence
  10. Coarsening process
  11. Adaptive results
  12. Conclusion

Key Result

Lemma 3.1

Let $\beta,\gamma$ be two non-negative numbers, $q\in [0,1]$, and $p(k)$ is chosen so that $p(k)>\max\{\beta,\gamma\}$. Then for any $u\in V^{\beta}$ and $v\in V^{\gamma}$, and arbitrary positive constants $\hat{C}, \tilde{C}$, the following inequalities hold for different cases of $\beta$ and $\gam

Figures (4)

  • Figure 1: Snapshots of the numerical solutions for scheme \ref{['eq:second order nss']}.
  • Figure 2: Semi-log plot of the energy $E$. Fitted line has the form $a \ln (t) + b$, with coefficients $a = -39.93$, $b = -50.33$.
  • Figure 3: The log-log plot of (1) the average surface height $h$ and (2) the average slope $m$. Fitted lines have the form $a t^b$, with coefficients (1) $a = 0.3319$, $b = 0.5414$ and (2) $a = 2.032$, $b = 0.2705$.
  • Figure 4: Solutions using large time steps $\tau=0.1$ (first line), adaptive time steps (second line) and small time steps $\tau=0.001$ (third line), and time steps curve and energy evolution (fourth line)

Theorems & Definitions (12)

  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Theorem 3.5
  • proof
  • ...and 2 more