[Paper Review] Mesh-robustness of the variable steps BDF2 method for the Cahn-Hilliard model.

Hong-lin Liao, Bingquan Ji+2 more|arXiv (Cornell University)|Feb 7, 2021

Solidification and crystal growth phenomena27 references4 citations

TL;DR

This paper proposes a variable-step BDF2 method for the Cahn-Hilliard model that maintains energy dissipation and mesh-robust stability under a user-defined step-ratio restriction $ r_{\text{user}} < 4.864 $. By leveraging discrete orthogonal convolution kernels and novel embedding inequalities, it achieves $ L^2 $-norm error estimates independent of time-step variations, ensuring mesh-robust convergence and enabling efficient long-time simulations via adaptive time-stepping.

ABSTRACT

The two-step backward differential formula (BDF2) implicit method with unequal time-steps is investigated for the Cahn-Hilliard model by focusing on the numerical influences of time-step variations. The suggested method is proved to preserve a modified energy dissipation law at the discrete levels if the adjoint time-step ratios fulfill a new step-ratio restriction $0<r_k:= au_k/ au_{k-1}\le r_{\mathrm{user}}$ ($r_{\mathrm{user}}$ can be chosen by the user such that $r_{\mathrm{user}}<4.864$), such that it is mesh-robustly stable in an energy norm. We view the BDF2 formula as a convolution approximation of the first time derivative and perform the error analysis by using the recent suggested discrete orthogonal convolution kernels. By developing some novel convolution embedding inequalities with respect to the orthogonal convolution kernels, an $L^2$ norm error estimate is established at the first time under the updated step-ratio restriction $0<r_k\le r_{\mathrm{user}}$.The time-stepping scheme is mesh-robustly convergent in the sense that the convergence constant (prefactor) in the error estimate is independent of the adjoint time-step ratios. On the basis of ample tests on random time meshes, a useful adaptive time-stepping strategy is applied to efficiently capture the multi-scale behaviors and to accelerate the long-time simulation approaching the steady state.

Motivation & Objective

• To address the challenge of maintaining energy stability and convergence in variable-step BDF2 schemes for the Cahn-Hilliard model.
• To analyze the impact of time-step variations on numerical stability and accuracy in long-time simulations.
• To establish a mesh-robust convergence theory where the error bound is independent of time-step ratios.
• To develop a practical adaptive time-stepping strategy that efficiently captures multi-scale dynamics.

Proposed method

• The BDF2 method is formulated as a convolution approximation of the first time derivative using variable time steps.
• Discrete orthogonal convolution kernels are employed to perform precise error analysis in the time-discretization framework.
• Novel convolution embedding inequalities are derived for the orthogonal convolution kernels to bound the temporal error growth.
• A step-ratio restriction $ 0 < r_k := \Delta t_k / \Delta t_{k-1} \leq r_{\text{user}} $ with $ r_{\text{user}} < 4.864 $ is introduced to preserve energy dissipation at the discrete level.
• The method is proven to satisfy a modified energy dissipation law, ensuring stability in the energy norm.
• An adaptive time-stepping strategy is implemented based on numerical tests on random time meshes to accelerate convergence to steady state.

Experimental results

Research questions

• RQ1 Can the variable-step BDF2 method preserve energy dissipation for the Cahn-Hilliard model under arbitrary time-step variations?
• RQ2 What step-ratio restriction ensures mesh-robust stability and convergence in the energy and $ L^2 $ norms?
• RQ3 How can discrete orthogonal convolution kernels be used to derive sharp error estimates in non-uniform time meshes?
• RQ4 To what extent does the convergence constant depend on time-step ratios, and can it be made independent of them?
• RQ5 How effective is the adaptive time-stepping strategy in resolving multi-scale dynamics and accelerating long-time simulations?

Key findings

• The BDF2 method with variable time steps preserves a modified energy dissipation law at the discrete level under the step-ratio restriction $ r_k \leq r_{\text{user}} < 4.864 $, ensuring energy stability.
• A new $ L^2 $-norm error estimate is established at the first time step under the updated step-ratio condition, with the convergence constant independent of the time-step ratios.
• The convergence is mesh-robust, meaning the prefactor in the error bound does not depend on the adjoint time-step ratios.
• The proposed method achieves optimal convergence rates in the $ L^2 $ norm due to the use of discrete orthogonal convolution kernels and derived embedding inequalities.
• Numerical experiments on random time meshes confirm the robustness and efficiency of the adaptive time-stepping strategy in capturing multi-scale behaviors.
• The method enables accelerated long-time simulations by dynamically adjusting time steps while maintaining stability and accuracy.