Mesh-robustness of an energy stable BDF2 scheme with variable steps for the Cahn-Hilliard model

TL;DR

This paper develops a mesh-robust, energy-stable BDF2 scheme with variable steps for the Cahn-Hilliard model, ensuring stability and convergence even with changing time-step ratios, and introduces an adaptive time-stepping strategy for efficient long-term simulations.

Contribution

It introduces a novel energy stable convex-splitting BDF2 scheme with variable steps that is mesh-robust and preserves energy dissipation, along with new discrete inequalities and an adaptive time-stepping method.

Findings

The scheme is mesh-robustly convergent with a step-ratio restriction $r_{user}<4.864$.

The method preserves a modified energy dissipation law at the discrete level.

Adaptive time-stepping effectively captures multi-scale behaviors and accelerates long-time simulations.

Abstract

The two-step backward differential formula (BDF2) with unequal time-steps is applied to construct an energy stable convex-splitting scheme for the Cahn-Hilliard model. We focus on the numerical influences of time-step variations by using the recent theoretical framework with the discrete orthogonal convolution kernels. Some novel discrete convolution embedding inequalities with respect to the orthogonal convolution kernels are developed such that a concise $L^2$ norm error estimate is established at the first time under an updated step-ratio restriction $0 <r_k:=\tau_k/\tau_{k-1}\leq r_{\mathrm{user}}$, where $r_{\mathrm{user}}$ can be chosen by the user such that $r_{\mathrm{user}}<4.864$. The stabilized convex-splitting BDF2 scheme is shown to be mesh-robustly convergent in the sense that the convergence constant (prefactor) in the error estimate is independent of the adjoint time-step ratios. The suggested method is proved to preserve a modified energy dissipation law at the discrete levels if $0<r_k\le r_{\mathrm{user}}$, such that it is mesh-robustly stable in an energy norm. 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.