[Paper Review] An implicit--explicit second order BDF numerical scheme with variable steps for gradient flows

Dianming Hou, Zhonghua Qiao|arXiv (Cornell University)|Apr 1, 2022

Advanced Numerical Methods in Computational Mathematics4 citations

TL;DR

This paper proposes a novel implicit–explicit (IMEX) second-order BDF scheme with variable time steps for gradient flows using the scalar auxiliary variable (SAV) method. The scheme achieves unconditional energy stability for time step ratios ≤ 4.8645 and provides a rigorous convergence analysis with optimal error estimates on nonuniform meshes, validated by numerical tests and adaptive time-stepping efficiency in coarsening dynamics simulations.

ABSTRACT

In this paper, we propose and analyze an efficient implicit--explicit (IMEX) second order in time backward differentiation formulation (BDF2) scheme with variable time steps for gradient flow problems using the scalar auxiliary variable (SAV) approach. We prove the unconditional energy stability of the scheme for a modified discrete energy with the adjacent time step ratio $γ_{n+1}:=\Dt_{n+1}/\Dt_{n}\leq 4.8645$. The uniform $H^{2}$ bound for the numerical solution is derived under a mild regularity restriction on the initial condition, that is $ϕ(\x,0)\in H^{2}$. Based on this uniform bound, a rigorous error estimate of the numerical solution is carried out on the temporal nonuniform mesh. Finally, serval numerical tests are provided to validate the theoretical claims. With the application of an adaptive time-stepping strategy, the efficiency of our proposed scheme can be clearly observed in the coarsening dynamics simulation.

Motivation & Objective

• Develop an efficient, unconditionally energy-stable numerical scheme for gradient flows on nonuniform time meshes.
• Address the challenge of constructing high-order, stable BDF2 schemes with variable time steps, particularly for stiff and long-time simulations.
• Enable adaptive time-stepping strategies in long-term simulations without sacrificing stability or accuracy.
• Establish rigorous convergence analysis and energy stability for the proposed scheme under mild regularity assumptions.
• Demonstrate the scheme’s efficiency and robustness through numerical experiments with adaptive time stepping in coarsening dynamics.

Proposed method

• Formulates a second-order BDF scheme with variable time steps (VBDF2) for gradient flows using the scalar auxiliary variable (SAV) approach.
• Introduces an IMEX (implicit–explicit) time discretization: implicit treatment of the linear diffusion term and explicit treatment of the nonlinear potential via a quadratic interpolation operator.
• Employs a modified discrete energy functional to prove unconditional energy stability under the condition that the adjacent time step ratio $\gamma_{n+1} = \Delta t_{n+1}/\Delta t_n \leq 4.8645$.
• Uses a quadratic interpolation operator $\Pi_{2,j}\phi(t)$ to construct second-order accurate approximations of time derivatives at intermediate points $t_{n+\sigma}$.
• Defines the auxiliary variable $r^{n+1}$ and $\xi^{n+1}$ to preserve the SAV structure, ensuring second-order accuracy in the phase variable $\phi$ despite first-order accuracy in $r$ and $\xi$.
• Derives a uniform $H^2$ bound for the numerical solution under the mild initial regularity condition $\phi(\mathbf{x},0) \in H^2$, enabling rigorous error analysis on nonuniform meshes.

Experimental results

Research questions

• RQ1 Can an unconditionally energy-stable second-order BDF scheme be constructed for gradient flows on nonuniform time meshes with variable time steps?
• RQ2 What is the maximal allowable time step ratio $\gamma_{n+1}$ for unconditional energy stability in a variable-step BDF2 scheme using the SAV method?
• RQ3 How can the SAV framework be extended to IMEX BDF2 schemes with variable time steps while preserving second-order accuracy and stability?
• RQ4 What is the convergence rate of the proposed scheme on nonuniform time meshes, and how does it depend on the regularity of the initial data?
• RQ5 Can the scheme support adaptive time-stepping strategies effectively in long-time simulations of coarsening dynamics?

Key findings

• The proposed IMEX VBDF2 scheme is unconditionally energy stable for time step ratios $\gamma_{n+1} \leq 4.8645$, significantly improving upon previous bounds in the literature.
• A uniform $H^2$ bound for the numerical solution is established under the mild initial condition $\phi(\mathbf{x},0) \in H^2$, which is essential for convergence analysis.
• A rigorous error estimate of optimal order is derived for the numerical solution on nonuniform time meshes, confirming second-order convergence in time.
• The scheme maintains second-order accuracy in the phase variable $\phi$ despite first-order accuracy in the auxiliary variables $r^{n+1}$ and $\xi^{n+1}$, due to the structure of the SAV formulation.
• Numerical experiments confirm the theoretical claims, showing excellent agreement with predicted convergence rates and demonstrating clear efficiency gains when using adaptive time stepping.
• The application of adaptive time-stepping in coarsening dynamics simulations confirms the scheme’s practical efficiency and robustness in long-time simulations.