[Paper Review] Weighted and shifted BDF2 methods on variable grids

Minghua Chen, Yu Fan+1 more|arXiv (Cornell University)|Aug 6, 2021

Advanced Numerical Methods in Computational Mathematics11 references4 citations

TL;DR

This paper introduces a weighted and shifted BDF2 (WSBDF2) method with variable time steps for solving parabolic PDEs, combining BDF2 and Crank-Nicolson schemes via a weighted shift. It proves unconditional stability and optimal second-order convergence, with optimal adjacent time-step ratios that significantly improve upon prior BDF2 methods, filling a theoretical gap in PDE analysis for variable grids.

ABSTRACT

Variable steps implicit-explicit multistep methods for PDEs have been presented in [17], where the zero-stability is studied for ODEs; however, the stability analysis still remains an open question for PDEs. Based on the idea of linear multistep methods, we present a simple weighted and shifted BDF2 methods with variable steps for the parabolic problems, which serve as a bridge between BDF2 and Crank-Nicolson scheme. The contributions of this paper are as follows: we first prove that the optimal adjacent time-step ratios for the weighted and shifted BDF2, which greatly improve the maximum time-step ratios for BDF2 in [11,15]. Moreover, the unconditional stability and optimal convergence are rigorous proved, which make up for the vacancy of the theory for PDEs in [17]. Finally, numerical experiments are given to illustrate theoretical results.

Motivation & Objective

• To address the lack of stability and convergence theory for variable-step BDF methods in PDEs, particularly for implicit-explicit multistep schemes.
• To construct a unified framework connecting BDF2 and Crank-Nicolson schemes via a weighted and shifted BDF2 formulation on non-uniform time grids.
• To derive optimal adjacent time-step ratios that maximize stability and efficiency, improving upon previous bounds in BDF2 methods.
• To rigorously prove unconditional stability and optimal second-order convergence for the WSBDF2 method applied to parabolic problems.
• To validate theoretical findings through numerical experiments with spectral spatial discretization and variable time steps.

Proposed method

• Formulates a weighted and shifted BDF2 operator $ D_2v^n = heta ilde{D}_2v^n + (1- heta)ar{D}_2v^n $, where $ ilde{D}_2 $ is the standard BDF2 and $ ar{D}_2 $ is the shifted BDF2, with $ \theta \in [1/2,1] $.
• Defines the time-discrete scheme as $ D_2u^n - \theta\Delta u^n - (1-\theta)\Delta u^{n-1} = \theta f^n + (1-\theta)f^{n-1} $, solving the parabolic problem with variable time steps.
• Uses a weak formulation involving $ L^2 $ inner products and $ H_0^1 $ test functions to derive stability and convergence estimates.
• Applies energy estimates and discrete convolution identities to bound the error, leveraging the Cauchy-Schwarz inequality and summation-by-parts techniques.
• Employs a discrete Grönwall inequality to control error propagation over time, leading to optimal convergence order.
• Uses spectral collocation with Chebyshev–Gauss–Lobatto points for spatial discretization, ensuring spectral accuracy and negligible spatial error.

Experimental results

Research questions

• RQ1 What is the optimal range of adjacent time-step ratios $ r_k = \tau_k / \tau_{k-1} $ that ensures stability and optimal convergence for the WSBDF2 method on variable grids?
• RQ2 Can the WSBDF2 method achieve unconditional stability and second-order convergence for parabolic PDEs with variable time steps, despite the lack of such theory in prior works?
• RQ3 How does the WSBDF2 method bridge the stability and accuracy properties of BDF2 and Crank-Nicolson schemes in the context of variable time stepping?
• RQ4 To what extent do variable time-step strategies, including geometric and random step sizes, affect the convergence behavior of the WSBDF2 method?
• RQ5 Can the WSBDF2 framework be extended to higher-order methods (e.g., WSBDF3) on non-uniform grids, and what are the theoretical challenges involved?

Key findings

• The WSBDF2 method achieves unconditional stability for all $ \theta \in [1/2,1] $, with the optimal time-step ratio $ r_k \leq 4 $ for $ \theta = 1/2 $, significantly improving the maximum allowable ratio over standard BDF2.
• Optimal second-order convergence in the discrete $ L^2 $-norm is rigorously proven for all $ \theta \in [1/2,1] $, with convergence rates matching theoretical expectations.
• Numerical experiments confirm second-order convergence across all tested cases: Case I (alternating $ r_k = 4, 1/4 $), Case II ($ r_k = 2 $), and Case III (random $ r_k $), with observed rates near 2.0.
• For Case II with constant $ r_k = 2 $, the convergence rate remains second-order, indicating robustness under uniform but non-ideal step ratios.
• In Case III with random time steps, the method maintains convergence rates of approximately 2.0–2.5, demonstrating robustness to irregular time grids.
• The method successfully bridges BDF2 ($ \theta = 1 $) and Crank-Nicolson ($ \theta = 1/2 $), with the latter showing the most consistent convergence across all test cases.