[Paper Review] Stability and error estimates for the variable step-size BDF2 method for linear and semilinear parabolic equations

Wansheng Wang, Mengli Mao+1 more|arXiv (Cornell University)|Mar 7, 2020

Numerical methods for differential equations20 references4 citations

TL;DR

This paper establishes the first equivalence between zero-stability and $ l^∞(0,T;H) $-stability for the variable step-size BDF2 method applied to linear and semilinear parabolic equations. It proves that the upper bound for step-size ratios ensuring $ l^∞(0,T;H) $-stability is identical to the zero-stability threshold $ R_0 = \sqrt{2}+1 \approx 2.414 $, and derives optimal error estimates in multiple norms, confirming second-order accuracy even with variable time steps when starting values are computed via the trapezoidal rule.

ABSTRACT

In this paper stability and error estimates for time discretizations of linear and semilinear parabolic equations by the two-step backward differentiation formula (BDF2) method with variable step-sizes are derived. An affirmative answer is provided to the question: whether the upper bound of step-size ratios for the $l^\infty(0,T;H)$-stability of the BDF2 method for linear and semilinear parabolic equations is identical with the upper bound for the zero-stability. The $l^\infty(0,T;V)$-stability of the variable step-size BDF2 method is also established under more relaxed condition on the ratios of consecutive step-sizes. Based on these stability results, error estimates in several different norms are derived. To utilize the BDF method the trapezoidal method and the backward Euler scheme are employed to compute the starting value. For the latter choice, order reduction phenomenon of the constant step-size BDF2 method is observed theoretically and numerically in several norms. Numerical results also illustrate the effectiveness of the proposed method for linear and semilinear parabolic equations.

Motivation & Objective

• To establish $ l^\infty(0,T;H) $-stability for the variable step-size BDF2 method applied to linear and semilinear parabolic equations.
• To determine whether the upper bound for step-size ratios ensuring $ l^\infty(0,T;H) $-stability matches the zero-stability threshold $ R_0 = \sqrt{2}+1 \approx 2.414 $.
• To derive global error estimates in multiple norms, including $ l^\infty(0,T;H) $, $ l^\infty(0,T;V) $, $ l^2(0,T;H) $, and $ l^2(0,T;V) $, under relaxed step-size ratio conditions.
• To analyze the impact of starting value computation methods (trapezoidal vs. backward Euler) on convergence order, particularly the order reduction phenomenon in constant step-size BDF2.
• To numerically validate the theoretical results and demonstrate the superiority of variable step-size BDF2 over constant step-size BDF2 in accuracy and efficiency.

Proposed method

• The variable step-size BDF2 method is applied to time discretization of linear and semilinear parabolic equations using a general time partition with variable step-sizes $ k_n = t^n - t^{n-1} $.
• The method employs the two-step backward differentiation formula with a modified time derivative operator $ \bar{\partial}^2_B U^n $, defined via $ s_n = k_n / (k_n + k_{n-1}) $, ensuring second-order accuracy.
• Stability is analyzed by testing the discrete equation with a generalized test function $ U^n_\delta = U^n + \delta k_n \bar{\partial}^1_B U^n $, enabling energy estimates in Hilbert space settings.
• The $ l^\infty(0,T;H) $-stability is proven under the condition $ r_n = k_n/k_{n-1} < \sqrt{2}+1 $, matching the zero-stability bound, using spectral and energy techniques.
• Error estimates are derived via consistency analysis and stability results, yielding global error bounds in $ l^\infty(0,T;H) $, $ l^\infty(0,T;V) $, $ l^2(0,T;H) $, and $ l^2(0,T;V) $ norms.
• The starting value $ U^1 $ is computed either by the trapezoidal rule or backward Euler scheme, with the latter shown to cause order reduction in certain norms.

Experimental results

Research questions

• RQ1 Is the upper bound for step-size ratios ensuring $ l^\infty(0,T;H) $-stability of the variable step-size BDF2 method identical to the zero-stability threshold $ R_0 = \sqrt{2}+1 $?
• RQ2 Can the $ l^\infty(0,T;V) $-stability of the variable step-size BDF2 method be established under a relaxed condition on step-size ratios compared to previous works?
• RQ3 Does the choice of starting value computation (trapezoidal vs. backward Euler) affect the convergence order of the BDF2 method, particularly in $ l^\infty(0,T;V) $ and $ l^2(0,T;H,H) $ norms?
• RQ4 Can the variable step-size BDF2 method maintain second-order convergence for both linear and semilinear parabolic equations on non-uniform grids?
• RQ5 Is the order reduction phenomenon observed in the constant step-size BDF2 method also present in the variable step-size variant when using backward Euler for $ U^1 $?

Key findings

• The upper bound for step-size ratios ensuring $ l^\infty(0,T;H) $-stability of the variable step-size BDF2 method is proven to be identical to the zero-stability threshold $ R_0 = \sqrt{2}+1 \approx 2.414 $, resolving a key theoretical question.
• The $ l^\infty(0,T;V) $-stability is established under the relaxed condition $ r_n < \sqrt{2}+1 $, improving upon earlier bounds such as $ 1.366 $ or $ 1.868 $.
• The method achieves second-order convergence in $ l^\infty(0,T;H) $, $ l^\infty(0,T;V) $, $ l^2(0,T;H) $, and $ l^2(0,T;V) $ norms for both linear and semilinear parabolic equations on variable time grids.
• Numerical experiments confirm second-order convergence for variable step-size BDF2 with $ \varpi = 3 $, and show that the method outperforms the constant step-size BDF2 in accuracy across all tested norms.
• When using backward Euler to compute $ U^1 $, the constant step-size BDF2 exhibits order reduction in $ l^\infty(0,T;V) $ and $ l^2(0,T;H,H) $ norms, but this can be mitigated in the variable step-size case by reducing $ k_1 $.
• The variable step-size BDF2 method maintains second-order accuracy and demonstrates superior performance compared to the constant step-size variant, especially in regions with rapidly varying solutions.