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.

本文利用具有可变步长的两步向后微分公式（BDF2）方法推导了线性和半线性抛物方程的时间离散化的稳定性和误差估计。这个问题的答案是肯定的：线性和半线性抛物方程的BDF2方法的\（l ^ {\ infty}（0，T; H）\） -稳定性的步长比的上限是否为与零稳定性上限相同。的\（L ^ {\ infty}（0，T; V）\）在更宽松的条件下，基于连续步长的比率，还建立了步长可变的BDF2方法的稳定性。根据这些稳定性结果，可以得出几种不同规范中的误差估计。为了利用BDF方法，梯形方法和后向Euler方案用于计算起始值。对于后一种选择，恒定步长BDF2方法的阶数减少现象在理论上和数值上都在几个规范中被观察到。数值结果也说明了该方法对线性和半线性抛物方程的有效性。