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

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