Sharp L2 Norm Convergence of Variable-Step BDF2 Implicit Scheme for the Extended Fisher–Kolmogorov Equation

Abstract

A variable-step BDF2 time-stepping method is investigated for simulating the extended Fisher-Kolmogorov equation. The time-stepping scheme is shown to preserve a discrete energy dissipation law if the adjacent time-step ratios $r_n≔\left({{\rm T}}_n/\left({{\rm T}}_{n-1}\right)\right)<\left(\left(3+\sqrt{17}\right)/2\right)\approx 3.561$ . With the aid of discrete orthogonal convolution kernels, concise $L^2$ norm error estimates are proved, for the first time, under the mild step ratios constraint $0<r_n<3.561$ . Our error estimates are almost independent of the step ratios $r_n$ so that the proposed numerical scheme is robust with respect to the variations of time steps. An adaptive time-stepping strategy based on solution accuracy is then applied to update the computational efficiency. Numerical examples are included to illustrate our theoretical results.