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

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) = (T-n/(Tn-1)) < ((3 + root 17/2) approximate to 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.