Sharp L2 Norm Convergence of Variable-Step BDF2 Implicit Scheme for the Extended Fisher–Kolmogorov Equation - Yang Li; Qihang Sun; Naidan Feng; Jianjun Liu (Journal of Function Spaces)
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 rn≔Τn/Τn−1<3+17/2≈3.561. With the aid of discrete orthogonal convolution kernels, concise L2 norm error estimates are proved, for the first time, under the mild step ratios constraint 0<rn<3.561. Our error estimates are almost independent of the step ratios rn 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.