Sharp L2 Norm Convergence of Variable-Step BDF2 Implicit Scheme for the Extended Fisher–Kolmogorov Equation. (21st March 2023)

• Record Type: Journal Article
• Authors: Li, Yang; Sun, Qihang; Feng, Naidan; Liu, Jianjun
• Other Names: Vetro Calogero Academic Editor.
• Is Part Of: Journal of function spaces. Volume 2023(2023)
• Publication Date: 2023-03-21
• Subjects: Function spaces -- Periodicals; 515.7305
• DOI: 10.1155/2023/1869660
• Languages: English
• ISSNs: 2314-8896

Abstract

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 ≔ Τ n / Τ n − 1 < 3 + 17 / 2 ≈ 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.