A convex splitting BDF2 method with variable time-steps for the extended Fisher–Kolmogorov equation

A variable time-step BDF2 scheme combined with the convex splitting strategy is proposed for the extended Fisher–Kolmogorov equation. Under the zero-stability condition r k : = τ k / τ k − 1 ≤ r user , ( r user < 4.864 ) , the suggested BDF2 scheme is shown to be uniquely solvable unconditionally, and preserve a discrete (modified) energy dissipation law. With the help of the recent discrete orthogonal convolution kernels technique, a concise L 2 norm error estimate for the convex splitting BDF2 scheme is established under the time-step ratios restriction 0 < r k ≤ r user . Numerical examples together with an adaptive time-stepping procedure are provided to demonstrate the robustness and effectiveness of the proposed methods.