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

Abstract

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 rk:=τk/τk−1≤ruser, (ruser<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 L2 norm error estimate for the convex splitting BDF2 scheme is established under the time-step ratios restriction 0<rk≤ruser. Numerical examples together with an adaptive time-stepping procedure are provided to demonstrate the robustness and effectiveness of the proposed methods.