Solvability and Convergence Analysis of an Adaptive BDF2 Scheme for the Two-dimensional Fisher Equation

Abstract

The two-step backward differentiation formula (BDF2) is applied to numerically solve the two-dimensional Fisher equation. The nonlinear term is handled skillfully by the extrapolation technique. On this basis, a finite difference scheme with the second-order accuracy in both time and space is constructed for the initial-boundary value problem of the two-dimensional Fisher equation. Based on the energy method with induction, the unique solvability and error analysis can be achieved under the condition that the adjacent maximum time step ratio r satisfies certain constraint: 0 < r < r* ≈ 4.8645. We first estimate the numerical error under the discrete H²-seminorm, and then obtain the error estimate under the maximum norm by Sobolev embedding inequality, thus obtain the uniform bound of the numerical solution. Numerical examples verify our theoretical results, and illustrate the computational efficiency of the proposed scheme under the adaptive strategy to cope with the strong reaction case as well as the maximum bound preserving (MBP) property of the scheme.