Abstract
In this paper, a two-step backward differentiation formula (BDF2) with variable time steps is applied to solve the Fisher equation. The proposed scheme is built by using the variable time-stepping BDF2 for linear terms and the extrapolation method for the nonlinear term in time combining with the finite difference method (FDM) in space. We show that the scheme is uniquely solvable under mild constraints on the time step sizes, and obtain the maximum error analysis under the constraint of some certain adjacent time-step ratios: \(0 < {{\tau }_{k}}{\text{/}}{{\tau }_{{k - 1}}}\;\leqslant \;r,\) where \(r\) can be chosen by the user such that \(r < r\text{*} \approx 4.8645,\) \({{\tau }_{k}}\) denotes the time step size of the kth temporal interval. The newly designed scheme boasts the excellent characteristic of maintaining the maximum bound principle (MBP). A key point to underscore is that the algorithm incorporating time-adaptive strategy can adeptly handle the transient and gradual dynamics in problems featuring a strong reaction term. To substantiate this claim and validate the precision of the algorithm, numerical experiments have been meticulously executed. These experiments reveal that the scheme exhibits second-order accuracy in both time and space. Furthermore, additional numerical example has been conducted, which corroborates that the constructed scheme adheres to the MBP. To the best of our knowledge, there is rare work to numerically solve the Fisher equation under nonuniform temporal meshes.
