Adaptive Algorithms for a Class of Nonlinear Evolution Systems

In this article, a numerical algorithm is proposed to solve a class of nonlinear evolution system: the initial-boundary value problem of Fisher equation. This algorithm is constructed by combining a two-step variable stepping backward differentiation formula (BDF2) and the extrapolation method. It can be proved that the numerical system admits a unique solution when the ratio of the maximum adjacent time steps satisfies certain conditions ($r \leq 4.865, r$ is the maximum adjacent time step ratio). The developed algorithm has a good property of preserving the maximum bound principle (MBP). It must be highlighted that the proposed adaptive time-stepping algorithm is well suited to capture the transient and gradual behaviours appearing in the problem with the strong reaction term. Numerical simulations are carried out to illustrate this point and to verify the accuracy of the constructed algorithm: it is of order two in both time and space. In addition, it is also illustrated that the constructed algorithm fulfils the MBP.