A class of compact boundary value methods applied to semi-linear reaction-diffusion equations

This paper deals with a class of compact boundary value methods (CBVMs) for solving semi-linear reaction-diffusion equations (SLREs). The presented CBVMs are constructed by combining a fourth-order compact difference method (CDM) with the p-order boundary value methods (BVMs), where the former is for the spatial discretization and the latter for temporal discretization. It is proven under some suitable conditions that the CBVMs are locally stable and uniquely solvable and have fourth-order accuracy in space and p-order accuracy in time. The computational effectiveness and accuracy of CBVMs are further testified by applying the methods to the Fisher equation. Besides these research, we also extend the CBVMs to solve the two-component coupled system of SLREs. The numerical experiment shows that the extended CBVMs are effective and can arrive at the high-precision.