High order methods for elliptic and time dependent reaction–diffusion singularly perturbed problems

摘要

The objective of this paper is to construct some high order uniform numerical methods to solve linear reaction–diffusion singularly perturbed problems. First, for 1D elliptic problems, based on the central finite difference scheme, a new HODIE method is defined on a piecewise uniform Shishkin mesh. Using this HODIE scheme jointly with a two stage SDIRK method, we solve a 1D parabolic singularly perturbed problem. In both cases we prove that the methods are third-order uniform convergent in the maximum norm. Finally, for a 2D parabolic problem of the same type, we show numerically that the combination of the HODIE scheme with a fractional step RK method gives again a third-order uniform convergent scheme.