An almost second-order robust computational technique for singularly perturbed parabolic turning point problem with an interior layer

Abstract

The subject of this paper is the numerical investigation of a singularly perturbed parabolic problem with a simple interior turning point on a rectangular domain in the x−t plane with Dirichlet boundary conditions. At the location of the turning point, the solution to this class of problems has an internal layer. By decomposing the solution into a regular and singular component, we may first establish stronger bounds for the regular and singular components, as well as their derivatives. The fitted mesh finite difference scheme, which consists of a hybrid finite difference operator on a non-uniform Shishkin mesh in the space direction and a backward implicit Euler scheme on a uniform mesh in the time variable, is used to identify the approximate numerical solution. The accuracy of the resulting scheme is then improved in the temporal direction using the Richardson extrapolation scheme, which reveals that the extrapolated scheme is almost second-order uniformly convergent in both space and time variables when ɛ≤CN−1. Two numerical examples are used to demonstrate the efficacy and accuracy of the finite difference approaches mentioned.