Parameter-uniform fitted mesh higher order finite difference scheme for singularly perturbed problem with an interior turning point

TL;DR

This paper develops a parameter-uniform higher order finite difference scheme on a fitted mesh for singularly perturbed problems with interior turning points, achieving second order uniform convergence and verified by numerical experiments.

Contribution

It introduces a hybrid finite difference scheme on a fitted mesh for interior turning point problems, providing second order uniform convergence analysis.

Findings

The scheme achieves second order uniform convergence.

Numerical results confirm the theoretical error bounds.

The method outperforms existing approaches in the literature.

Abstract

In this paper, a parameter-uniform fitted mesh finite difference scheme is constructed and analyzed for a class of singularly perturbed interior turning point problems. The solution of this class of turning point problem possess two outflow exponential boundary layers. Parameter-explicit theoretical bounds on the derivatives of the analytical solution are given, which are used in the error analysis of the proposed scheme. The problem is discretized by a hybrid finite difference scheme comprises of midpoint-upwind and central difference operator on an appropriate piecewise-uniform fitted mesh. An error analysis has been carried out for the proposed scheme by splitting the solution into regular and singular components and the method has been shown second order uniform convergent except for a logarithmic factor with respect to the singular perturbation parameter. Some relevant numerical examples are also illustrated to verify computationally the theoretical aspects. Numerical experiments show that the proposed method gives competitive results in comparison to those of other methods exist in the literature.