Higher order robust numerical computation for singularly perturbed problem involving discontinuous convective and source term

Abstract

In this article, we considered a singularly perturbed convection–diffusion equation having discontinuous convective and source terms. Due to these discontinuities, an expected behavior appears in the solution known as the interior layer behavior at the point of discontinuity. Layers are some small narrow regions of the domain where the steep variations in the solution occur as singular perturbation parameter ε approaches zero. Classical central finite difference schemes and finite element methods with piecewise polynomial basis functions on the uniform mesh are known to be inadequate to solve such problems unless an unacceptably large number of mesh points are used when ε is very small. Thus, the problem is solved numerically first by a first‐order accurate, simple upwind scheme on a specially designed piecewise uniform Shishkin mesh. Further, to enhance the order of convergence and to achieve better accuracy, we constructed a Richardson extrapolation scheme (RE scheme) whichis almost second‐order accurate. An extensive analysis has been carried out to establish the almost second‐order ε‐uniform convergence of the proposed RE scheme. Lastly, some numerical examples are considered to verify our theoretical results, and comparisons of numerical results are performed with the numerical results that exist in the literature.

A robust numerical scheme for singularly perturbed differential equations with spatio-temporal delays

“…In [24], a singularly perturbed problem with time lag is treated by constructing a numerical method using the standard finite difference operators centered in space and implicit in time on a piece-wise uniform mesh. Sahoo and Gupta [25] solved a singularly perturbed problem involving discontinuous convective and source terms by developing a numerical scheme using a first-order accurate, simple upwind scheme on specially designed piece-wise uniform Shishkin meshes. Appadu and Tijani [26] treated a one-dimensional generalized Burgers-Huxley equation by proposing two solutions using the classical finite difference scheme and nonstandard finite difference scheme and obtained that one of the proposed solutions contains a minor error.…”