Construction Of Difference Schemes For Nonlinear Singular Perturbed Equations By Approximation Of Coefficients

TL;DR

This paper develops difference schemes for nonlinear singularly perturbed equations with boundary layers, ensuring uniform convergence despite small coefficients at higher derivatives, which classical schemes fail to achieve.

Contribution

It introduces new difference schemes tailored for nonlinear singular perturbation problems with boundary layers, improving accuracy and convergence.

Findings

Schemes effectively handle exponential boundary layers.

Schemes maintain accuracy as perturbation parameter approaches zero.

Numerical experiments confirm uniform convergence.

Abstract

Mathematical modeling of many physical processes such as diffusion, viscosity of fluids and combustion involves differential equations with small coefficients of higher derivatives. These may be small diffusion coefficients for modeling the spreading of impurities, small coefficients of viscosity in fluid flow simulation etc. The difficulty with solving such problem is that if you set the small parameter at higher derivatives to zero, the solution of the degenerate problem doesn't correctly approximate the original problem, even if the small parameter approaches zero; the solution of the original problem exhibits the emergency of a boundary layer. As a result, the application of classical difference schemes for solving such equations produces great inaccuracies. Therefore, numerical solution of differential equations with small coefficients at higher derivatives demands special difference schemes exhibiting uniform convergence with respect to the small parameters involved. In this article author investigates two nonlinear boundary value problems on a finite interval, resulting in exponential and power-law boundary layers.