Numerical Analysis For Singularity Perturbed Differential Equations and Its Applications

Advancements in Numerical Analysis for Singularly Perturbed Differential Equations

Abstract

In this talk, I will discuss the role of numericalanalysis in the design of numerical al­gorithms to approximately solve certainclasses of singularly perturbed differential equations. The solutions ofsingularly perturbed differential equation have narrow layer regions in thedomain, where the solution exhibits steep gradients. Classical numericalmethods suffer major defects in these regions. Alternative computationalapproaches will be discussed and the central issues in the associated numericalanaly­sis of these layer-adapted algorithms will be outlined. We present new results in the numerical analysis ofsingularly perturbed convection-diffusion- reaction problems that have appearedin the last five years. Mainly discussing layer-adapted meshes, we present alsoa survey on stabilization methods, adaptive methods, and on systems ofsingularly perturbed equations. In this paper a singularly perturbed reaction-diffusionequation with a dis­continuous source term is examined. A numerical method isconstructed for this problem which involves an appropriate piecewise-uniformmesh. The method is shown to be uniformly convergent with respect to thesingular perturba­tion parameter. An exponentially-fitted method for singularly perturbed,one-dimensional parabolic equations and ordinary differential equations both ofthe convection-diffusion-reaction type in equally-spaced grids is presented. Singular perturbation problems with turning points ariseas mathematical models for various physical phe­nomena. The problem withinterior turning point represent one-dimensional version of stationaryconvection- diffusion problems with a dominant convective term and a speedfield that changes its sign in the catch basin.