Adaptive numerical schemes for the solution of singularly perturbed reaction diffusion equations

Abstract

Singularly perturbed reaction-diffusion differential equations arise in modelling of various physical phenomena. These types of problems depend on a small perturbation parameter and#949;, that multiplies some or all of the highest order derivative terms. On limiting the value of perturbation parameter to zero, the solutions to such problems approach a discontinuous limit. Due to presence of steep boundary layers, theoretical methods are unable to approximate solution accurately. They are applicable only for a restrictive class of problems. These methods provide qualitative information about the family of problems and semi-quantitative information about the particular member of the family. Moreover, to apply these methods effectively, user must have some priori information about behavior of solution. This motivated us to develop robust numerical methods for solving such types of problems. In this thesis, we have developed adaptive numerical schemes for the solution of five different types of reaction-diffusion problems of varying complexity. The proposed discretization techniques combine anefficient spline difference scheme or a high-order compact difference scheme with an adaptive grid refinement strategy. For reaction-diffusion problems with Dirichlet boundary conditions, the compact difference scheme yields parameter-uniform and global uniform convergence of fourth-order. To deal with the more general class of robin boundary conditions, the spline difference scheme confirms parameter-uniform and global uniform convergence of second-order. The compact finite difference scheme generalizes well for fourth-order reaction-diffusion problems and yields parameter-uniform fourth-order convergence on the adaptive grid. To solve a coupled system of reaction-diffusion problems, an efficient hybrid spline difference scheme of second-order is also developed. The highly efficient hybrid spline difference scheme is also extended to solve class of singularly perturbed parabolic reaction diffusion problems with robinboundaryconditions