Fitted Numerical Methods for Two-Parameter Singularly Perturbed Problems

Abstract

This dissertation focuses on a class of second-order two-parameter singularly perturbed parabolic problems, with and without time delay. These problems exhibit narrow boundary layers near the domain boundaries due to the presence of the perturbation parameters, as well as an interior layer because of the presence of discontinuous data in the convection term and/or source term. The rapid changes observed within these layer regions make analytical solutions challenging to obtain, while standard numerical methods often fail to capture the layer behaviors effectively. Consequently, the development of parameter-uniform numeri- cal schemes for such singularly perturbed problems has become a prominent research area. Based on our literature review, we have encountered two major difficulties regarding the so- lutions to singularly perturbed problems. These difficulties include standard numerical meth- ods failing to capture the behavior of the problem’s solution, as well as the need for improve- ment in the accuracy of the solution obtained using existing numerical schemes. Therefore, in order to address these challenges, the main objective of this dissertation is to formulate and analyze fitted numerical schemes that exhibit convergence independent of the parame- ters. Two effective approaches are explored: the fitted operator finite difference method and the fitted mesh finite difference method. The fitted operator numerical methods are obtained by replacing the perturbation parameter with positive functions. Alternatively, in fitted mesh numerical methods, the successful approach of formulating a parameter-uniform numerical scheme involves employing standard finite difference operators on a specialized piece-wise uniform mesh condensed around the layer regions. These approaches yield uniformly con- vergent numerical results, overcoming the limitations of standard numerical methods. In this dissertation work, five uniformly convergent numerical schemes are formulated. Again, the stability estimate of each scheme is established and the parameter-uniform convergence of the schemes are investigated and proved. Various numerical experiments are conducted to demonstrate the validity and applicability of the formulated numerical schemes, and nu- merical results using Python and/or MATLAB tools are presented in tables and graphs. The theoretical and numerical results confirm that the formulated schemes are more accurate and uniformly convergent as compared to other existing literature.