Parameter Uniform Numerical Methods for Singularly Perturbed Parabolic Partial Differential Equations

Abstract

This thesis deals with some efficient and higher order numerical methods for solving singularly perturbed parabolic partial differential equations (SPPDEs) in one and two dimensions. The model problems includes one dimensional SPPDEs, time delay SPPDEs, two dimensional SPPDEs and mixed type of parabolic-elliptic problems. In general, these problems are described by partial differential equations in which the highest order derivative is multiplied by a small parameter ε, known as the “singular perturbation parameter” (0 < ε ≪ 1). If the parameter ε tends to 0, the problem has a limiting solution, which is called the solution of the reduced problem. The regions of nonuniform convergence lie near the boundary of the domain, which are known as boundary/interior layers. Due to this layer phenomena, it is a very difficult and challenging task to provide parameter uniform numerical methods for solving SPPDEs. The term “parameter uniform” is meant to identify those numerical methods, in which the approximate solution converges (measured in the supremum norm) independently with respect to the perturbation parameter. The purpose of the thesis is to analyze, develop and optimize the parameter uniform fitted mesh methods for solving SPPDEs on Shishkin-type meshes like the standard Shishkin mesh (S-mesh), the Bakhvalov-Shishkin mesh (B-S mesh), and the modified Bakhvalov-Shishkin mesh (M-B-S mesh) in the spacial direction.
This thesis contains eight chapters. It begins with introduction along with the objective and the motivation for solving SPPDEs. Next, Chapter 2 contains a time delay SPPDE which is solved using a hybrid scheme (combination of the midpoint upwind scheme and the central difference scheme) on S-mesh and B-S mesh in space direction and the implicit trapezoidal scheme on uniform mesh in time direction. We have obtained an optimal global second order accuracy with respect to space and time. In Chapter 3, a monotone hybrid scheme for discretization in space and the implicit Euler, then the implicit trapezoidal scheme for discretization in time are used to solve a SPPDE. A monotone hybrid scheme is different from the usual hybrid scheme, which is a combination of the midpoint and the central difference scheme with variable weights. Here, an optimal second order accuracy is obtained in space and time. Again, the same monotone hybrid scheme in spatial direction and the Euler scheme in time direction are used in Chapter 4 for solving the model problem considered in Chapter 2. Chapter 5 contains a singularly perturbed time delay reaction-diffusion problem and it is solved by using the central difference scheme on S-mesh and B-S mesh in space direction and the implicit Euler, the implicit trapezoidal scheme in time direction to get a second order accuracy.
Then, the post-processing technique (Richardson extrapolation) is used to improve the accuracy from second order to fourth order. Chapter 6 presents the hybrid scheme on Shishkin-type meshes in spatial direction and the implicit Euler, the Crack-Nicolson schemes in time direction for solving the singularly perturbed mixed type of parabolic-elliptic problem. Next, we extend from 1D problem to 2D problem SPPDE in Chapter 7, in which a 2D SPPDE is solved by using the central difference scheme on Shishkin-type meshes in space direction and the Peaceman-Rachford scheme in time direction. Here, we have achieved an optimal second order accuracy in both spatial and temporal direction. In all cases, Thomas algorithm is used throughout this thesis to reduce the computational time over the usual matrix inverse method. Finally, Chapter 8 summarizes the results made by this thesis.
Extensive numerical results are presented in support of the theoretical findings and also to demonstrate the accuracy of the proposed methods. The corresponding numerical results are presented in the numerical section of each chapter of the thesis in shape of tables and figures. Some comparison results are also provided which confirms the efficiency of the proposed methods made in this thesis over the existing methods in literature.