Uniformly Convergent Numerical Methodsfor Singularly Perturbed Initial and Boundary Value Problems

Abstract

This thesis provides some efficient numerical methods for solving a various class of singularly perturbed initial and boundary value problems. These types of problems are
described by differential equations in which the highest order derivative is multiplied by a small parameter. This small parameter is known as ``singular perturbation parameter".As the perturbation parameter approaches to zero, it leads to occurrence of a thin narrow region in the neighborhood of the boundary of the domain, known as the boundary layers where solution varies very rapidly. Due to this layer phenomena,it is a very difficult and challenging task to develop parameter uniform numerical methods for solving SPPs. The parameter uniform means to develop such numerical methods in which the approximate solution converges independently with respect to the perturbation parameter. The purpose of this thesis is, therefore, to develop, analyze, improve and optimize the parameter-uniform
numerical methods for solving singularly perturbed initial and boundary-value problems. This purpose is fulfilled by constructing an appropriate non-uniform mesh which can resolve the boundary layers. At first, a post-processing technique (Richardson extrapolation), which improves the accuracy of the upwind scheme, is analyzed on a Shishkin type meshes and on an adaptive grid for a parameterized singularly perturbed problem.Then,a weighted uniform numerical method is analyzed for this problem which automatic switches from the backward Euler scheme to a monotone hybrid scheme depending on the weight parameters.
The monotone hybrid scheme utilizes a proper combination of the midpoint upwind scheme and the upwind scheme. The analogous study of a similar kind of monotone hybrid
scheme with weight parameters is also made for a system of nonlinear singularly perturbed problems. A modified monotone hybrid scheme is proposed and analyzed on layer resolving
nonuniform meshes for singularly perturbed scalar and system of convection–diffusion problems. The scheme utilizes appropriate weight parameters which automatically switched
the midpoint upwind scheme to the classical central difference scheme as mesh points go from the inner region to the outer region. Further, the efficiency of the monotone hybrid scheme is tested by extending it for solving one-dimensional singularly perturbed parabolic convection–diffusion IBVP with a regular boundary layer on a nonuniform spatial mesh, the backward-Euler scheme for discretizing the time derivative on a uniform mesh. In all the cases, the newly proposed monotone hybrid schemes provide second-order accuracy on the adaptive grid generated via equidistribution principle also it can be easily implemented to any arbitrary nonuniform mesh. Finally, a system of singularly perturbed parabolic reaction–diffusion problems is analyzed where the perturbation parameters are of the same magnitude. A combination of a modified implicit-Euler method in the time direction and central difference scheme for the spatial derivative is used for the discretization on a space
adaptive mesh.