Uniformly Convergent Numerical Schemes for Singularly Perturbed Parabolic Partial Diffrential Equations on Adaptive Grid
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This thesis provides some efficient numerical techniques for solving singularly perturbed parabolic initial-boundary-value problems of convection-diffusion and reaction-diffusion types with boundary layers. These types of problems are identified by partial differential equations in which the highest spatial derivative is multiplied by an arbitrarily small parameter The perturbation is in the sense that, as 0, the problem becomes ill-posed since the order of the differential equation is reduced, but the number of boundary conditions remain the same. It is a well-known fact that the solution of singularly perturbed boundary-value problem exhibits a multiscale character. That is, there is a thin layer where the solution varies rapidly, while away from the layer the solution behaves regularly and varies slowly. The study of singular perturbation problems (SPPs) is exceptionally useful because they describe the physics of many event of academic and practical interest. This class of problems has recently gained importance in the literature because of its application nature. These problems have been treated numerically by means of exponential-fitting, adaptive meshes, and ideas based on the method of matched asymptotic expansions. Due to this layer phenomena, it is a very difficult and challenging task to provide numerical methods for solving SPPs. The term is meant to identify those numerical methods in which the approximate solution converges in some norm (preferably the supremum norm) independently with respect to the parameter to the corresponding exact solution of SPP. It is well-known that uniform meshes with classical schemes fail to converge uniformly with respect to the singular perturbation parameter. It is desired to develop methods which converge uniformly. In this thesis we develop and analyze the numerical methods for solving singularly perturbed parabolic PDEs of convection-diffusion and reaction-diffusion initial-boundary-value problems (IBVPs) on a nonuniform mesh, which is obtained by equidistribution of a positive monitor function. We begin the thesis with an introduction followed by a section describing the motivation. Then, we introduce the equidistribution principle, the terminology used throughout the thesis. The monitor functions are introduced, further, the numerical algorithm for generating adaptive nonuniform grids are given. Then, numerical schemes are developed for reaction-diffusion and convection-diffusion parabolic IBVPs exhibiting parabolic and regular boundary layers. Numerical experiments are carried out to validate theoretical error estimates. Convergence rates are calculated for the numerical solutions and the flux for linear and semilinear parabolic PDEs. We, then extended the method to wider class of problems of singularly perturbed parabolic convection-diffusion and reaction-diffusion nature. Subsequently the numerical experiment results are presented to verify the theoretical results. Then, the analysis for singularly perturbed delay parabolic reaction-diffusion and convection-diffusion are given using equidistribution and the piecewise-uniform Shishkin mesh, respectively. We also compared the results produced in this thesis with the corresponding layer adapted nonuniform meshes like the piecewise-uniform Shishkin mesh and the Bakhvalov mesh...
Supervisor: Natesan Srinivasan