Robust numerical schemes for singularly perturbed boundary-value problems on adaptive meshes
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In this thesis, our primary interest is to provide some efficient and higher-order numerical techniques for solving singularly perturbed convection-diffusion and reaction-diffusion boundary-value problems exhibiting boundary layers. These singular perturbation problems (SPPs) are described by differential equations in which the highest-order derivative is multiplied by an arbitrarily small parameter "" (say) known as singular perturbation parameter. This leads to the existence of boundary layers which are basically narrow regions in the neighbourhood of the boundary of the domain, where the gradient of the solution becomes steep as the perturbation parameter tends to zero. Due to the appearance of the layer phenomena, it is a challenging task to provide ""-uniform numerical methods. The term D""-uniformD refers to identify those numerical methods in which the approximate solution converges to the corresponding exact solution (measured to the supremum norm) independently with respect to the perturbation parameter "". The purpose of this thesis is to develop, analyze and improve the ""-uniform numerical methods for solving SPPs. These methods are mainly based on two types of nonuniform meshes. They are the well-known layer resolving piecewise-uniform Shishkin mesh and the equidistributed layer-adapted meshes, which are obtained by moving fixed number mesh points to equidistribute a positive monitor function, depending on the solution or/and its derivatives. At first, a uniformly convergent hybrid numerical scheme is proposed and analyzed on the equidistributed mesh for singularly perturbed Robin type reaction-diffusion problems. This scheme uses a proper combination of central difference and cubic spline approximation for the second-order derivative. In addition, the proposed hybrid scheme is extended for a system of Robin type reaction-diffusion problems on apriori chosen piecewise-uniform Shishkin mesh. In all these cases, the newly proposed hybrid scheme attains almost second-order accuracy. Thereafter, the mesh equidistribution technique is extended for a class of fourth-order ordinary differential equations (ODEs), where the adaptively generated mesh is obtained by equidistribution of a curvature type monitor function. Moreover, we derive theoretically apriori monitor function to study the effect of mesh equidistribution for a general singularly perturbed system of reaction-diffusion problems. Next, Richardson extrapolation technique, which improves the first-order accuracy of the standard upwind scheme to second-order convergence is analyzed for singularly perturbed convection-diffusion problems using moving mesh methods. Finally, the equidistribution of a monitor function which works for scalar form of convection-diffusion problem is extended for a system singularly perturbed convection-diffusion problems to obtain an optimal first-order accurate parameter-uniform convergent solution. Extensive numerical experiments are conducted which support all of our theoretical findings. A concise conclusion with possible further work is provided at the end of this thesis..
Supervisor: S. Natesan