Upwind Based Numerical Methods for Time-Dependent singularly perturbed problems with boundary and interior layers
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This thesis provides some efficient numerical techniques for solving time-dependent singularly perturbed problems (SPPs) possessing boundary and interior layers. These types of problems are described by partial differential equations in which the highest spatial derivative is multiplied by an arbitrarily small parameter "", known as Dsingular perturbation parameterD. This leads to the occurrence of boundary (or interior) layers, which are basically thin regions in the neighbourhood of the boundary (or interior) of the domain, where the gradients of the solutions steepen as the perturbation parameter "" tends to zero. Due to this layer phenomena, it is a very difficult and challenging task to provide ""-uniform numerical methods for solving SPPs. The term D""-uniformD is meant to identify those numerical methods in which the approximate solution converges (measured in the supremum norm) independently with respect to the parameter "" to the corresponding exact solution of SPP . The purpose of this thesis is therefore to develop, analyze, improve and optimize the ""-uniform upwind based numerical methods for solving time-dependent singularly perturbed initial-boundary-value problems (IBVPs) with smooth and non-smooth data. This is accomplished by constructing spacial non-uniform meshes resolving boundary and interior layers. At first, a uniformly convergent hybrid numerical scheme is proposed and analyzed on a layer resolving piecewise-uniform Shishkin mesh for singularly perturbed one-dimensional parabolic convection-diffusion IBVP with a regular boundary layer as well as a class of parabolic convection-diffusion IBVPs with strong interior layers. The scheme utilizes a proper combination of the midpoint upwind scheme and the classical central difference scheme for the spatial discretization and the backward-Euler scheme for discretizing the time derivative. The analogous study of a similar kind of hybrid scheme is also made for a class of singularly perturbed mixed parabolic-elliptic IBVPs exhibiting both boundary and interior layers. Further, the efficiency of the hybrid scheme (proposed for 1D parabolic IBVP with smooth data ) is tested by extending it for solving two-dimensional singularly perturbed parabolic convection-diffusion IBVP on a spacial rectangular mesh, utilizing the Peaceman and Rachford method for the time discretization. In all the cases, the newly proposed hybrid schemes attain an almost second-order spatial accuracy. Moreover, a unified theory is derived to obtain an optimal order of convergence of the classical implicit upwind finite difference scheme on Shishkin-type meshes (including the piecewise-uniform Shishkin mesh and the Bakhalov-Shishkin mesh), for a class of singularly perturbed parabolic IBVPs exhibiting strong interior layers. Finally, a post-processing technique (Richardson extrapolation), which improves the accuracy of the standard upwind scheme, is analyzed on a piecewise-uniform Shishkin mesh for singularly perturbed parabolic convection-diffusion IBVP exhibiting a regular boundary layer...
Supervisor: S Natesan