Robust Numerical Techniques Based on Layer Resolving Meshes for Time-delayed Singularly Perturbed Parabolic Problems

Abstract

This thesis enunciates some robust numerical schemes for singularly perturbed parabolic problems with a large time lag/delay in both one and two dimensions. In every problem, a tiny “singular perturbation parameter” (0 < ε ≪ 1) is multiplied by the highest-order spatial derivative term. This small parameter plays a crucial role in determining the nature of solutions to these problems. As ε approaches zero, the solution undergoes a sharp change at some regions namely, the boundary layer and/or interior layer region of the domain. The conventional numerical methods (finite difference methods on uniform mesh) fail to tackle this abrupt change inside the layer regions. Further, the time lag makes finding solutions to the considered problems quite challenging and time-consuming. So, the primary concern is to develop some computationally efficient “parameter-uniform schemes” that provide approximate solutions converging to the exact solution, being independent with respect to both parameters. The “parameter-uniform schemes” are developed via some efficient fitted mesh methods for one and two-dimensional time-delayed singularly perturbed parabolic problems with one or two perturbation parameters along with their possible extensions to problems with semilinearity. The thesis starts with some preliminaries of time-delayed singularly perturbed problems and the numerical strategies to be used for their solutions. All the proposed schemes are applied to two different non-uniform meshes. The well-known Shishkin mesh is used that provide a leading-order approximation with the drawback of a logarithmic term in the convergence. This effect is further rectified by the use of Bakhvalov-Shishkin mesh. The widely-used upwind scheme is tested on some semilinear problems as well as problems with interior layers and problems of higher dimensions with multiple perturbation parameters. In some cases, the Richardson extrapolation is used to elevate the accuracy of the upwind scheme. A second-order accurate hybrid scheme and its modified versions are studied thoroughly to be used for problems with small spatial shifts, problems with interior layers, and problems of higher dimensions. Further, a weighted-variable-based monotone hybrid scheme is also discussed that overrides the need for two separate spatial schemes. For time discretization, along with the conventional implicit Euler scheme, two other efficient approaches are discussed. In the case of second-order accurate schemes, the Crank-Nicolson scheme is used in time to match up the accuracy in the spatial domain. Again, a generalized scheme namely the θ-scheme is applied that can provide first-order as well as second-order accuracy by varying θ. The idea of splitting in time is discussed for the problems of higher dimensions. Finally, the highly efficient Thomas’ algorithm is used to solve the resulting system of linear difference equations. The convergence analysis for all the schemes is thoroughly discussed using the truncation error and barrier function approach. The efficacy of the schemes is proved through numerical experiments. Numerical outputs are presented in the form of various plots and tables to support the theoretical claims. These simulations prove proposed schemes to be leading-order accurate and beneficial over many existing schemes in the literature. A summary of all the major contributions and possible ideas for future enhancement are discussed at the end of this thesis.