NUMERICAL METHODS FOR THE SOLUTION OF SINGULARLY PERTURBED PARABOLIC CONVECTION-DIFFUSION PROBLEMS WITH DISCONTINUOUS COEFFICIENT AND DELAY

Abstract

In the present thesis, an attempt has been made to construct, apply, analyse and optimise some simple and efficient parameter-uniform finite difference methods for solving singu larly perturbed parabolic convection-diffusion problems with discontinuous coefficients, source term and delay. These problems commonly arise in the different fields of applied mathematics, for example, edge layers in solid mechanics, aerodynamics, oceanography, rafted-gas dynamics, transition points in quantum mechanics, shock and boundary layers in fluid dynamics, magnetohydrodynamics, drift-diffusion equations of semiconductor de vices, plasma dynamics, skin layers in electrical applications, Stoke’s line in mathematics and rarefied-gas dynamics. These types of problems depend on a small perturbation pa rameter ε, that multiplies some or all of the highest-order derivative terms. On limiting the value of the perturbation parameter to zero, the solutions to such problems approach a discontinuous limit and exhibit a multiscale character. Often these mathematical problems are extremely difficult (or even impossible) to solve exactly, and in these circumstances, approximate solutions are necessary. Asymptotic analysis and numerical analysis are two principal approaches for solving singular perturbation problems. The classical numerical methods have been known to be effective for solving most problems that arise in appli cations, but they failed when applied to singular perturbation problems. That is, for the solutions to these problems, classical numerical methods fail to provide good approxima tions. This motivated us to develop robust numerical methods for solving such types of problems with an emphasis on non-uniform grids. In this thesis, we have provided numerical schemes for solving three different types of convection-diffusion problems of varying complexity. The thesis consists of six chap ters. A brief outline of the chapters is as follows: Chapter one provides an overview of the fundamentals of singular perturbation the ory. Besides, it presents concepts and a historical assessment of the related literature. This chapter also provides a detailed literature survey of various state-of-the-art techniques de veloped in the recent past. In addition, the chapter illustrates the aim and objectives of the research work. xi xii Abstract Chapter two presents an adaptive finite difference method to solve a class of sin gularly perturbed parabolic delay differential equations with discontinuous convection coefficient and source. The simultaneous presence of discontinuity and the delay makes the problem stiff. The solution to the problem considers the present state of the physi cal system and its history. The numerical scheme based on the upwind finite difference method is presented on a specially generated mesh to solve the problem. The adaptive mesh is chosen so that most of the mesh points remain in regions with rapid transitions. The proposed numerical method is analysed for consistency, stability and convergence. Extensive theoretical analysis is performed to obtain consistency and error estimates. The proposed method is unconditionally stable, and the convergence obtained is parameter uniform with first-order convergence in space and first-order convergence in time. The chapter ends with numerical illustrations for the method suggested. Chapter three extends the idea further and aims to provide a better numerical ap proximation of the solution to the model problem considered in Chapter two. The chapter presents a higher-order hybrid difference method over an adaptive mesh to solve the prob lem. The proposed method is a composition of a central difference scheme and a midpoint upwind scheme on a specially generated mesh. Moreover, the time variable is discretised using an implicit finite difference method. The error estimates of the proposed numerical method satisfy parameter-uniform second-order convergence in space and first-order con vergence in time. The rigorous numerical analysis of the proposed method on a Shishkin class mesh establishes the supremacy of the proposed scheme. Chapter four presents a high-order finite difference scheme to solve singularly per turbed parabolic convection-diffusion problems with a large delay and an integral bound ary condition. The solution of the problem features a weak interior layer besides a boundary layer. This chapter presents a higher-order accurate numerical method on a specially designed non-uniform mesh. The technique employs the Crank-Nicolson dif ference scheme in the temporal variable, whereas an upwind difference scheme in space. It is proved that the proposed method is unconditionally stable and converges uniformly, independent of the perturbation parameter. The error analysis indicates that the numerical solution is uniformly stable and shows parameter-uniform second-order convergence in time and first-order convergence in space. Chapter five presents a robust computational technique to solve a class of two parameter parabolic convection-diffusion problems with a large delay. The presence of perturbation parameters leads to the twin boundary layers and interior layers in the so lution, whose appropriate numerical approximation is the main goal of this chapter. The numerical method is composed of an upwind difference scheme in space, and a Crank- Abstract xiii Nicolson scheme in time is used to find the approximate solution of the problem. It is proved that the method is parameter-uniform with second-order accuracy in time and al most first-order accuracy in space. Numerical examples are provided in support of the theory. Chapter six concludes the work done and provides insight into the author’s thoughts on the future direction of the research.