ROBUST COMPUTATIONAL METHODS FOR SINGULAR PERTURBATION PROBLEMS WITH SHIFTS AND INTEGRAL BOUNDARY CONDITIONS

Abstract

In the present thesis, an attempt has been made to construct, apply, analyse and optimise higher-order hybrid parameter-uniform finite difference methods for solving singular perturbation problems involving a system of reaction-diffusion equations with shifts and integral boundary conditions. These problems commonly arise in the different fields of applied mathematics, for example, edge layers in solid me- chanics, aerodynamics, oceanography, rafted-gas dynamics, transition points in quantum mechanics, shock and boundary layers in fluid dynamics, magnetohydrodynamics, drift-diffusion equations of semi- conductor devices, plasma dynamics, skin layers in electrical applications, Stoke’s line in mathematics, and rarefied-gas dynamics. These problems depend on a small perturbation parameter ε, which mul- tiplies the highest-order derivative terms. When the value of the perturbation parameter is limited to zero, the solutions to such problems approach a discontinuous limit and exhibit a multiscale charac- ter. Often, these mathematical problems are extremely difficult (or even impossible) to solve exactly and approximate solutions are necessary in certain circumstances. Asymptotic and numerical analy- sis are two principal approaches to solving singular perturbation problems. Although asymptotic and numerical methods offer valuable tools for tackling singularly perturbed systems of reaction-diffusion equations, they also have limitations. Asymptotic methods struggle to provide accurate solutions in regions where multiple lengths or time scales interact. Additionally, these methods often rely on an- alytical approximations, which may not fully capture the system’s behaviour. Numerical methods also have limitations when applied on uniform meshes. They require excessively fine meshes to capture the solution behaviour within the boundary layers, leading to computationally expensive simulations. The analysis and solution of these systems require specialised mathematical techniques tailored to handle stiffness and boundary layer phenomena. The thesis provides higher-order hybrid numerical methods over an adaptive mesh for solving different classes of reaction-diffusion problems. The thesis consists of six chapters. A brief outline of the chapters is as follows: Chapter 1 recalls an overview of the fundamentals of singular perturbation theory. It also presents concepts and a historical assessment of the related literature. This chapter also provides a detailed literature review of various state-of-the-art techniques developed in the recent past. In addition, the chapter illustrates the purpose and objectives of the thesis. Chapter 2 presents a higher-order adaptive hybrid difference method to solve a singularly perturbed system of reaction-diffusion problems with Dirichlet boundary conditions. The numerical method com- bines a Hermite difference method with the classical central difference method on a layer-adapted ix mesh. The equidistribution principle generates the mesh using a nonnegative monitor function. The mesh generation procedure automatically detects the thickness and steepness of any boundary lay- ers present in the solution and does not require prior information about its analytical behaviour. The chapter presents a rigorous theoretical analysis and numerical results for model problems to support theoretical findings. The method is almost fourth-order accurate, converges uniformly, and is uncon- ditionally stable. Moreover, the convergence obtained is optimal, as the estimates are free from any logarithmic term compared to the difference methods over the piecewise uniform Shishkin mesh. Chapter 3 presents a higher-order hybrid approximation over an adaptive mesh designed to solve a coupled system of singularly perturbed reaction-diffusion equations with a shift on an equidistributed mesh. The difference method combines an exponential spline difference method for the outer layer and a cubic spline difference method for the boundary layer on the adaptive mesh generated. The mesh relies on the equidistribution principle, a nonnegative monitor function, and the second-order derivatives of the layer components of the solution. The proposed numerical method improves the accuracy of numerical solutions while maintaining computational efficiency. The proposed numerical method is consistent, stable, and converges regardless of the size of the perturbation parameter. The numerical results and illustrations support the theoretical findings. Chapter 4 presents a semi-analytical approach to solving a system of singularly perturbed convection- diffusion equations with shifts. A careful factorisation handles complex multiscale systems by splitting them into two explicit parts: one capturing smooth solutions and the other addressing boundary layer solutions. The strategy involves factoring a coupled system of equations into explicit systems of first- order initial value problems and second-order boundary value problems. The solutions to the degener- ate system correspond to the regular component. In contrast, those of the system of boundary value problems represent the singular component. The process combines the regular and singular compo- nents to obtain the complete solution. The q-stage Runge-Kutta method computes the outer solution, and an analytical approach derives the inner solution. The proposed method is unconditionally stable and converges independently of the perturbation parameters. Unlike numerical methods, the proposed technique does not require adaptive mesh generation to sustain approximation and consequently has lower computational complexity. The process is straightforward, and interdisciplinary researchers can quickly adapt the method to solve problems related to chemical kinetics, mathematical physics, and biology. The method is highly accurate, free from directional bias, and the estimates are free from logarithmic terms. The results demonstrate that the numerical method outperforms many existing methods. Chapter 5 presents a highly efficient hybrid difference approximation for a time-dependent singularly perturbed reaction-diffusion equation with shift and integral boundary conditions. The technique utilises a modified backward difference discretisation in time on a uniform mesh and a suitable combination of the exponential and cubic spline difference methods over a layer adaptive moving mesh in space. The layer-adapted mesh in space is generated by equidistributing a nonnegative monitor function, and the modified backward difference discretisation ensures alignment with the mesh at each subsequent time level. The presented method demonstrates second-order spatial uniform convergence and first-order temporal convergence. The method improves the accuracy of numerical solutions while maintaining x computational efficiency. The method is unconditionally stable and free from directional bias. The numerical experiments validate the theoretical estimates. Chapter 6 concludes the work done and provides insight into the author’s thoughts on the future direction of the research.