Exploration of moving transformation methods for boundary value ordinary differential equations and one-dimensional time-dependent partial differential equations

Abstract

Rapid advances in computing power have given computational analysis and simulation a prominent role in modern scientific exploration. Differential equations are often used to model complex scientific phenomena. In practice, these equations can not be solved exactly and numerical approximations which accurately preserve the characteristics of the modelled phenomena must be employed. This has motivated the development of accurate and efficient numerical methods and software for these problems. This thesis explores a class of adaptive methods for accurately computing numerical solutions for two common classes of differential equations, boundary value ordinary differential equations and time-dependent partial differential equations in one spatial dimension. These adaptive methods, referred to as moving transformation (MT) methods, are used to improve the accuracy of standard numerical methods for these problem classes and can be extended to higher dimensions. MT methods improve the accuracy of these standard numerical algorithms by transforming the differential equation into a related differential equation on a computational domain where it is easier to solve. The solution to this transformed differential equation can then be transformed back to the original physical domain to obtain a solution to the original differential equation. Software implementing MT methods is developed and computational experiments performed to determine the effectiveness of these methods compared to traditional adaptation approaches. We also investigate the suitability of these methods for implementation in adaptive error control algorithms.