Performance Analysis of Error-Control B-spline Gaussian Collocation Software for PDEs

Pre-printB-spline Gaussian collocation software has been widely used in the numerical solution of boundary value ordinary differential equations (BVODEs) and partial differential equations (PDEs) in one space dimension (1D) for many years. The software package, BACOL, developed over a decade ago, was one of the first 1D PDE packages to provide both temporal and spatial error control. A new package, BACOLI, improves upon the efficiency of BACOL through the use of new types of spatial error estimation and control. The complexity of the interactions among the component numerical algorithms used by these packages implies that extensive testing and analysis of the test results is an essential factor in their development. In this paper, we investigate the performance of the BACOL and BACOLI packages with respect to several important machine independent algorithmic measures and examine the effectiveness of the new error estimation and error control strategies. We also investigate the influence of the choice of the degree of the B-splines on the efficiency and reliability of the solvers. These results will provide new insights into how to improve BACOLI, lead to improvements in the Gaussian collocation BVODE solvers, COLSYS and COLNEW, and guide the further development of B-spline Gaussian collocation software with error control for 2D PDEs

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

1 online resource (112, 4 unnumbered pages) : colour illustrationsIncludes abstract.Includes bibliographical references (pages [113-116]).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