Superconvergant interpolants for the collocation solution of boundary value ordinary differential equations

Publisher's version/PDFA long-standing open question associated with the use of collocation methods for boundary value ordinary differential equations is concerned with the development of a high order continuous solution approximation to augment the high order discrete solution approximation, obtained at the mesh points which subdivide the problem interval. It is well known that the use of collocation at Gauss points leads to solution approximations at the mesh points for which the global error is O(h[superscript 2k]), where k is the number of collocation points used per subinterval and h is the subinterval size. This discrete solution is said to be superconvergent. The collocation solution also yields a C[superscript 0] continuous solution approximation that has a global error of O(h[supercript k+1]). In this paper, we show how to efficiently augment the superconvergent discrete collocation solution to obtain C[superscript 1] continuous "superconvergent" interpolants whose global errors are O(h[superscript 2k]). The key ideas are to use the theoretical framework of continuous Runge-Kutta schemes and to augment the collocation solution with inexpensive monoimplicit Runge-Kutta stages. Specific schemes are derived for k = 1, 2, 3, and 4. Numerical results are provided to support the theoretical analysis

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

B-spline collocation for two dimensional, time-dependent, parabolic PDEs

vi, 177 leaves : ill. ; 29 cm.Includes abstract and appendices.Includes bibliographical references (leaves 82-88).In this thesis, we consider B-spline collocation algorithms for solving two-dimensional in space, time-dependent parabolic partial differential equations (PDEs), defined over a rectangular region. We propose two ways to solve the problem: (i) The Method of Surfaces: Discretizing the problem in one of the spatial domains, we obtain a system of one-dimensional parabolic PDEs, which is then solved using a one-dimensional PDE system solver. (ii) Two-dimensional B-spline collocation: The numerical solution is represented as a bi-variate piecewise polynomial with unknown time-dependent coefficients. These coefficients are determined by requiring the numerical solution to satisfy the PDE at a number of points within the spatial domain, i.e., we collocate simultaneously in both spatial dimensions. This leads to an approximation of the PDE by a large system of time-dependent differential algebraic equations (DAEs), which we then solve using a high quality DAE solver