Asymptotically correct defect control software for boundary value ordinary differential equations

xii, 109 leaves : col. ill. ; 29 cm.Includes abstract.Includes bibliographical references (leaves 106-109).BVP_SOLVER II [Boisvert, Muir, Spiteri, 2013] is an efficient software package for the numerical solution of systems of boundary value ordinary differential equations. It employs discrete mono-implicit Runge-Kutta (MIRK) schemes to transform the ODEs into nonlinear systems which are solved by modified Newton iterations. Continuous MIRK interpolants then augment the discrete solutions from the nonlinear system, to obtain a continuous solution approximation across the problem domain. The code monitors solution quality through defect analysis and employs an adaptive mesh refinement strategy as a means of controlling the defect, which is the amount by which the computed solution fails to satisfy the ODEs. This thesis describes the development of new Hermite-Birkhoff interpolants and modifications to the BVP_SOLVER II software in order to implement a new defect estimation strategy called “Asymptotically Correct Maximum Defect Estimation”, based on the new interpolants. Numerical results which demonstrate the robustness and efficiency of the new strategy are presented

Efficient continuous Runge-Kutta methods for asymptotically correct defect control

xii, 124 leaves : coloured illustrations ; 29 cmIncludes abstract.Includes bibliographical references (leaves 120-124).Mono-Implicit Runge-Kutta (MIRK) methods and continuous MIRK (CMIRK) methods, are used in the numerical solution of boundary value ordinary differential equations (ODEs). One way of assessing the quality of the numerical solution is to estimate its maximum defect, which is the amount by which the solution fails to satisfy the ODE. The standard approach is to perform two point sampling of the defect on each subinterval of a mesh which partitions the problem domain to estimate the maximum defect. However, the location of the maximum defect on each subinterval typically varies from subinterval to subinterval, and from problem to problem. Thus sampling at only two points typically leads to an underestimate of the maximum defect. In this thesis, we will derive a new class of CMIRK interpolants for which the location of the maximum defect on each subinterval is the same over all subintervals and problems

Generalized mono-implicit Runge-Kutta methods for stiff ordinary differential equations

ix, 100 leaves ; 29 cmIncludes abstract and appendices.Includes bibliographical references (leaves 58-61).Ordinary differential equations (ODEs) arise in many applications. Typically these ODEs are sufficiently complicated that they must be solved using numerical methods. One of the well-known classes of numerical methods for ODEs is the class of Mono- Implicit Runge-Kutta (MIRK) methods. An important property of a MIRK method is its order; a method is of order p if its global error is O(hp). An issue with MIRK methods, when applied to certain ODEs, known as stiff ODEs, is that when they should be of order p, they perform as if their order is q, where q < p. This is called order reduction. This means that the MIRK methods will be inefficient when the ODE is stiff because the amount of computation that is performed is not consistent with the accuracy obtained. In this thesis, we derive generalizations of MIRK methods that can avoid order reduction when the ODE is stiff

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

Almost Block Diagonal Linear Systems: Sequential and Parallel Solution Techniques, and Applications

Almost block diagonal (ABD) linear systems arise in a variety of contexts, specifically in numerical methods for two-point boundary value problems for ordinary differential equations and in related partial differential equation problems. The stable, efficient sequential solution of ABDs has received much attention over the last fifteen years and the parallel solution more recently. We survey the fields of application with emphasis on how ABDs and bordered ABDs (BABDs) arise. We outline most known direct solution techniques, both sequential and parallel, and discuss the comparative efficiency of the parallel methods. Finally, we examine parallel iterative methods for solving BABD systems. Copyright (C) 2000 John Wiley & Sons, Ltd

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

A problem-solving environment for the numerical solution of boundary value problems

Boundary value problems (BVPs) are systems of ordinary differential equations (ODEs) with boundary conditions imposed at two or more distinct points. Such problems arise within mathematical models in a wide variety of applications. Numerically solving BVPs for ODEs generally requires the use of a series of complex numerical algorithms. Fortunately, when users are required to solve a BVP, they have a variety of BVP software packages from which to choose. However, all BVP software packages currently available implement a specific set of numerical algorithms and therefore function quite differently from each other. Users must often try multiple software packages on a BVP to find the one that solves their problem most effectively. This creates two problems for users. First, they must learn how to specify the BVP for each software package. Second, because each package solves a BVP with specific numerical algorithms, it becomes difficult to determine why one BVP package outperforms another. With that in mind, this thesis offers two contributions. First, this thesis describes the development of the BVP component to the fully featured problem-solving environment (PSE) for the numerical solution of ODEs called pythODE. This software allows users to select between multiple numerical algorithms to solve BVPs. As a consequence, they are able to determine the numerical algorithms that are effective at each step of the solution process. Users are also able to easily add new numerical algorithms to the PSE. The effect of adding a new algorithm can be measured by making use of an automated test suite. Second, the BVP component of pythODE is used to perform two research studies. In the first study, four known global-error estimation algorithms are compared in pythODE. These algorithms are based on the use of Richardson extrapolation, higher-order formulas, deferred corrections, and a conditioning constant. Through numerical experimentation, the algorithms based on higher-order formulas and deferred corrections are shown to be computationally faster than Richardson extrapolation while having similar accuracy. In the second study, pythODE is used to solve a newly developed one-dimensional model of the agglomerate in the catalyst layer of a proton exchange membrane fuel cell