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

Modifications to a Runge-Kutta type software package for the numerical solution of boundary value ordinary differential equations

ix, 90 leaves : ill. ; 28 cm.Includes abstract.Includes bibliographical references (leaves 86-90).MIRKDC [Enright, Muir, 1996] is a software package for the numerical solution of systems of first order, nonlinear, boundary value ordinary differential equations (ODEs), with separated boundary conditions. It employs mono-implicit Runge-Kutta methods for the discretization of the ODEs and monitors the quality of the numerical solution using defect control. The discrete systems are solved by modified Newton iterations and extensive use of adaptive mesh refinement is employed. This thesis describes modifications to the MIRKDC software package in order to incorporate a number of performance enhancements including computational derivative approximation, analytic derivative assessment, problem sensitivity (conditioning) assessment, introduction of new optimized Runge-Kutta formulas, improvement of the defect control strategy and the introduction of an auxiliary global error indicator. Numerical results to demonstrate the impact of these enhancements are presented