1,224 research outputs found

    Robust and reliable defect control for Runge-Kutta methods

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    On global error estimation and control for initial value problems

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    This paper addresses global error estimation and control for initial value problems for ordinary differential equations. The focus lies on a comparison between a novel approach based on the adjoint method combined with a small sample statistical initialization and the classical approach based on the first variational equation. Control is achieved through tolerance proportionality. Both approaches are found to work well and to enable estimation and control in a reliable manner. However, the novel approach is not found to be competitive with the classical approach, mainly because of its huge storage demand for large problems

    The numerical solution of nonlinear two-point boundary value problems using iterated deferred correction - a survey

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    The use of iterated deferred correction has proved to be a very efficient approach to the numerical solution of general first order systems of nonlinear two-point boundary value problems. In particular the two high order codes TWPBVP.f, based on mono-implicit Runge-Kutta (MIRK) formulae, and TWPBVPL.f based on Lobatto Runge-Kutta formulae as well as the continuation codes ACDC.f and COLMOD.f are now widely used. In this survey we describe some of the problems involved in the derivation of efficient deferred correction schemes. In particular we consider the construction of high order methods which preserve the stability of the underlying formulae, the choice of the mesh choosing algorithm which is based both on local accuracy and conditioning, and the computation of continuous solutions

    Asymptotically correct defect control software for boundary value ordinary differential equations

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    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

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    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

    Starting step size for an ODE solver

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    AbstractOne of the more critical issues in solving ordinary differential equations by a step-by-step process occurs in the starting phase. Somehow the procedure must be supplied with an initial step size that is on scale for the problem at hand. It must be small enough to yield a reliable solution by the process, but not so small as to significantly affect the efficiency of solution. In this paper, we discuss an algorithm for obtaining a good starting step size and present a subroutine which can be readily used in most ODE solvers
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