Asymptotically correct defect control software for boundary value ordinary differential equations

Abstract

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.