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

Abstract

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.