A Runge-kutta Type Boundary Value Ode Solver with Defect Control

A popular approach for the numerical solution of boundary value ODE problems involves the use of collocation methods. Such methods can be naturally implemented so as to provide a continuous approximation to the solution over the entire problem interval. On the other hand, several authors have suggested as an alternative, certain subclasses of the implicit Runge-Kutta formulas, known as mono-implicit Runge-Kutta (MIRK) formulas, which can be implemented substantially more eeciently than the collocation methods. These latter formulas do not have a natural implementation that provides a continuous approximation to the solution; rather only a discrete approximation at certain points within the problem interval is obtained. However recent work in the area of initial value problems has demonstrated the possibility of generating inexpensive interpolants for any explicit Runge-Kutta formula. These ideas have recently been extended to develop inter-polants for the MIRK formulas. In this paper, we describe our investigation of the use of continuous MIRK formulas in a code for the numerical solution of boundary value ODE problems. A primary thrust of this investigation is to consider defect control, based on these interpolants, as an alternative to the standard use of global error estimates, as the basis for termination and mesh redistribution criteria.

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