An Adaptive Boundary Value Runge–Kutta Solver for First Order Boundary Value Problems

Abstract

The use of finite difference methods which combine features of both Runge–Kutta processes and Gap schemes is proposed for developing adaptive codes for the solution of first order differential equations with two-point boundary conditions. The order conditions for the coefficients of these processes are given. A set of conditions, by which these order conditions can be reduced in number, are developed. An eighth order A-stable method which has second, fourth and sixth order A-stable methods embedded in it is given as an example. A variable order, variable step finite difference solver using embedded methods of this kind is described.