Improving performance when solving high-order and mixed-order boundary value problems in ODEs

Abstract

Solving high-order or mixed-order boundary value problems by general purpose software often requires the system to be first converted to a larger equivalent first-order system. The cost of solving such problems is generally O(m ³), where m is the dimension of the equivalent first-order system. In this paper, we show how to reduce this cost by exploiting the special structure the “equivalent” first-order system inherits from the original associated mixed-order system. This technique applies to a broad class of boundary value methods. We illustrate the potential benefits by considering in detail a general purpose Runge–Kutta method and a multiple shooting method.