ODE Methods for the Solution of Differential/Algebraic Systems

TLDR

Many differential/algebraic systems can be solved conveniently and economically using a range of ODE methods, while others require only a small subset or are intractable for current methods. The paper investigates which ODE methods are best for the solvable groups, analyzes why the hard group fails, and proposes a reduction technique. The authors examine the first two groups, describe a reduction technique that transforms hard systems into solvable ones, and provide a tool for analytical study of system structure.

Abstract

In this paper we study the numerical solution of the differential/algebraic systems $F(t,y,y') = 0$. Many of these systems can be solved conveniently and economically using a range of ODE methods. Others can be solved only by a small subset of ODE methods, and still others present insurmountable difficulty for all current ODE methods. We examine the first two groups of problems and indicate which methods we believe to be best for them. Then we explore the properties of the third group which cause the methods to fail. We describe a reduction technique which allows systems to be reduced to ones that can be solved. It also provides a tool for the analytical study of the structure of systems.