Differential/Algebraic Equations are not ODE’s

Abstract

This paper outlines a number of difficulties which can arise when numerical methods are used to solve systems of differential/algebraic equations of the form F(t, y, y') 0. Problems which can be written in this general form include standard ODE systems as well as problems which are substantially different from standard ODE's. Some of the differential/algebraic systems can be solved using numerical methods which are commonly used for solving stiff systems of ordinary differential equations. Other problems can be solved using codes based on the stiff methods, but only after extensive modifications to the error estimates and other strategies in the code. A further class of problems cannot be solved at all with such codes, because changing the stepsize causes large errors in the solution. We describe in detail the causes of these difficulties and indicate solutions in some cases. used to solve systems of differential/algebraic equations (DAE) of the form F(t, y, y') 0. These problems look much like standard ordinary differential equation (ODE) systems of the form y'= f(t, y) (and of course include these systems as a special case), and many of the DAE systems can be solved using numerical methods which are commonly used for solving stiff systems of ODE's. However, the class ofDAE systems also includes problems with properties that are very different from those of standard ODE's. Some of these problems cannot be solved using variable-stepsize stiff methods such as backward differentiation formulas (BDF). Others can be solved using such methods but only after substantial modifications to the strategies usually used in codes implementing those methods. In this paper we explore the causes of the difficulties and describe modifications which enable codes based on BDF to solve a wider class of problems than were previously possible. Additionally, we suggest strategies for detecting the problems which cannot be solved with this technique. Several authors (1), (2), (3), (4), (5), (6), (7) have written codes designed to deal