Simultaneous Numerical Solution of Differential-Algebraic Equations

Abstract

A unified method for handling the mixed differential and algebraic equations of the type that commonly occur in the transient analysis of large networks or in continuous system simulation is discussed. The first part of the paper is a brief review of existing techniques of handling initial value problems for stiff ordinary differential equations written in the standard form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y' f(y, t)</tex> . In the second part one of these techniques is applied to the problem <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F(y, y', t)=0</tex> . This may be either a differential or an algebraic equation as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\partial F/ \partial y'</tex> is nonzero or zero. It will represent a mixed system when vectors F and y represent components of a system. The method lends itself to the use of sparse matrix techniques when the problem is sparse.