Behavior of the nonunique terms in general DAE integrators

Abstract

Differential algebraic equations (DAEs) are implicit systems of ordinary differential equations F( y′, y, t) = 0. DAEs arise in many applications and a variety of numerical methods have been developed for solving DAEs. Numerical methods have been proposed for integrating general higher index DAEs and successfully applied to test problems. These methods require solving a nonlinear system of equations which is larger than the original DAE at each time step. For fully implicit problems part of the solution of the nonlinear system is not uniquely determined. This poses questions about the effects of predictors and also a possible instability in the growth of these terms during a numerical integration. In this paper it is shown that the nonunique component is actually the numerical solution of an auxiliary DAE which depends not only on the original DAE but also on the predictor being used in the Gauss-Newton iteration. As an important consequence we both establish a basis for the design of low order integrators for high index DAEs and develop guidelines for the use of predictors in integrating general high index DAEs.