Design and Implementation of a Solver for High-Index Differential-Algebraic Equations

Abstract

<p> Systems of differential-algebraic equations (DAEs) arise in numerious applications, and there has been considerable research on solving DAE initial value problems (IVPs). Existing methods and software for solving DAEs usually handle at most index-three problems. However, DAE problems of index three and higher do arise, for example, in actuator dynamics, multi-stage processes, and optimization.</p> <p> We present the method of J. Pryce and N. Nedialkov for solving DAEs, which can be of high index, fully implicit, and contain derivatives of order higher than one. We solve such DAEs by expanding their solution in Taylor series (TS). To compute Taylor coefficients, we employ J. Pryce's structural analysis and automatic differentiation. Then we compute an approximate TS solution with appropriate stepsize and project this solution to satisfy the constraints (explicit and hidden) of the problem.</p> <p> This thesis discusses the algorithms involved in this method, including the algorithms for Taylor coefficients computation, consistent point projection, error estimation, stepsize control, and the overall integration process. The author has implemented a software package named HIDAETS (High-Index DAE by Taylor Series). In this thesis, we present the specification, design, implementation, and usage of HIDAETS. Numerical results on several high-index DAEs are reported. These results demonstrate that HIDAETS is efficient and accurate for solving IVP in DAEs.</p>