Solution of Differential-Algebraic Equations Through Symbolic Index Reduction

Differential-Algebraic Equations (DAEs) are crucial for modeling and simulating dynamical systems such as mechanisms, electrical circuits, and chemical reactions. However, the numerical solution of DAEs can be challenging due to their high index, which indicates the minimum number of differentiations needed to transform the DAE system into an equivalent system of Ordinary Differential Equations (ODEs) with invariants. Index reduction is a preliminary step in the numerical integration of high-index DAEs that reduces the system’s complicatedness and enables the use of standard ODE solvers. This research introduces an index reduction algorithm based on symbolic computation, which utilizes matrix factorization and differentiation to reduce DAEs’ index. Special attention is given to the mitigation of expression swell. A hierarchical representation technique is employed not only to address this issue but also to improve the reduction processes with the introduction of index-1 variables. The algorithm is implemented in the Indigo library, which combines Maple®’s symbolic manipulation capabilities with Matlab®’s numerical computation capabilities. Validation across various benchmarks demonstrates the algorithm’s effectiveness in reducing high-index systems and showcases its implementation robustness in numerical integration. Overall, the proposed methodology provides a reliable means for addressing generic DAE systems linear in the states’ derivatives.