Improved Structural Methods for Nonlinear Differential-Algebraic Equations via Combinatorial Relaxation

DOI10.1145/3326229.3326236zbMATH Open1467.34009arXiv1907.04511OpenAlexW2964055522WikidataQ131128386 ScholiaQ131128386MaRDI QIDQ5121123

Taihei Oki

Publication date: 10 September 2020

Published in: Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation (Search for Journal in Brave)

Abstract: Differential-algebraic equations (DAEs) are widely used for modeling of dynamical systems. In numerical analysis of DAEs, consistent initialization and index reduction are important preprocessing prior to numerical integration. Existing DAE solvers commonly adopt structural preprocessing methods based on combinatorial optimization. Unfortunately, the structural methods fail if the DAE has numerical or symbolic cancellations. For such DAEs, methods have been proposed to modify them to other DAEs to which the structural methods are applicable, based on the combinatorial relaxation technique. Existing modification methods, however, work only for a class of DAEs that are linear or close to linear. This paper presents two new modification methods for nonlinear DAEs: the substitution method and the augmentation method. Both methods are based on the combinatorial relaxation approach and are applicable to a large class of nonlinear DAEs. The substitution method symbolically solves equations for some derivatives based on the implicit function theorem and substitutes the solution back into the system. Instead of solving equations, the augmentation method modifies DAEs by appending new variables and equations. The augmentation method has advantages that the equation solving is not needed and the sparsity of DAEs is retained. It is shown in numerical experiments that both methods, especially the augmentation method, successfully modify high-index DAEs that the DAE solver in MATLAB cannot handle.

Full work available at URL: https://arxiv.org/abs/1907.04511

zbMATH Keywords

differential-algebraic equationsimplicit function theoremindex reductioncombinatorial relaxationconsistent initializationcombinatorial scientific computing

Mathematics Subject Classification ID

Symbolic computation and algebraic computation (68W30) Implicit ordinary differential equations, differential-algebraic equations (34A09)

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