Projected explicit and implicit Taylor series methods for DAEs

• Published in: Numerical algorithms Vol. 88; no. 2; pp. 615 - 646
• Main Authors: Estévez Schwarz, Diana, Lamour, René
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2021; Springer Nature B.V
• Subjects: Algebra; Algorithms; Approximation; Computer Science; Constraints; Linear algebra; Methods; Multibody systems; Numeric Computing; Numerical Analysis; Optimization; Original Paper; Systems simulation; Taylor series; Theory of Computation; Consistent initial value; Integration; Projector-based analysis; DAE; Index; Taylor series methods; Derivative array; Differential-algebraic equation; Nonlinear constrained optimization; Automatic differentiation
• ISSN: 1017-1398; 1572-9265; 1572-9265
• DOI: 10.1007/s11075-020-01051-z

The recently developed new algorithm for computing consistent initial values and Taylor coefficients for DAEs using projector-based constrained optimization opens new possibilities to apply Taylor series integration methods. In this paper, we show how corresponding projected explicit and implicit Taylor series methods can be adapted to DAEs of arbitrary index. Owing to our formulation as a projected optimization problem constrained by the derivative array, no explicit description of the inherent dynamics is necessary, and various Taylor integration schemes can be defined in a general framework. In particular, we address higher-order Padé methods that stand out due to their stability. We further discuss several aspects of our prototype implemented in Python using Automatic Differentiation. The methods have been successfully tested on examples arising from multibody systems simulation and a higher-index DAE benchmark arising from servo-constraint problems.