Projected explicit and implicit Taylor series methods for DAEs

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 Pade 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.

Automated summary

The newly developed algorithm for computing consistent initial values and Taylor coefficients for DAEs offers new possibilities for applying Taylor series integration methods. By formulating the problem as a constrained optimization problem constrained by the derivative array, explicit description of the inherent dynamics is not required, allowing for various Taylor integration schemes to be defined within a general framework. The methods have been successfully tested on examples from multibody systems simulation and a higher-index DAE benchmark from servo-constraint problems.