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Stochastic dynamics for inextensible fibers in a spatially semi-discrete setting

: Lindner, Felix; Marheinke, Nicole; Stroot, Holger; Vibe, Alexander; Wegener, Raimund


Stochastics and Dynamics 17 (2017), Nr.2, Art. 1750016, 29 S.
ISSN: 0219-4937
Fraunhofer ITWM ()
stochastic elastic beam dynamics; stochastic differential algebraic equation; manifold-valued stochastic differential equation; nonlinear constraint; global solution theory

We investigate a spatially discrete surrogate model for the dynamics of a slender, elastic, inextensible fiber in turbulent flows. Deduced from a continuous space-time beam model for which no solution theory is available, it consists of a high-dimensional second order stochastic differential equation in time with a nonlinear algebraic constraint and an associated Lagrange multiplier term. We establish a suitable framework for the rigorous formulation and analysis of the semi-discrete model and prove existence and uniqueness of a global strong solution. The proof is based on an explicit representation of the Lagrange multiplier and on the observation that the obtained explicit drift term in the equation satisfies a one-sided linear growth condition on the constraint manifold. The theoretical analysis is complemented by numerical studies concerning the time discretization of our model. The performance of implicit Euler-type methods can be improved when using the explicit representation of the Lagrange multiplier to compute refined initial estimates for the Newton method applied in each time step.