Article:
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Tobias Neckel, Alfredo Parra Hinojosa & Florian Rupp (2017): Path-Wise Algorithms for Random &
Stochastic ODEs with Applications to Ground-Motion-Induced Excitations of Multi-Storey Buildings,
TU München Technical Report: TUM-I1758, DOI: http://dx.doi.org/10.14459/2017md1350560.


Abstract:
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Random effects play a vital role in various physical phenomena and have to be addressed in the
mathematical modeling and numerical simulations of such processes. In this contribution, the
concepts of random (ordinary) differential equations (RODEs) and stochastic (ordinary) differential
equations (SODEs) are reviewed. Details concerning the numerical solution of problems formulated
via SODEs and RODEs are summarised. In particular, key features of the averaged Euler and Heun as
well as K-RODE Taylor methods for RODEs are compared with numerical methods for SODEs such as the
Euler-Maruyama and the Milstein scheme. Finally, the studied RODE and SODE methods are applied to
one- and multi-dimensional examples, including the problem of ground-motion-induced excitations of
multi-storey buildings subject to the Kanai-Tajimi earthquake model. Being paradigms for additive
white noise driven systems, these applications illustrate on the one hand the applicability of the
aforementioned numerical methods on an extremely wide class of problems. On the other hand, the
coupled oscillator structure of the earthquake induced motion of multi-storey buildings naturally
gives rise to the development of hybrid numerical schemes that combine averaged and deterministic
schemes to fully exploit their benefits.


Keywords:
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random differential equation, stochastic differential equation, path-wise solution,
numerical methods, Kanai-Tajimi earthquake model