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Volume 14, Issue 2
Stochastic Runge-Kutta–Munthe-Kaas Methods in the Modelling of Perturbed Rigid Bodies

Michelle Muniz, Matthias Ehrhardt, Michael Günther & Renate Winkler

DOI: 10.4208/aamm.OA-2021-0176

Adv. Appl. Math. Mech., 14 (2022), pp. 528-538.

Published online: 2022-01

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  • Abstract

In this paper we present how nonlinear stochastic Itô differential equations arising in the modelling of perturbed rigid bodies can be solved numerically in such a way that the solution evolves on the correct manifold. To this end, we formulate an approach based on Runge-Kutta–Munthe-Kaas (RKMK) schemes for ordinary differential equations on manifolds. 

Moreover, we provide a proof of the mean-square convergence of this stochastic version of the RKMK schemes applied to the rigid body problem and illustrate the effectiveness of our proposed schemes by demonstrating the structure preservation of the stochastic RKMK schemes in contrast to the stochastic Runge-Kutta methods.

  • Keywords

Stochastic Runge-Kutta method, Runge-Kutta–Munthe-Kaas scheme, nonlinear Itô SDEs, rigid body problem.

  • AMS Subject Headings

60H10, 70G65, 91G80

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-14-528, author = {Muniz , MichelleEhrhardt , MatthiasGünther , Michael and Winkler , Renate}, title = {Stochastic Runge-Kutta–Munthe-Kaas Methods in the Modelling of Perturbed Rigid Bodies}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2022}, volume = {14}, number = {2}, pages = {528--538}, abstract = {

In this paper we present how nonlinear stochastic Itô differential equations arising in the modelling of perturbed rigid bodies can be solved numerically in such a way that the solution evolves on the correct manifold. To this end, we formulate an approach based on Runge-Kutta–Munthe-Kaas (RKMK) schemes for ordinary differential equations on manifolds. 

Moreover, we provide a proof of the mean-square convergence of this stochastic version of the RKMK schemes applied to the rigid body problem and illustrate the effectiveness of our proposed schemes by demonstrating the structure preservation of the stochastic RKMK schemes in contrast to the stochastic Runge-Kutta methods.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2021-0176}, url = {http://global-sci.org/intro/article_detail/aamm/20208.html} }
TY - JOUR T1 - Stochastic Runge-Kutta–Munthe-Kaas Methods in the Modelling of Perturbed Rigid Bodies AU - Muniz , Michelle AU - Ehrhardt , Matthias AU - Günther , Michael AU - Winkler , Renate JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 528 EP - 538 PY - 2022 DA - 2022/01 SN - 14 DO - http://doi.org/10.4208/aamm.OA-2021-0176 UR - https://global-sci.org/intro/article_detail/aamm/20208.html KW - Stochastic Runge-Kutta method, Runge-Kutta–Munthe-Kaas scheme, nonlinear Itô SDEs, rigid body problem. AB -

In this paper we present how nonlinear stochastic Itô differential equations arising in the modelling of perturbed rigid bodies can be solved numerically in such a way that the solution evolves on the correct manifold. To this end, we formulate an approach based on Runge-Kutta–Munthe-Kaas (RKMK) schemes for ordinary differential equations on manifolds. 

Moreover, we provide a proof of the mean-square convergence of this stochastic version of the RKMK schemes applied to the rigid body problem and illustrate the effectiveness of our proposed schemes by demonstrating the structure preservation of the stochastic RKMK schemes in contrast to the stochastic Runge-Kutta methods.

Muniz , MichelleEhrhardt , MatthiasGünther , Michael and Winkler , Renate. (2022). Stochastic Runge-Kutta–Munthe-Kaas Methods in the Modelling of Perturbed Rigid Bodies. Advances in Applied Mathematics and Mechanics. 14 (2). 528-538. doi:10.4208/aamm.OA-2021-0176
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