Stochastic Runge-Kutta–Munthe-Kaas Methods in the Modelling of Perturbed Rigid Bodies (Advances in Applied Mathematics and Mechanics, 2022)
In this paper we present how nonlinear stochastic Itô differential equations arising in the modelling of perturbed rigid bodies can be solved numerically in sucha way that the solution evolves on the correct manifold. To this end, we formulate anapproach 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 stochasticversion of the RKMK schemes applied to the rigid body problem and illustrate theeffectiveness of our proposed schemes by demonstrating the structure preservation ofthe stochastic RKMK schemes in contrast to the stochastic Runge-Kutta methods.