Strong stochastic Runge-Kutta–Munthe-Kaas methods for nonlinear Itô SDEs on manifolds

In this paper we present numerical methods for the approximation of nonlinear Itô stochastic differential equations on manifolds. For this purpose, we extend Runge-Kutta–Munthe-Kaas (RKMK) schemes for ordinary differential equations on manifolds to the stochastic case and analyse the strong convergence of these schemes. Since these schemes are based on the application of a stochastic Runge-Kutta (SRK) scheme in a corresponding Lie algebra, we address the question under which circumstances the stochastic RKMK method has the same strong order of convergence as the applied SRK scheme. To illustrate our answer to this question and the effectiveness of our schemes, we show some numerical results of applying these methods to a problem with an autonomous underwater vehicle.