Discrete Gradient Approach to Stochastic Differential Equations with a Conserved Quantity
SIAM Journal on Numerical Analysis
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We present Lie group integrators for nonlinear stochastic differential equations with noncommutative vector fields whose solution evolves on a smooth finite-dimensional manifold. Given a Lie group action that generates transport along the manifold, we pull back the stochastic flow on the manifold to the Lie group via the action, and subsequently pull back the flow to the corresponding Lie algebra via the exponential map. We construct an approximation to the stochastic flow in the Lie algebra via closed operations, and then push back to the Lie group and then to the manifold, thus ensuring that our approximation lies in the manifold. We call such schemes stochastic Munthe-Kaas methods after their deterministic counterparts. We also present stochastic Lie group integration schemes based on Castell-Gaines methods. These involve using an underlying ordinary differential integrator to approximate the flow generated by a truncated stochastic exponential Lie series. They become stochastic Lie group integrator schemes if we use Munthe-Kaas methods as the underlying ordinary differential integrator. Further, we show that some Castell-Gaines methods are uniformly more accurate than the corresponding stochastic Taylor schemes. Lastly we demonstrate our methods by simulating the dynamics of a free rigid body such as a satellite and an autonomous underwater vehicle both perturbed by two independent multiplicative stochastic noise processes.