Midpoint projection algorithm for stochastic differential equations on manifolds

Stochastic differential equations projected onto manifolds occur in physics, chemistry, biology, engineering, nanotechnology, and optimization, with interdisciplinary applications. Intrinsic coordinate stochastic equations on the manifold are sometimes computationally impractical, and numerical projections are therefore useful in many cases. In this paper a combined midpoint projection algorithm is proposed that uses a midpoint projection onto a tangent space, combined with a subsequent normal projection to satisfy the constraints. We also show that the Stratonovich form of stochastic calculus is generally obtained with finite bandwidth noise in the presence of a strong enough external potential that constrains the resulting physical motion to a manifold. Numerical examples are given for a wide range of manifolds, including circular, spheroidal, hyperboloidal, and catenoidal cases, higher-order polynomial constraints that give a quasicubical surface, and a ten-dimensional hypersphere. In all cases the combined midpoint method has greatly reduced errors compared to other methods used for comparison, namely, a combined Euler projection approach and a tangential projection algorithm. We derive intrinsic stochastic equations for spheroidal and hyperboloidal surfaces for comparison purposes to verify the results. Our technique can handle multiple constraints, which allows manifolds that embody several conserved quantities. The algorithm is accurate, simple, and efficient. A reduction of an order of magnitude in the diffusion distance error is found compared to the other methods and an up to several orders of magnitude reduction in constraint function errors.