A 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 often computationally impractical, and numerical projections are useful in many cases. We show that the Stratonovich interpretation of the stochastic calculus is obtained using adiabatic elimination with a constraint potential. We derive intrinsic stochastic equations for spheroidal and hyperboloidal surfaces for comparison purposes, and review some earlier projection algorithms. 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. Numerical examples are given for a range of manifolds, including circular, spheroidal, hyperboloidal, and catenoidal cases, as well as higher-order polynomial constraints and a ten-dimensional hypersphere. We show that in all cases the combined midpoint method has greatly reduced errors compared to methods using a combined Euler projection approach or purely tangential projection. Our technique can handle multiple constraints. This allows manifolds that embody several conserved quantities. The algorithm is accurate, simple and efficient. An order of magnitude error reduction in diffusion distance is typically found compared to the other methods, with reductions of several orders of magnitude in constraint errors.