Numerical Analysis and Scientific Computing Seminar | Andrey Shkerin, High-Order Stochastic Symplectic Scheme for the Langevin Equation and Field Theory

We propose a new numerical scheme to solve stochastic differential equations. The scheme combines the 3rd order regularization of stochastic forces with a general operator-splitting symplectic method to solve the resulting regular equation, at each time step. For the Langevin-type equation with additive white noise the scheme has 3rd order of strong convergence, as we demonstrate analytically and numerically. We apply the scheme to simulate stochastic dynamics of Hamiltonian systems with many degrees of freedom, in particular, classical field theories. To the best of our knowledge, this is the first application of high-order stochastic numerical schemes to problems in field theory. We demonstrate the advantage of using the scheme in the cases when high accuracy or fast convergence are required.