Variable-stepsize Runge–Kutta methods for stochastic Schrödinger equations

Abstract

Stochastic wave equations of Schrödinger type are widely employed in physics and have numerous potential applications in chemistry. While some accurate numerical methods exist for particular classes of stochastic differential equations they cannot generally be used for Schrödinger equations. Efficient and accurate methods for their numerical solution therefore need to be developed. Here we show that existing Runge–Kutta methods for ordinary differential equations (odes) can be modified to solve stochastic wave equations provided that appropriate changes are made to the way stepsizes are selected. The order of the resulting stochastic differential equation (sde) scheme is half the order of the ode scheme. Specifically, we show that an explicit 9th order Runge–Kutta method (with an embedded 8th order method) for odes yields an order 4.5 method for sdes which can be implemented with variable stepsizes. This method is tested by solving systems of equations originating from master equations and from the many-body Schrödinger equation.