Stochastic B-series and order conditions for exponential integrators

We will discuss B–series for the solution of a stochastic differential equation. Based on this, we will derive an order theory for exponential integrators for such problems. We will then discuss how the order theory for exponential integrators derived can be simplified and adjusted to a semilinear SDE with time-dependent additive noise, and present a mean square order 1.5 method based on these condi- tions. The method has been applied to a semi-discretized diffusion-reaction PDE with bounday noise, and some implementation issues will be discussed.