Before revision:  Further, we introduce a new algorithm in this setting, the weak trapezoidal algorithm, which has been studied previously only in the diffusive setting , and prove that it is second order accurate in the size of the time discretization, making it the first of its kind. In the multi-scale setting it is typically an extremely difficult task to perform approximations, such as Langevin approximations or law of large number type arguments, to simplify or reduce a system. Therefore, numerical methods oftentimes are the only reasonable means by which such models can be understood in real time .
After revision:  Further, we introduce a new algorithm in this setting, the weak trapezoidal algorithm, which has been studied previously only in the diffusive setting , and prove that it is second order accurate in the size of the time discretization, making it the first second order method in this setting .