Euler-Maruyama

Denoting r = A /2, the update scheme (7.38) becomes the Euler-Maruyama scheme... 

The benefit of (7.40) over the Euler-Maruyama scheme is that we can get second order accuracy in averages with only a single evaluation of the force at each timestep. However, our analysis presented here is only for the infinite time case. For averages computed in finite time, work in [224] has shown that (7.40) does indeed give a first order error, albeit one that vanishes exponentially fast in time as the simulation progresses. Hence for suitably long simulations, the observed error will be 0(h ). 

Write out the limiting schemes for the OAB] and OBA] methods and verify that they are inconsistent with the dynamics by comparing them to the Euler-Maruyama scheme. 

Now we need to solve the stochastic differential equation given in eq. (3b) using stochastic version of Euler-Maruyama s method using the iterative method as follows [Card, 1998] ... 

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