  Stochastic differential equations, especially the one called Langevin
equation, play an important role in many fields of modern science. In this
paper, we use the bicolour rooted tree method, which is based on the stochastic
Taylor expansion, to get the systematic pattern of the high order algorithm for
Langevin equation. We propose a popular test problem, which is related to the
energy relaxation in the double well, to test the validity of our algorithm and
compare our algorithm with other usually used algorithms in simulations. And we
also consider the time-dependent Langevin equation with the Ornstein-Uhlenbeck
noise as our second example to demonstrate the versatility of our method.

Computational Physics