For calculating the transition amplitude, especially, the density matrix, important for the quantum mechanical chemical reaction dynamics, the stochastic path-integral method (SPIM) was developed and then realized by solving the stochastic partial differential equation (SPDE) in use of a corresponding set of difference equations (M. Nagaoka, Int. J. Quantum Chem. 30, 91(1996)). We discritize the SPDE spatially, then we obtain a system of stochastic differential equations. Three kind of numerical schemes, i.e. the Euler-Maruyama, Heun and explicit order 1.5 strong schemes, whose strong orders of convergence are 0.5, 1.0 and 1.5, respectively, are examined by analyzing the averaged coordinate, the averaged coordinate correlation function and the density matrix. Comparing the latter with the analytical result obtained for a typical quantum harmonic system at temperature 100K, it was curiously found that the Heun scheme brings about rather worse transition amplitude than the other two schemes in spite of its strong order of convergence 1.0.

MINISIMPOSIUM SESSION: 7. Numerical methods for Stochastic equations