Numerical integration of stochastic differential equations: weak second-order mid-point scheme for application in the composition PDF method

In this paper, a weak second-order accurate mid-point scheme for the stochastic differential equations (SDEs) arising in the composition PDF method for turbulent reactive flows is proposed and tested. The results are compared with two other schemes which are commonly used in the composition PDF method. In contrast to most higher-order schemes for SDEs, the present scheme uses a mid-point, which makes it especially suitable for the implementation of the position-advance fractional step in the composition PDF method. The scheme can also be applied to the PDF method used in conjunction with large eddy simulation (LES), since the SDEs considered in this paper include explicit time dependence of the drift and diffusion coefficients. Test calculations are reported, including a 2D unsteady case, to demonstrate the weak second-order accuracy of the scheme and to compare its performance to that of two first-order schemes. A new methodology for developing higher-order scheme for SDEs (by comparing the moments of the increments of the numerical approximation with those of the exact increments) is used to develop this scheme.