ROBUST ALGORITHMS FOR SOLVING STOCHASTIC PARTIAL-DIFFERENTIAL EQUATIONS

A robust semi-implicit central partial difference algorithm for the nu merical solution of coupled stochastic parabolic partial differential equations (PDEs) is described. This can be used for calculating correl ation functions of systems of interacting stochastic fields. Such fiel d equations can arise in the description of Hamiltonian and open syste ms in the physics of nonlinear processes, and may include multiplicati ve noise sources. The algorithm can be used for studying the propertie s of nonlinear quantum or classical field theories. The general approa ch is outlined and applied to a specific example, namely the quantum s tatistical fluctuations of ultra-short optical pulses in chi((2)) para metric waveguides. This example uses a non-diagonal coherent state rep resentation, and correctly predicts the sub-shot noise level spectral fluctuations observed in homodyne detection measurements. It is expect ed that the methods used wilt be applicable for higher-order correlati on functions and other physical problems as well. A stochastic differe ncing technique for reducing sampling errors is also introduced. This involves solving nonlinear stochastic parabolic PDEs in combination wi th a reference process, which uses the Wigner representation in the ex ample presented here. A computer implementation on MIMD parallel archi tectures is discussed. (C) 1997 Academic Press.