Particle method for turbulent flows: Integration of stochastic model equations
(From author's summary.) A numerical method is developed to integrate the stochastic differential equations that arise in a particle method for modelling turbulent flows. These equations present several challenges, the foremost being the presence of multiple time scales, the smallest of which can be significantly less than an acceptable time-step size, \(\Delta t\). The essence of the approach adopted is to transform and decompose the equations so that the stochastic components (which contain the small time scales) appear as strictly linear stochastic differential equations. Analytic solutions to these equations (with frozen coefficients) are then exploited to produce a stable and accurate scheme. When the method is used to advance the properties of \(N\) particles, the resulting numerical error can be decomposed into three contributions: statistical error, bias, and time-stepping error. Comprehensive tests to study these errors are reported for two test cases. A novel variance- reduction technique is described that significantly reduces the statistical error, which scales as \(N^{- 1/2}\). In general, the bias is smaller, and scales as \(N^{-1}\) (in accord with a simple analysis). The time-stepping error is less than 1\% for a non-dimensional time step of 1/32 which may be several times larger than the smallest time scale. Over the range of time-step size investigated, the dominant time-stepping error varies as \(\Delta t^{3/2}\). The method has the requisite stability, accuracy, and efficiency for incorporation in multidimensional particle methods.
linear stochastic differential equations
scientific article; zbMATH DE number 758607