Estimating the statistical error in calculating velocity and temperature components by the direct simulation Monte Carlo method

The direct simulation Monte Carlo (DSMC) method is widely used to solve problems in rarefied gas dynamics. In solving steady state problems, a special feature of the method is in using dependent sample values of random variables to calculate the macroparameters of gas flow. In this paper, the possibilities of methods of statistical physics to estimate the statistical error of the DSMC method are theoretically analyzed. A simple approach to approximate estimation of the statistical error in calculating temperature and velocity components is proposed. The approach is tested on a number of problems.     

Numerical estimation of heterogeneous reaction constants in the flow of rarefied gas through a cylindrical channel

The Direct Simulation Monte Carlo method is used to study the influence of the coefficients of heterogeneous dissociation and recombination reactions on the rarefied gas flow through a cylindrical channel. It is established that the degree of dissociation of the flow coming out of the channel is significantly dependent on the relationship between the dissociation and recombination coefficients. The technique for determining the dissociation and recombination coefficients on the basis of the experimental data is proposed.     

Error Estimation and Optimization of the Functional Algorithms of a Random Walk on a Grid Which Are Applied to Solving the Dirichlet Problem for the Helmholtz Equation

We consider the algorithms of a random walk on a grid which are applied to global solution of the Dirichlet problem for the Helmholtz equation (the direct and conjugate methods). In the metric space C we construct some upper error bounds and obtain optimal values (in the sense of the error bound) of the parameters of the algorithms (the number of nodes and the sample size).     

Estimation of the statistical error of the direct simulation Monte Carlo method

The statistical error of the direct simulation Monte Carlo method for numerical solution of the rarefied gas dynamics problems is investigated. Based on the central limit theorem for Markov processes, asymptotic confidence intervals for the errors connected with the number of time steps are obtained for estimates of the three main macroparameters of the flow (density, velocity, and temperature). For the quantities involved in the expressions for the confidence intervals, practical recommendations are given concerning their numerical evaluation simultaneously with the calculation of the flow macroparameters. The proposed approaches to constructing the confidence intervals are illustrated using the classical problem of heat transfer between two infinite parallel plates as an example.