Interdisciplinary Applied Mathematics

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5    I mkBT    16 Ay2


16d2 V 7Г    45-7Г A2


where d is the molecular diameter. Therefore, if the cell size is equal to the mean free path, the error is on the order of 15%.


2. Finite Time Step: Due to the time splitting of the molecular motion and collisions, the maximum allowable time step should be smaller that the    local    collision    time    Atc.    Values    of    time    steps    larger    than


Atc will result in traveling of molecules through several cells prior to a cell-based (even subcell-based) collision calculation. In (Hadjicon-stantinou, 2000), it was found that the error in transport coefficients is proportional to the square of the time step; e.g., the viscosity has


the form    _


_ 5 jmkBT    16 (vmAt)2


^    16d2 V 7Г    75-7Г A2    ’


where vm = a/2RT is the most probable velocity.


The time-step and cell-size restrictions stated above are not a Courant Friederichs Lewy (CFL) stability restriction of a typical wave equation formulation. The DSMC method is always stable. Violation of the physical restrictions stated in 1 and 2 may result in highly diffused numerical results.


3.    Ratio of the Simulated Molecules to the Real Molecules: Due


to the vast number of molecules in simulations and limited computational resources, one always has to choose a sample of molecules to simulate. If the ratio of the actual to the simulated molecules becomes too high, the statistical scatter of the solution is increased. The details for the statistical error can be found in (Oran et al., 1998; Bird, 1994; Chen and Boyd, 1996).

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