Numerical grid equidistant

In the present contribution, particles (i.e. primary particles as well as agglomerates) are fully resolved by the numerical grid whereby details of the flow around the particles are captured and the forces on the particles follow from the boundary condition of the LBM. These forces are used to update the particle position within the stationary equidistant grid. This approach also imphes that the presence of particles and agglomerates affect the flow field and the fluid dynamic interaction between the particles (i.e. primary particles and agglomerates) is captured automatically. 

The discrete Fourier transform can also be used for differentiating a function, and this is used in the spectral method for solving differential equations [Gottlieb, D., and S. A. Orszag, Numerical Analysis of Spectral Methods Theory and Applications, SIAM, Philadelphia (1977) Trefethen, L. N., Spectral Methods in Matlab, SIAM, Philadelphia (2000)]. Suppose we have a grid of equidistant points... 

The numerical solution of the initial-boundary-value problem based on the equation system (44) can be performed (Winkler et al, 1995) by applying a finite-difference method to an equidistant grid in energy U and time t. The discrete form of the equation system (44) is obtained using, on the rectangular grid, second-order-correct centered difference analogues for both distributions f iU, i)/n and f U, t)/n and their partial derivatives of first order. 

Finite-difference methods operating on a grid consisting of equidistant points ( Xi, Xi = ih + Xq) are known to be one of the most accurate techniques available [496]. Additionally, on an equidistant grid all discretized operators appear in a simple form. The uniform step size h allows us to use the Richardson extrapolation method [494,497] for the control of the numerical truncation error. Many methods are available for the discretization of differential equations on equidistant grids and for the integration (quadrature) of functions needed for the calculation of expectation values. 

In the numerical analysis the radial and axial derivatives of the parabolic equations are replaced by the central difference approximations and the backward difference approximations, respectively. Thus N-1 sets of finite parabolic difference equations are obtained, N being the number of radial steps. The number of equidistant grid points in radial direction amounted to 40, while for the finite difference increment of ( 2z/Re) a value of 1.25 10 was used. Details of the numerical solution procedure are given elswhe-re (J 2). Figure I is a graphical representation of the development of the axial velocity obtained by the numerical solution procedure. This result agrees quite well with that obtained by Vrentas et.al. 

As the radial concentration gradient at the tube wall determines the mass flux supplied by diffusion a good approximation of this gradient is necessary to obtain an accurate description of the concentration profiles in the tube. In the numerical analysis this is effected by reduction of the radial step width near the tube wall. During the calculations it turned out, however, that a non equidistant radial grid resulted in persistent numerical instabilities. 

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