Diffusion, numerical

Fig. 7. Comparison of various transport schemes for advecting a cone-shaped puff in a rotating windfield after one complete rotation (a), the exact solution (b), obtained by an accurate numerical technique (c), the effect of numerical diffusion where the peak height of the cone has been severely tmncated and (d), where the predicted concentration field is very bumpy, showing the effects of artificial dispersion. In the case of (d), spurious waves are...

The effect of using upstream derivatives is to add artificial or numerical diffusion to the model. This can be ascertained by rearranging the finite difference form of the convective diffusion equation... 

Approaches used to model ozone formation include box, gradient transfer, and trajectoty methods. Another method, the particle-in-cell method, advects centers of mass (that have a specific mass assigned) with an effective velocity that includes both transport and dispersion over each time step. Chemistry is calculated using the total mass within each grid cell at the end of each time step. This method has the advantage of avoiding both the numerical diffusion of some gradient transfer methods and the distortion due to wind shear of some trajectory methods. 

The term numerical diffusion describes the effect of artificial diffusive fluxes which are induced by discretization errors. This effect becomes visible when the transport of quantities with small diffusivities [with the exact meaning of small yet to be specified in Eq. (42)] is considered. In macroscopic systems such small diffusivities are rarely found, at least when being looked at from a phenomenological point of view. The reason for the reduced importance of numerical diffusion in many macroscopic systems lies in the turbulent nature of most macro flows. The turbulent velocity fluctuations induce an effective diffusivity of comparatively large magnitude which includes transport effects due to turbulent eddies [1]. The effective diffusivity often dominates the numerical diffusivity. In contrast, micro flows are often laminar, and especially for liquid flows numerical diffusion can become the major effect limiting the accuracy of the model predictions. 

The second term on the right-hand side is the numerical diffusivity, depending on the flow velocity in the x-direction, u, and the length of the control volume in the %-direction, h. Similar expressions hold for the other coordinate directions. Apparently, the numerical diffusion increases with increasing flow velocity and... 

In order to minimize numerical diffusion, Boris and Book [131] formulated the idea of blending a low-order stable differencing scheme with a higher order, potentially unstable, scheme in such a way that steep concentration gradients are maintained as well as possible. The algorithm they proposed consists of the following steps ... 

Two numerical methods have been used for the solution of the spray equation. In the first method, i.e., the full spray equation method 543 544 the full distribution function / is found approximately by subdividing the domain of coordinates accessible to the droplets, including their physical positions, velocities, sizes, and temperatures, into computational cells and keeping a value of / in each cell. The computational cells are fixed in time as in an Eulerian fluid dynamics calculation, and derivatives off are approximated by taking finite differences of the cell values. This approach suffersfrom two principal drawbacks (a) large numerical diffusion and dispersion... 

As described above, spatial transport in an Eulerian PDF code is simulated by random jumps of notional particles between grid cells. Even in the simplest case of one-dimensional purely convective flow with equal-sized grids, so-called numerical diffusion will be present. In order to show that this is the case, we can use the analysis presented in Mobus et al. (2001), simplified to one-dimensional flow in the domain [0, L (Mobus et al. 1999). Let X(rnAt) denote the random location of a notional particle at time step m. Since the location of the particle is discrete, we can denote it by a random integer i X(mAt) = iAx, where the grid spacing is related to the number of grid cells (M) by Ax = L/M. For purely convective flow, the time step is related to the mean velocity (U) by16... 

Numerical diffusion for the purely convective case is measured by the variance of the particle location. In the absence of numerical diffusion, X(t) = (X(t), and the variance is zero. From (7.20), the location variance can be computed as... 

At t = Tpfr = L/(U) (which corresponds to the convective front reaching the end of the domain), the relative magnitude of the numerical diffusion will depend only on a and M ... 

For this case (Ax constant), it would be possible to eliminate numerical diffusion by setting a = I (Roekaerts 1991). However, in more general cases, the value of a < 1 will be controlled by the smallest characteristic flow time in (7.13), and thus numerical diffusion cannot be eliminated in an Eulerian PDF code. 

Spatial transport is limited to first-order, up-wind schemes, and is thus strongly affected by numerical diffusion. 

While some of the disadvantages listed above can be overcome by modifying the algorithm, the problem of numerical diffusion remains as the principal shortcoming of all Eulerian PDF codes. 

However, use of the grid-cell kernel induces a deterministic error similar to numerical diffusion due to die piece-wise constant approximation. 

Upwind differences are typical for convective flux, where the upstream concentration is important to determine the convective flux at the upstream interface. Upwind differences have a lower numerical diffusion than central differences... 

The breakthrough curve for different values of the Courant number is given in Figure E7.3.1. A lower Courant number, less than 1, adds more numerical diffusion to the solution. If the Courant number is greater than 1, the solution is unstable. This Cou > 1 solution is not shown in Figure E7.3.1 because it dwarfs the actual solution. Thus, for a purely convective problem, the Courant number needs to be close to 1, but not greater than 1, for an accurate solution. In addition to the value of the Courant number, the amount of numerical diffusion depends on the value of the term UAz, which is the topic discussed next. 

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