Dispersion numerical

J. Song. A New Flux Correcting Method for Reducing Numerical Dispersion—Application to FOR (Enhanced Oil Recovery) By Chemical Processes (Reduction de la Dispersion Numerique par Correction desflux Massiques—Application au Probleme de la Recuperation d Hydrocarburespar Procedes Chimiques). PhD thesis, Paris VI Univ, 1992. 

Truncation error arises from approximating each of the various space and time derivatives in the transport equation. The error resulting from the derivative in the advection term is especially notable and has its own name. It is known as numerical dispersion because it manifests itself in the calculation results in the same way as hydrodynamic dispersion. 

To see why numerical dispersion arises, consider solute passing into a nodal block, across its upstream face. Over a time step, the solute might traverse only a fraction of the block s length. In the numerical solution, however, solute is distributed evenly within the block. At the end of the time step, some of it has in effect flowed across the entire nodal block and is in position to be carried into the next block downstream, in the subsequent time step. In this way, the numerical procedure advances some of the solute relative to the mean groundwater flow, much as hydrodynamic dispersion does. 

For the transport of a non-reacting solute in one dimension, a coefficient of numerical dispersion can be defined,... 

From this result, we can be assured a solution reflects hydrodynamic rather than numerical dispersion - that is, D > Dnum - wherever Pe < 2. 

Numerical dispersion can be minimized in several ways. The nodal block spacing Ax can be set small by dividing the domain into as many blocks as practical. The value specified for Dl can be reduced to account for the anticipated numerical dispersion. And a time step can be chosen to give a grid Courant number Co as close to one as allowed by the stability criterion (Eqn. 20.32). 

We consider a 100-m length of an aquifer with a porosity of 30% and a nominal dispersivity of 10 cm the dispersivity reflected in the calculation results will be somewhat larger than this value, due to the effects of numerical dispersion. The domain is divided into 100 nodal blocks, each 1 m long. We assume local equilibrium, so time enters into the calculation only as a measure of the cumulative volume of fluid that has passed through the aquifer. Specifying the aquifer s pore volume be replaced 30 times over the course of the simulation, and setting the time span to 30 years, each year in the simulation corresponds to a single replacement of the aquifer s pore fluid. 

Numerous disperse dyes are marketed in a metastable crystalline form that gives significantly higher uptake than the corresponding more stable modification. The molar free enthalpy difference can be used as a criterion of the relative thermodynamic stabilities of two different modifications [53]. Certain dyes can be isolated in several different morphological forms. For example, an azopyrazole yellow disperse dye (3.52) was prepared in five different crystal forms and applied to cellulose acetate fibres. Each form exhibited a different saturation limit, the less stable modifications giving the higher values [54]. 

We could operate our computational transport model with only numerical dispersion (i.e., as a tanks-in-series model). It is often inconvenient to do so in environmental transport applications, however, because the cross-sectional mean velocity, U, can... 

We perform nonlinear fitting using the Levenberg-Marquardt method implemented in the MRQMIN routine [75], From the experimental end, eight families of data are involved, namely, x (T) and x jT) at four frequencies, taken from Ref. 64. From the theory end, we employ formulas (4.121)-(4.124) with the numerical dispersion factors. The results of fitting are presented in Figures 4.6 and 4.7 and Table I. 

Numerous dispersion models have been documented over the last two decades. These models can be divided into four general classes ... 

A final remark in this Section concerns axially symmetric problems. We usually treat these in radial coordinates and apply a numerical Hankel transform instead of the Fourier transform. This is a slow transform with a dense matrix, but due to the relatively small computational domain radially symmetric problems require, this is not a big problem. Alternatively, one could treat such situations by finite differencing in the radial dimension, but it would mean accepting additional (paraxial) approximation, and would introduce artificial numerical dispersion into the algorithm. 

At metallographic research of structure melted of sites 2 mechanisms of education of spherical particles of free carbon are revealed. In one of them, sold directly at the deformed graphite the formed particles became covered by a film austenite, that testifies to development abnormal eutectic crystallization. In other sites containing less of carbons and cooled less intensively, eutectic crystallization the education numerous dispersed dendrites austenite preceded. Crystallization of thin layers smelt, placed between branches austenite, occured to complete division of phases, that on an example of other materials was analyzed in job [5], Thus eutectic austenite strated on dendrites superfluous austenite, and the spherical inclusions of free carbon grew in smelt in absence austenite of an environment. Because of high-density graphite-similar precipitates in interdendritic sites the pig-iron is characterized by low mechanical properties. 

Coupled methods (transport model coupled with hydrogeochemical code) For coupled models solving the transport equation can be done by means of the finite-difference method (and finite volumes) and of the finite-elements method. Algorithms based on the principle of particle tracking (or random walk), as for instance the method of characteristics (MOC), have the advantage of not being prone to numerical dispersion (see 1.3.3.4.1). 

Both the spatial discretisation and the choice of the type of differences (e.g. uplift differences, central differences) have a strong influence on the result. This fuzziness caused by the application of different methods is subsumed as numeric dispersion . 

Numeric dispersion can be eliminated largely by a high-resolution discretisation. The Grid-Peclet number helps for the definition of the cell size. Pinder and Gray (1977) recommend the Pe to be < 2. The high resolution discretisation, however, leads to extremely long computing times. Additionally the stability of the numeric finite-differences method is influenced by the discretisation of time. The Courant number (Eq. 104) is a criterion, so that the transport of a particle is calculated within at least one time interval per cell. 

Methods applying reverse differences in time are called implicit. Generally these implicit methods, as e.g. the Crank-Nicholson method, show high numerical stability. On the other side, there are explicit methods, and the methods of iterative solution algorithms. Besides the strong attenuation (numeric dispersion) there is another problem with the finite differences method, and that is the oscillation. 

With the finite-elements method the discretisation is more flexible, although, as with the finite-differences method, numeric dispersion and oscillation effects can... 

Fig. 26 Numeric dispersion and oscillation effects for the numeric solution of the transport equation (after Kovarik, 2000)...

In physics, the random walk method has already been in use for decades to understand and model diffusion processes. Prickett et al. (1981) developed a simple model for groundwater transport to calculate the migration of contamination. An essential advantage of the methods of random walk and particle tracking is that they are free of numeric dispersion and oszillations (Abbot 1966). 

Numerical dispersion or oscillation effects can occur as accidental source of error when using finite differences and finite element methods while modeling mass transport. Utilizing the criteria of numerical stability (Grid-Peclet number or Courant number) or the random walk procedure, these errors can be either reduced or even eliminated. 

In the next example the PHREEQC job is presented that simulates the experiment. To adjust the model to the data observed, the exchange capacity (X under EXCHANGE, here 0.0015 mol per kg water), the selectivity coefficients in the data set WATEQ4F.dat and the chosen dispersivity (TRANSPORT, dispersivity, here 0.1 m) are decisive besides the spatial discretisation (number of cells, here 40). If one sets the dispersivity to a very small value (e.g. T10"6) in the input file Exchange and rerun the job, one will see that no numerical dispersion occurs showing that numerical stability criteria are maintained properly. 

For the solution of sophisticated mathematical models of adsorption cycles including complex multicomponent equilibrium and rate expressions, two numerical methods are popular. These are finite difference methods and orthogonal collocation. The former vary in the manner in which distance variables are discretized, ranging from simple backward difference stage models (akin to the plate theory of chromatography) to more involved schemes exhibiting little numerical dispersion. Collocation methods are often thought to be faster computationally, but oscillations in the polynomial trial function can be a problem. The choice of best method is often the preference of the user. 

Second, we know that numerical errors (such as a grid that is too coarse in a finite difference computation) can mimic some of the effects of dispersion. It may be that such "numerical dispersion" is responsible for unexpectedly stable displacements in some cases. This may be supported by the experience of Settari et al. (50) and of Russell (51), who found that their displacements were stable, unless the effects of dispersion were sufficiently small. 

A simple way to solve Eqs. (7.22)-(7.29) is with the use of the finite-difference algorithm (Eq. (7.30)), assuming that the right-hand side of Eq. (7.22) is zero, and the numerical dispersion of the finite-difference algorithm gives the plate count for that component. 

An example of the. .errors" introduced by the approximation is the effect of. .numerical dispersion", which leads to an additional artificial band broadening. 

Second Method Replacement of Axial Dispersion by Numerical Dispersion. . . 497... 

In this first method, we can write a large number of possible combinations of terms. The number of useful combinations is much lower because of our requirements that (1) the solution should be stable and (2) the error term should be of the second order, 0(h - - - T ), so that it is small enough and can be neglected. With a calculation scheme that gives a first-order error term, a second-order partial differential term equivalent to a numerical dispersion term would appear. The contribution of this term to the band profile could be difficult to control or cancel. 

The error analysis of this calculation procedure can be done using the equations in the previous section. It shows that the error made in using this scheme is of the order of 0(h + t). Thus, the scheme introduces an error term equivalent to a second-order partial differential term, which would add up to the RHS of Eq. 10.61, t.e., would decrease the apparent column efficiency. This procedure should not be used, unless very small values of the time increment t are selected. This, in turn, would make the computation time very long. In order to overcome this type of problem. Lax and Wendroff have suggested the addition to the axial dispersion term of an extra term, equivalent to the numerical dispersion term but of opposite sign [51]. This term compensates the first-order error contribution. In linear chromatography, the new finite difference equation, or Lax-Wendroff scheme, can be written as follows ... 

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