Finite difference method dispersion

Interestingly, if one Taylor series expands Eq. (36) and equates the terms of the same order in kj with Eq. (37) one can derive the standard Lagrangian FD approximations (i.e., require the coefficient of kj to be —1, and require the coefficient of all other orders in kj up to the desired order of approximation to be 0.) A more global approach is to attempt to fit Eq. (36) to Eq. (37) over some range of Kj = kjA values that leads to a maximum absolute error between Eq. (36) and Eq. (37) less than or equal to some prespecrfied value, E. This is the essential idea of the dispersion-fitted finite difference method [25]. 

The basic principles are described in many textbooks [24, 26]. They are thus only sketchily presented here. In a conventional classical molecular dynamics calculation, a system of particles is placed within a cell of fixed volume, most frequently cubic in size. A set of velocities is also assigned, usually drawn from a Maxwell-Boltzmann distribution appropriate to the temperature of interest and selected in a way so as to make the net linear momentum zero. The subsequent trajectories of the particles are then calculated using the Newton equations of motion. Employing the finite difference method, this set of differential equations is transformed into a set of algebraic equations, which are solved by computer. The particles are assumed to interact through some prescribed force law. The dispersion, dipole-dipole, and polarization forces are typically included whenever possible, they are taken from the literature. 

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). 

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... 

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. 

Gray. S.K. and Goldfield. E.M. (2001) Dispersion fitted finite difference method with applications to molecular quantum mechanics J. Chem. Phys. 115, 8331-8344. 

As we have shown above (Eqs. 10.99 to 10.101), the finite difference methods im-derestimate or overestimate the extent to which axial dispersion affects the profiles of single-component bands. There are cases in which we need a more exact profile that the one calculated with the finite difference schemes. This is of special importance in the case of multicomponent bands (Chapter 11). As we show in the next chapter, the use of finite difference methods causes more important errors in the calculation of the individual band profiles of the components of mixtures than in the calculation of the profiles of single component bands. A more accurate method may be necessary to account for the elution profiles of multicomponent mixtures, especially when the column efficiency is modest [59]. 

Chapter 10, which provides satisfactory accuracy and is the simplest and fastest calculation procedure. This method consists of neglecting the second-order term (RHS of Eq. 11.7) and calculating numerical solutions of the ideal model, using the numerical dispersion (which is equivalent to the introduction in Eq. 11.7 of a first-order error term) to replace the neglected axial dispersion term. Since we know that any finite difference method will result in truncation errors, the most effective procedure is to control them and to use them to simplify the calculation. The results obtained are excellent, as demonstrated by the agreement between experimental band profiles recorded with single-component samples and profiles calculated [2-7]. Thus, it appears reasonable to use the same method in the calculation of solutions of multicomponent problems. However, in the multicomponent case a new source of errors appears, besides the errors discussed in detail in Chapter 10 (Section 10.3.5). 

Thus, in the calculation of the individual band profiles in the case of multicomponent mixtures, there is a third source of errors, besides the two classical error sources observed with finite difference methods, which we have discussed in the study of the single-component problem. Obviously, these two sources are also found in the calculations of solutions of the multicomponent problems. As can be seen from Eqs. 11.17 to 11.19, this new error increases with the difference between the retention factors of the two components, and it decreases with decreasing Courant number. The error would disappear with the second and third schemes (Eqs. 11.18 and 11.19) and the numerical dispersion for the two solutes would become equal and correspond to the proper value of H if a was dose enough to 0. This observation is important because, for these two schemes (Eqs. 10.87 and 10.88), we can always select low values of the Courant raunber if needed by combining a large space increment h and a suitably small time increment t (Eq. 11.10). 

Figure 11.26 compares the individual elution profiles calculated using the same two finite difference methods, the forward-backward and the backward-forward schemes, for 2-phenylethanol and 3-phenyl-l-propanol [22]. The bands obtained with the backward-forward scheme are taller and narrower and their front sharper than those derived from the forward-backward scheme, in agreement with the result derived in Chapter 10 regarding the amotmt of dispersion introduced by the different numerical methods. The agreement between calculated and experimental results is slightly better with the backward-forward scheme than with the other one. Note, however, that the forward-backward calculations are carried out with 3 = 2, while the backward-forward calculations are done with a - 0.5. 

Numerical solutions of the former [50,51] and latter [52] models have been calculated with finite difference methods. Any of the schemes discussed in Chapter 10 can be used for the calculation of solutions of these kinetic models. However, the generation of the error due to the numerical dispersion is more complex and it is more difficult to control, because two differential equations are now involved for each component, instead of one with the equilibrium-dispersive model. 

In the case of a step input, the numerical solution of the system of Eqs. 16.30 and 16.31 has been discussed in the literature for multicomponent mixtures [16]. The numerical solution of Eqs. 16.30 and 16.31 without an axial dispersion term i.e., with Di = 0) has been described by Wang and Tien [17] and by Moon and Lee [18], in the case of a step input. These authors used a finite difference method. A solution of Eq. 16.31 with D, = 0, combined with a liquid film linear driving force model, has also been described for a step input [19,20]. The numerical solution of the same kinetic model (Eqs. 16.30 and 16.31) has been discussed by Phillips et al. [21] in the case of displacement chromatography, using a finite difference method, and by Golshan-Sliirazi et al. [22,23] in the case of overloaded elution and displacement, also using finite difference methods. 

Agarwal and Jayaraman Spectral method with finite difference method 100 < Aoe Asc < Iff For the range of A e Asc from Iffff to Iff the axial dispersion in a circular curved tube is markedly less than that in a straight tube... 

K. Asami, Dielectric dispersion in biological cells of complex geometry simulated by the three-dimensional finite difference method. J. Phys. D Appl. Phys., 39, 492-499 (2006). 

For the dynamic simulation of the SMB-SFC process a plug-flow model with axial dispersion and linear mass-transfer resistance was used. The solution of the resulting mass-balance equations was performed with a finite difference method first developed by Rouchon et al. [69] and adapted to the conditions of the SMB process by Kniep et al. [70]. The pressure drop in the columns is calculated with the Darcy equation. The equation of state from Span and Wagner [60] is used to calculate the mobile phase density. The density of the mobile phase is considered variable. 

The equations of motion (10.2) and (10.3) include the dispersion terms in the right-hand side. These dispersion terms cause numerical difficulties in practice because of the mixed form of differentiations with respect to both time and space. Consequently, it calls for the use of implicit scheme, which solves a matrix system. The implicit scheme requires very hne grids to reduce the numerical dispersion errors inherent in the numerical scheme such as finite-difference method. A fine grid... 

A consideration of axial dispersion is essential in any realistic description of extraction column behaviour. Here a dynamic method of solution is demonstrated, based on a finite differencing of the column height coordinate. Figure 1 below shows the extraction column approximated by N finite-difference elements. 

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... 

Pollutants emitted by various sources entered an air parcel moving with the wind in the model proposed by Eschenroeder and Martinez. Finite-difference solutions to the species-mass-balance equations described the pollutant chemical kinetics and the upward spread through a series of vertical cells. The initial chemical mechanism consisted of 7 species participating in 13 reactions based on sm< -chamber observations. Atmospheric dispersion data from the literature were introduced to provide vertical-diffusion coefficients. Initial validity tests were conducted for a static air mass over central Los Angeles on October 23, 1968, and during an episode late in 1%8 while a special mobile laboratory was set up by Scott Research Laboratories. Curves were plotted to illustrate sensitivity to rate and emission values, and the feasibility of this prediction technique was demonstrated. Some problems of the future were ultimately identified by this work, and the method developed has been applied to several environmental impact studies (see, for example, Wayne et al. ). 

To understand the behavior of the movement of the contaminant in ground-water, people solve Eq. (1) forward in time. In solving this equation forward in time, one assumes that the plume is originated from somewhere and will travel through the porous media due to advection and dispersion. The conventional procedure to solve Eq. (1) is to use finite difference or finite element methods. For simple cases, closed-form solutions exist. Quantitative descriptions of the processes forward in time are well understood. Multidimensional models of these processes have been used successfully in practice [50]. Numerical solute transport models were first developed about 25 years ago. When properly applied, these models can provide useful information about transport processes and can assist in the design of remedial programs. 

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. 

The numerical solution to the advection-dispersion equation and associated adsorption equations can be performed using finite difference schemes, either in their implicit and/or explicit form. In the one-dimensional MRTM model (Selim et al., 1990), the Crank-Nicholson algorithm was applied to solve the governing equations of the chemical transport and retention in soils. The web-based simulation system for the one-dimensional MRTM model is detailed in Zeng et al. (2002). The alternating direction-implicit (ADI) method is used here to solve the three-dimensional models. 

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