Flux-corrected method

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. 

A variety of other methods are used to obtain a good solution without using extremely fine meshes. The flux correction methods keep track of the flux or material into and out of a cell (from one node to another) and put limits on the flux to make sure that no more material... 

An additional consideration when using in situ techniques is the volume of the luminal drug solution as water absorption and secretion during the perfusion may introduce errors in the lumenal concentrations and therefore in the calculated absorption. Various water flux correction methods have been published [42] including the co-perfusion of a non-absorbed marker such... 

The concentration profile is steeper for the MacCormack method than for the upstream derivatives, but oscillations can still be present. The flux-corrected transport method can be added to the MacCormack method. A solution is obtained both with the upstream algorithm and the MacCormack method and then they are combinea to add just enough diffusion to ehminate the oscillations without smoothing the solution too much. The algorithm is comphcated and lengthy but well worth the effort (Refs. 37, 107, and 270). 

A drawback of the Lagrangean artificial-viscosity method is that, if sufficient artificial viscosity is added to produce an oscillation-free distribution, the solution becomes fairly inaccurate because wave amplitudes are damped, and sharp discontinuities are smeared over an increasing number of grid points during computation. To overcome these deficiencies a variety of new methods have been developed since 1970. Flux-corrected transport (FCT) is a popular exponent in this area of development in computational fluid dynamics. FCT is generally applicable to finite difference schemes to solve continuity equations, and, according to Boris and Book (1976), its principles may be represented as follows. 

Boris, J. P., and Book D. L. 1976. Solution of continuity equations by the method of Flux-Corrected Transport. Meth. Computat. Phys. Vol. 16. New York Academic Press. 

Guirao and Bach (1979) used the flux-corrected transport method (a finite-difference method) to calculate blast from fuel-air explosions (see also Chapter 4). Three of their calculations were of a volumetric explosion, that is, an explosion in which the unbumed fuel-air mixture is instantaneously transformed into combustion gases. By this route, they obtained spheres whose pressure ratios (identical with temperature ratios) were 8.3 to 17.2, and whose ratios of specific heats were 1.136 to 1.26. Their calculations of shock overpressure compare well with those of Baker et al. (1975). In addition, they calculated the work done by the expanding contact surface between combustion products and their surroundings. They found that only 27% to 37% of the combustion energy was translated into work. 

Boris, J. P., and D. L. Book. 1973. Flux-corrected transport I SHASTA-A fluid transport method that works. J. Comp. Phys. 11 38. 

Two approaches have been suggested for calculating the finite domain correction the multiple reflection method [58] and the flux balance method [59]. Both give essentially the same result. Suppose to be the finite domain... 

The parameter 3 in this expression accounts for the nonlinearity of the composition profiles it is, in fact, a high flux correction factor as is [3] in the matrix methods described above. Equation 8.5.18 involves no iteration because the rate factor <1> and the correction factor 3 can be calculated from Eqs. 8.5.15 all we need to know are the boundary conditions y Q and y Q. 

The deferred correction method can also be employed to improve the convergence properties of high order schemes, rewriting a high order flux approximation as the sum of a stable low order flux and a higher order correction in the following manner ... 

One such approach is based on the concepts of non-linear flux limiters introduced by van Leer [193] and Boris and Book [13]. The work of Boris and Book [13] and Zalesak [213] determine the basis for a group of methods called flux correction transport (FCT) schemes. The schemes of Smolarkiewicz [175] is representative for this group. In the FCT schemes a first order accurate monotone scheme is converted to a high resolution scheme by adding limited amounts of an anti-diffusive flux. The work of van Leer [193, 195], on the other hand, represents an extension of the ideas of Gudunov [64] to higher order accuracy. 

The second term in the mass flux correction relation (12.175), is due to the compressibility effects. It involves the correction to the density at the grid cell volume face. If the SIMPLE method is to be extended to compressible flows, we must thus express the density correction in terms of the pressure correction to enable an efficient update of the density corrections in the pressure correction equation. 

All the problems encountered during application of the algorithms presented above illustrate the difficulty of solution of the advection problem in atmospheric transport models. A number of techniques have been developed to treat advection accurately, including flux-corrected transport (FCT) algorithms (Boris and Book, 1973), spectral and finite element methods [for reviews, see Oran and Boris (1987), Rood (1987), and Dabdub and Seinfeld (1994). Bott (1989, 1992), Prather (1986), Yamartino (1992), Park and Liggett (1991), and others have developed schemes specifically for atmospheric transport models. 

Sounart, T.L. and Baygents, J.C., Simulation of electrophoretic separations by the flux-corrected transport method, J. Chromatogr. A, 890, 321, 2000. 

The symbols are p, gas mass density u, gas velocity p, gas pressure e, gas internal energy T, gas temperature Cf, friction coefficient Ch, heat transfer coefficient (Stanton number) Ty, wall temperature Cp, specific heat at constant pressure and w, channel radius or width. These equations were solved by the explicit Flux-Corrected Transport finite-difference method described by Oran and Boris.Computations were carried out on a coarse 100-point uniform mesh when the shock was in the vessel interior and on a 250-point variable mesh when the shock was near the walls or the center of symmetry. The variable mesh was used to increase the spatial resolution by a factor of 10 near the shock reflection or implosion point. 

The density here refers to the spatial coordinate, i.e. the concentration of the reaction product, and is not to be confused with the D(vx,vy,vz) in previous sections which refers to the center-of-mass velocity space. Laser spectroscopic detection methods in general measure the number of product particles within the detection volume rather than a flux, which is proportional to the reaction rate, emerging from it. Thus, products recoiling at low laboratory velocities will be detected more efficiently than those with higher velocities. The correction for this laboratory velocity-dependent detection efficiency is called a density-to-flux transformation.40 It is a 3D space- and time-resolved problem and is usually treated by a Monte Carlo simulation.41,42... 

Astronomical Observatory, were used to carry out the calculations of theoretical equivalent widths of lines, synthetic spectra and a set of plane parallel, line-blanketed, flux constant LTE model atmospheres. The effective temperatures of the stars were determined from photometry, the infrared flux method and corrected, if needed, in order to achieve the LTE excitation balance in the iron abundance results. The gravities were found by forcing Fe I and Fe II to yield the same iron abundances. The microturbulent velocities were determined by forcing Fe I line abundances to be independent of the equivalent width. For more details on the method of analysis and atomic data see Tautvaisiene et al. (2001). 

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