Advection numerical scheme

Long, P. E., and Pepper, D. W., A comparison of six numerical schemes for calculating the advection of atmospheric pollution, in "Proceedings of the Third Symposium on Atmospheric Turbulence, Diffusion and Air Quality." American Meteorological Societv, Boston, 1976, pp. 181-186. 

Physical parametrizations and numerical schemes (e.g. for advection) are the same No inconsistencies... 

There are many choices for numerical schemes for the advection of tracers. Generally, a compromise between numerical accuracy and computational effort is chosen. Especially, for ecosystem models it is important, that the advection scheme is positive definite. Positive... 

A third variant of the VOF method calculates the interface tension force by the CSS method and perform an independent FLIC reconstruction of the interface to improve the design of the advection schemes. In this way the tailored advection discretization schemes prevent numerical smoothing of the interface [149]. 

An important aspect of Eulerian reactor models is the truncation errors caused by the numerical approximation of the convection/advection terms [82], Very different numerical properties are built into the various schemes proposed for solving these operators. The numerical schemes chosen for a particular problem must be consistent with and reflect the actual physics represented by the model equations. 

The first class is a generalization of the advection-diffusion problem discussed in Section 8.3, and much of the material developed there can be reformulated to develop a realizable scheme for the GPBE. The second class is a generalization of the kinetic equation considered in Section 8.4, and has been referred to in the literature as the semi-kinetic model (Laurent et al, 2004 Laurent Massot, 2001). In the following, we will treat each class separately, although the reader will undoubtedly note many similarities between the numerical schemes. A third class of GPBE, lying between the two listed above, can also be identified wherein the scalar-dependent convection velocity has a parametric form such as... 

The computational approach is based on a colocated, finite-volume, energy-con-serving numerical scheme on unstructured grids [10] and solves the low-Mach number, variable density gas-phase flow equations. Numerical solution of the governing equations of continuum phase and droplet phase are staggered in time to maintain time-centered, second-order advection of the fluid equations. Denoting the time level by a superscript index, the velocities are located at time level f and... 

The CSF and CSS based versions of the VOF method have been used to calculate improved estimates of the single particle drag and lift coefficients and for simulating breakage and coalescence of dispersed flows containing a few fluid particles [20, 53, 54, 150, 232]. A third variant of the VOF method calculates the interface tension force by the CSS method and perform an independent PLIC reconstruction of the interface to improve the design of the advection schemes. In this way the tailored advection discretization schemes prevent numerical smoothing of the interface [160]. 

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...

Errors in advection may completely overshadow diffusion. The amplification of random errors with each succeeding step causes numerical instability (or distortion). Higher-order differencing techniques are used to avoid this instability, but they may result in sharp gradients, which may cause negative concentrations to appear in the computations. Many of the numerical instability (distortion) problems can be overcome with a second-moment scheme (9) which advects the moments of the distributions instead of the pollutants alone. Six numerical techniques were investigated (10), including the second-moment scheme three were found that limited numerical distortion the second-moment, the cubic spline, and the chapeau function. 

The advection—diffusion equation with a source term can be solved by CFD algorithms in general. Patankar provided an excellent introduction to numerical fluid flow and heat transfer. Oran and Boris discussed numerical solutions of diffusion—convection problems with chemical reactions. Since fuel cells feature an aspect ratio of the order of 100, 0(100), the upwind scheme for the flow-field solution is applicable and proves to be very effective. Unstructured meshes are commonly employed in commercial CFD codes. 

Dabdub, D., and J. H. Seinfeld, Numerical Advective Schemes Used in Air Quality Models—Sequential and Parallel Implementation, Atmos. Environ., 28, 3369-3385 (1994b). 

In the first stage of the solution process, the advective control model seeks a pumping scheme in which the capture zone fully encompasses all control points representing the contaminant plume. The capture zone is simulated by tracking particles from extraction wells backwards through the velocity field. To represent the plume capture constraints numerically, a distance measure is used in which the minimum distance between each plume control point and all particles (see Figure 1) is constrained. When the distance between a control point and particle pathline equals zero then the plume control point lies within the capture zone. To ensure capture of the entire plume, the constraint function must equal zero for all control points. The reverse tracking formulation is stated as... 

Hwang D, Byun DW, Odman MT (1997) An automatic differentiation technique for sensitivity analysis of numerical advection schemes in air quality models. Atmospheric Environment, 31(6) 879-888. 

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. 

Further tests and previous experiences in other geographic locations confirmed that minimum Kz is a relevant (and often neglected) parameter to model properly the dispersion during weak wind and very stable conditions. Unfortunately no general value for minimum Kz can be defined, while proper values depend on season and local climatology, as well as on numerical diffusion in the advection scheme. 

Numerical advection schemes in the meteorological part are not meeting all the requirements for ACTMs, so they should be improved and harmonized in NWP and ACT models. 

Because of the success encountered by finite elements in the solution of elliptic problems, it was extended (in the 80s) to the advection or transport equation which is a hyperbolic equation with only one real characteristic. This equation can be solved naturally for an analytical velocity field by solving a time differential equation. It appeared important, when the velocity field was numerically obtained, to be able to solve simultaneously propagation and diffusion equations at low cost. By introducing upwinding in test functions or in the discretization scheme, the particular nature of the transport equation was considered. In this case, a particular direction is given at each point (the direction of the convecting flow) and boundary conditions are only considered on the part of the boundary where the flow is entrant. 

Harvie, D. J. E., and Fletcher, D. F. (2001) A New Volume of Fluid Advection Algorithm The Defined Donating Region Scheme, International Journal for Numerical Methods in Fluids, Vol. 35(2), pp. 151-172. 

Numerical methods constructed based on the advective form (non-conservative form) of the transport operator are shape preserving, but not conservative [82]. Schemes constructed based on the conservative form (or flux form) of the transport operator are preferable when strict conservation is required. 

Another method for analyzing the truncation error of advection schemes is the Fourier (or von Neumann method) [135, 174, 136]. This method is used to study the effects of numerical diffusion on the solution. 

A large number of explicit numerical advection algorithms were described and evaluated for the use in atmospheric transport and chemistry models by Rood [162], and Dabdub and Seinfeld [32]. A requirement in air pollution simulations is to calculate the transport of pollutants in a strictly conservative manner. For this purpose, the flux integral method has been a popular procedure for constructing an explicit single step forward in time conservative control volume update of the unsteady multidimensional convection-diffusion equation. The second order moments (SOM) [164, 148], Bott [14, 15], and UTOPIA (Uniformly Third-Order Polynomial Interpolation Algorithm) [112] schemes are all derived based on the flux integral concept. 

Dabdub D, Seinfeld JH (1994) Numerical Advective Schemes Used in Air Quality Models - Sequential and Parallell Implementation. Atm Env 28(20) 3369-3385... 

Smith GD (1985) Numerical Solution of Partial Differential Equations Finite Difference Methods. Third edition. Clarendon Press, Oxford Smolarkiewicz PK (1983) A simple positive definite advection scheme with small implicit diffusion. Mon Wea Rev 11 479-486... 

Thuburn J (1995) Dissipation and Cascades to Small Scales in Numerical Models Using a Shape-Preserving Advection Scheme. Mon Wea Rev 123(6) 1888-1903... 

In summary, in this section we have shown that the realizable scheme for mixed advection is very similar to those introduced for advection and free transport. However, the numerical properties of the scheme are very dependent on the functional form used for A. In order for the standard models for mixed advection to be well conditioned, the functional form for A must compensate for velocity abscissas located too close to the origin. With multivariate FQMOM, the model given in Fq. (B.56) can be used for this purpose. The potential for singular behavior with mixed advection makes it nevertheless problematic, and alternative schemes may be required to properly handle such behavior. 

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