Finite difference simulations

Consider a spatial coordinate, x, divided into NK points at a distance Ax apart k is used to index a particular point (k = 1... NK). The concentration gradient at point k in the grid depicted in Fig. 44 may be represented in three ways as shown in (100)-(102). 

The upwind and downwind differences may be combined to give expression (103) for the second derivative which is centred at point k. 

Consider a typical time-dependent mass transport equation, such as (104) for a microdisc electrode. 

If this is approximated using (central) finite differences it becomes (105), 

There is a choice as to whether the concentrations on the right-hand side are chosen to be at / or t -I-1. If concentrations at the old time (/) are used, we have an explicit equation (107). 

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Hiebcr, C, A. and Shen, S.F., 1980. A finite element/finite difference simulation of the injection-moulding filling process. J. Non-Newtonian Fluid Mech. 7, 1-32. 

This revision does not attempt to take many of these recent advances into account, even though some of them are cited in this chapter. Rather, it continues to provide a rigorous foundation for writing programs that will perform explicit finite difference simulations. In learning how to do this, the reader develops an appreciation of the method and, more importantly, its limitations. 

Bartholmai M, Schartel B. Assessing the performance of intumescent coatings using bench-scaled cone calorimeter and finite difference simulations. Fire Mater. 2007 31 187-205. 

Matsunaga, K., Mijata, H., Aoki, K., and Zhu, M., Finite-difference simulation of 3D vortical flows past road vehicles. Vehicle Hydrodynamics, SAE Special Publication 908,65 (1992). 

The use offictitious points as a means of locally modifying the differential stencils near laborious media interfaces in finite-difference simulations has been initially developed in [17, 18] and extended in [21, 28]. The specific method, which matches the problematic boundaries with physical derivative conditions, enhances the flexibility of higher order FDTD schemes and facilitates the discretization of difficult geometries. 

Figure 4.1 (a) Instantaneous number density of the Monte-Carlo particles as obtained by LES/FMDF in a near-equilibrium methane jet flame and (6) radial (r/d) variations of the Reynolds-averaged values of the filtered fluid density as obtained by LES/FMDF and by finite difference simulation at two streamwise (xfd) locations (J — x/d = 7.5 (finite difference) 2 — x/d = 15 (finite difference) 3 — x/d = 7.5 (Monte-Carlo) and 4 — x/d = 15 (Monte-Carlo)). 

The explicit finite difference simulation methods described here are quite straightforward and have been employed for a variety of electrochemical problems. Nevertheless, there are cases when the computation times required for accurate solutions become excessive, and more efficient numerical methods are appropriate. This is generally the case when very rapid coupled homogeneous reactions occur. As discussed in Section B.3, when is large, the reaction layer thickness may be small compared to the box thickness... 

S. W. Feldberg, M. L. Bowers, and F. C. Anson. Hopscotch-finite-difference simulation of the rotating ring-disc electrode, J. Electroanal. Chem. 215, 11-28 (1986). 

Figure 7. Discretization of a cylindrical battery for finite differences simulation and reduction to a radial ID heat transfer problem.

Manwart, C., Aaltosalmi, U., Koponen, A., Hilfer R., and Timonen, J., Lattice-Boltzmann and finite-difference simulations for the permeability for three-dimensional porous media, Phys. Rev. E., 66, 016702, 2002. 

A history matching package consisting of a one-dimensional, two-phase, finite difference simulator and Levenberg-Marquardt optimization algorithm was developed. The capillary end effect is considered in the simulator. 

C. A. Hieber and S. P. Shen, A Finite Element/Finite Difference Simulation of the Irijeotion Molding Pilhng Process, Journal of Non-Newtonian Fluid Mech., vol. 7, 1980, pp. 1-32. 

A numerical model has been developed by Viola etaL (1990) to optimize the operating parameters of a pultrusion line. Their model incorporates a onedimensional finite difference simulation of the pultrusion process. Based on given target parameters, the model primarily optimizes the temperature set points along the die. 

As we have seen, the various parts of an electrode process can be treated sequentially in a finite difference simulation, and it is possible to write a program to cover most eventualities. A flow diagram for such a program is shown in Fig. A.8. However, since programming is relatively straightforward it is usual to write a program for each new system and to include only the required parts. 

The capacitive current and IR drop are interrelated. Since capacitive current depends on sweep rate, and IR drop causes distortions in linearity of sweep, the capacitance will not follow exactly from theory of linear potential sweep. This effect can be incorporated into finite difference simulations. 

Equation (6) is the framework of an explicit finite difference simulation. The electrochemical experiment can be described by a discrete time (of the experiment) and space (distance from the electrode) grid (Pigure 4-2), where t = 0 at the beginning of the experiment and x = 0 at the electrode surface. If the concentration of every species is known for every space and time grid point, then the experiment is completely described. A point on the grid represents the concentration for an entire volume element, the boundaries of which are the midpoints between the grid points. 

A program to illustrate simulation by explicit finite differences—Simulates an EC mechanism) N + enables math coprocessor)... 

Feldberg SW (1981) Optimization of explicit finite-difference simulation of electrochemical phenomena utilizing an exponentially expanding space grid. Refinement of the Joslin-Pletcher algorithm. 

Hanafey MK, Scott RL, Ridgway TH, Reilley CN (1978) Analysis of electrochemical mechanisms by finite difference simulation and simplex fitting of double potential step current, charge and absorbance responses. Anal Chem 50 116. 

Tait RJ, Bury PC, Finnin BC, Reed B, Bond AM (1993) An explicit finite difference simulation for chronoamperometry at a disk microelectrode in a channel flow solution. J... 

Alden JA, Hakoura S, Compton RG (1999) Finite difference simulations of steady-state voltammetry at the wall-jet electrode. Effects of radial diffusion and working curves for common electrochemical mechanisms. Anal Chem 71 827-836... 

A series of numerical, finite difference simulations were carried out to investigate the behavior of the system as a function of X]v[, Fiyi and Pq for given values of the other parameters. An interesting case in the present context is shown in Fig. 1. 

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