Finite-difference discretization

Figure 6.5 illustrates a stencil for the finite-difference discretization described by Eqs. 6.42 and 6.43. A spreadsheet that implements the finite-difference solution is described in detail in Appendix D (Section D.2). 

Fig. 6. 5 A stencil that illustrates the finite-difference discretization of the semi-infinite-domain axisymmetric stagnation flow problem.

The numerical solution for the solute-humic cotransport model was obtained by an unconditionally stable, fully implicit finite difference discretization method. The three governing transport Eqs. (38), (48), and (54) in conjunction with the initial and boundary conditions given by Eqs. (39)-(41), (51)—(53), (58) and (59) were solved simultaneously [57]. All flux boundary conditions were estimated using a second-order accurate one sided approximation [53]. 

Several hybrid simulations on crystal growth can be found in recent literature. Examples include dendritic solidification by coupling finite-different discretization of a phase field model to a MC simulation (Plapp and Karma, 2000), coupling a finite difference for the melt with a cellular automata for the solidification (Grujicic et al., 2001), a DSMC model for the fluid phase with a Metropolis-based MC for the surface to address cluster deposition onto substrates (Hongo et al., 2002 Mizuseki et al., 2002), a step model for the surface processes coupled with a CFD simulation of flow (Kwon and Derby, 2001) (two continuum but different feature scale models), an adaptive FEM CVD model coupled with a feature scale model (Merchant et al., 2000), and one-way coupled growth models in plasma systems (Hoekstra et al., 1997). Some specific applications are discussed in more detail below. 

The moderate and low Peclet regimes are characterized by significant magnitudes of all the terms of eq. (2), which should therefore be solved numerically. A non-uniform finite -difference discretization scheme has been chosen for the system, estimating the overall adsorption efficiency, Xg, as follows ... 

The numerical Method of Lines as implemented in the routine NDSolve of the Mathematica system deals with system (32) by employing the default fourth order finite difference discretization in the spatial variable Z, and creating a much larger coupled system of ordinnary equations for the transformed dimensionless temperature evaluated on the knots of the created mesh. This resulting system is internally solved (still inside NDSolve routine) with Gear s method for stiff ODE systems. Once numerical results have been obtained and automatically interpolated by NDSolve, one can apply the inverse expression (31.b) to obtain the full dimensionless temperature field. 

Figure 1 displays the variations of these properties and their second-order finite differences (discrete derivatives) with atomic number Z. As shown earlier [24], the second derivative possesses two advantages over the first one a stronger signal (but also noise) enhancement and a maximum located as in the function itself. 

Wissink JG (2004) On unconditional conservation of kinetic energy by finite-difference discretizations of the linear and non-linear convection equation. Computers Fluids 33 315-343... 

For the two vertically stacked dies, significant thermal gradients may exist due to non-uniform on-chip power distributions. To capture such non-uniform on-chip profiles, detailed thermal modeling and analysis is employed to achieve good accuracy. For this purpose, we adopt an efficient full-chip thermal simulator 231 that is based on detailed finite difference discretization of the following governing heat transfer partial differential equation... 

It has recently become possible to compute rather accurate dielectric boundary forces from finite difference solutions of the PB equations.50 The method is based on the finite difference discretization of the second term of the force density expression (Eq. [35]), together with the use of dielectric boundary smoothing. A full description of the method and accuracy testing is beyond the scope of the present chapter. However, the results of a sample calculation may be instructive. 

FIGURE 3.13 Solution of the DPM isothermal drying model of one-dimensional plate by pdepe solver of MATLAB . Finite difference discretization by uniform mesh both for space and time, 5. is dimensionless time, xlL is dimensionless distance. 

Bieniasz LK (2004) A fourth-order accurate, Numerov-type, three-point finite-difference discretization of electrochemical reaction-diffusion equations on nonuniform (exponentially expanding) grids in one-dimensional space geometry. J Comput Chem 25 1515-1521... 

Finite-difference discrete ordinate codes provide the most rigorous solution of the transport equation. However, truly 3D finite-difference Sn methods are still too inefficient to be used as routine tools for fast reactor analysis. Therefore, nodal transport methods have been developed intensively in recent years, because these methods, although approximate, provide the... 

In contrast to Chapter 1, we have explicitly introduced q(x,y,x,t), representing the local source volume flow rate per unit volume produced by any infinitesimal element of a general well. It is a three-dimensional, point singularity that applies to both injector and producer applications. For example, when q is a semi-infinite line, cylindrical radial flow is obtained over most of the source distribution, while spherical flow effects apply at the tip. In other words, partial penetration and spherical flow are modeled exactly. In this section, subscripts are used in three different contexts. First, they represent partial derivatives for example, Px is the partial derivative of p(x,y,z,t) with respect to the spatial coordinate x. Second, they are used as directional markers for example, ky (x,y,z) is the anisotropic permeability in the y direction. Finally, subscript indexes (i,j,k) in pijrepresent the centers of grid block volumes used in our finite difference discretizations. As usual. Ax, Ay, Az, and At are used to denote grid sizes for the independent variables x, y, z, and t. 

The resulting finite difference discretization would generate sets of nonlinear algebraic equations. The solution of this problem would require the application of Newton s method for simultaneous nonlinear equations (see Chap. 1). 

Through this approach, we can employ a finite difference discretization on a regular grid in (f, ri) space however, the differential eqnation now involves more complex derivatives. The finite element method, described below, allows us to solve BVPs in complex geometries without performing such coordinate transformations (which are not always possible anyway). 

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