Finite difference grid

Because of the use of the focusing method [18], more than four calculations are actually carried out for each group. However, the focusing method saves computer time by permitting the use of less extensive finite-difference grids. 

FIG. 3-50 Finite difference grid with variable spacing. 

Larger elements can produce comparable accuracy to smaller mesh elements of a finite difference grid, which is especially useful in handling elliptic partial differential matrices. 

Fig. 20.2. Indexing of nodal blocks in a finite difference grid, showing point (7,./) and it nearest neighbors. Properties of the nodal blocks are projected onto nodal points located at the center of each block.

Figure 8. Finite difference grid with 5-point stencil.

In addition to the above ad hoc approximations there are many procedures to develop finite difference approximations for given PDE s with their respective finite difference grids. The most commonly used are... 

Figure 8.5 Schematic diagram and finite difference grid of a fin cooling problem.

Alternatively, the problem can be solved numerically by applying a finite differences grid on the stagnant zone and determining the diffusive exchange iteratively (Parkhurst and Appelo, 1999 Appelo and Postma, 1994). 

Figure 4. Schematic of Westinghouse cold flow 30-cm diameter semicircular model and finite difference grid for numerical calculations...

For the case cited above, the ponderomotive energy is approximately 1 eV. For typical short pulse experiments today, this energy can easily be hundreds of electron volts. Therefore the wave function of a photoelectron in an intense laser field does not resemble that of the normal field-free Coulomb state, but is dressed by the field, becoming, in the absence of a binding potential, a Volkov state [5], This complex motion of the photoelectrons in the continuum is very difficult to reproduce in terms of the field-free atomic basis functions, so that we have chosen to define our electron wave functions on a finite difference grid. These numerical wave functions have the flexibility to represent both the bound and continuum states in the laser field accurately. 

PB models36 37 solve Equation 7.8 or its nonlinear extension for stronger fields by using a finite difference grid and treating the shape of the protein in detail, while continuing to use macroscopic dielectric constants for both the protein and the solvent. 

In geochemical models, these quantities represent the smallest time period for incremental steps in a simulated titration, or the smallest distance between grid points in a finite element or finite difference grid, if LEQ is to be a valid assumption. Or, as Knapp puts it, reactive transport calculations assuming LEQ are good approximations only if teq is less than the size of the time step, and /eq is less than the distance between adjacent grid points. 

Nl,° and yy., are the number of points used in the representation of the second derivative operator in the main body of the finite difference grid and at the last grid point respectively. 

NT is the number of points appended to the main part of the finite difference grid with geometrically decreasing spacing. 

V,R is the number of extra points inserted between neighboring points in the third step of the finite difference grid generation scheme explained in Ref. 61. 

Figure 4.3 Discretization of the forecast area and indexing of nodal blocks in finite differences grid relative point I, J) (Domenico P.A. et al, 1997).

Figure 2 Popular discretization schemes for numerical solution of the Poisson-Boltzmann equation. The solid black line and circles denote a model protein other lines denote the mesh on which the system is discretized, (a) Finite difference, (b) Boundary element, (c) Finite Element, (d) Focusing on finite difference grids. See color insert.

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