Grid step

Example 1. An equidistant grid on a segment. The segment [0,1] of unit length is splitted into N equal intervals. The spacing between the adjacent nodes = h = is termed a grid step or simply step. 

The nodes lying on one and the same straight line (horizontal or vertical), the distance between which is equal to the grid step (h or r), are called adjacent grid nodes. 

The natural replacement of the central difference derivative u x) by the first derivative Uo leads to a scheme of second-order approximation. Such a scheme is monotone only for sufficiently small grid steps. Moreover, the elimination method can be applied only for sufficiently small h under the restriction h r x) < 2k x). If u is approximated by one-sided difference derivatives (the right one for r > 0 and the left one % for r < 0), we obtain a monotone scheme for which the maximum principle is certainly true for any step h, but it is of first-order approximation. This is unacceptable for us. 

The explicit scheme is stable only under condition (25) relating the grid steps h and r (a conditionally stable scheme). 

Since the elimination method reciuires several operations at one node, the total number of which is independent of the grid step, the algorithm just established will be economical if one succeeds in showing that scheme (9)-(14) is absolutely stable. The following sections place a special emphasis on stability and convergence of the scheme concerned. 

In this regard, it should be noted that for the difference elliptic problems posed in Section 3, item 7, ro(e) is independent of the grid step. What is more, when G is a rectangle and = h-2 = h, the work in doing this is... 

Because of the enormous range of difference approximations to an equation having similar asymptotic properties with respect to a grid step (the same order of accuracy or the number of necessary operations), their numerical realizations resulted in the appearance of different schemes for solving the basic problems in mathematical physics. 

BUSTER chooses the minimal grid necessary to avoid aliasing effects, based on the prior prejudice used and on the fall-off of the structure factor amplitudes with resolution for the 23 K L-alanine valence density reconstruction the grid was (64 144 64). The cell parameters for the crystal are a = 5.928(1)A b = 12.260(2)A c = 5.794(1) A [45], so that the grid step was shorter than 0.095 A along each axis. 

Refined finite-difference schemes were obtained by introducing corrections at the interface between the core and the cladding according to the method proposed in" (the value of the transverse grid step Ax was chosen to meet the condition d = NAx where N is an integer). 

In [50], the mean annual wind field compiled according to the data of the Russian Climatic Reference Book was used. The mean wind speeds became two to threefold higher. The maximums of the velocity and cyclonic vorticity of the wind were confined to the eastern part of the Black Sea. The almost twofold decrease in the horizontal grid step (11 km) as compared to [48] allowed one to reproduce in [50] a system of subbasin cyclonic and anticyclonic eddies quasiperiodic over the longitude it clearly dominated over the large-scale BSGC. The latter is represented in [50] only in the weaker mean annual current fields. 

The molecular interaction fields (MIF) in the binding site of the cytochromes were obtained using a grid step size of 0.5 A and a self-accommodating dielectric constant [16]. The grid box size for the five isoforms was placed around the active site cavities and carefully refined using the tools available in the GRID software. 

The resulting field calculated as a 3D grid map with 0.5 A grid step size is contoured using an in-house algorithm to produce envelopes, whose location, shape, and volume are indicative of the ligand binding pockets. 

Now we have to move down one grid step in the z direction, and in order to do this we must use the finite difference versions of conservation equations (i) and (ii). There are two ways to do this, using either the so-called explicit or implicit approximations. We will use the explicit method for illustration here. 

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