Discrete Formulation

These properties carry back to the discrete formulation. We shall use both discrete and continuous formulations in this volume, changing back and forth as needs require. The continuous regime allows us to avoid consideration of sampling effects when such consideration is not of immediate concern. Deconvolution algorithms, on the other hand, are numerically implemented on sampled data, and we find the discrete representation indispensable in such cases. 

This is called point-simultaneous overrelaxation. If we set k = [s]nn, we have obtained the discrete formulation of Van Cittert s method. This connection between Van Cittert s method and the classic iterative methods of solving simultaneous equations was demonstrated in an earlier work (Jansson, 1968, 1970). 

The radiative source term is a discretized formulation of the net radiant absorption for each volume zone which may be incorporated as a source term into numerical approximations for the generalized energy equation. As such, it permits formulation of energy balances on each zone that may include conductive and convective heat transfer. For K—> 0, GS —> 0, and GG —> 0 leading to S —> On. When K 0 and S = 0N, the gas is said to be in a state of radiative equilibrium. In the notation usually associated with the discrete ordinate (DO) and finite volume (FV) methods, see Modest (op. cit., Chap. 16), one would write S /V, = K[G - 4- g] = Here H. = G/4 is the average flux... 

The radiative source term is a discretized formulation of the net radiant absorption for each volume zone which may be incorporated as a source term into numerical approximations for the generalized energy equation. As such, it permits formulation of energy balances on each zone that may include conductive and convective heat transfer. Eor K—> 0, GS —> 0, and GG —> 0 leading to S —> On. When and... 

David R, Marchal P, Klein JP, Villermaux J. Crystallization and precipitation engineering. III. A discrete formulation of the agglomeration rate of crystals in a crystallization process. Chem Eng Sci 1991 46(1) 205-213. 

There are a number of approaches to discrete formulation some are illustrated here in terms of a steady one-dimensional fin problem (recall Ex. 2.9). For an infinitely long fin, with a specified base temperature To, transferring heat with a coefficient h to an ambient at temperature T00, an exact solution for temperature is... 

We shall use Eq. (4.3) as a base for the following approximate discrete formulations. 

Table 4.1 compares the results of the foregoing discrete formulations for a typical inner node. Since Eqs. (4.3), (4.7), and (4.9) differ only in the coefficient of 0,-, these coefficients, their percent error relative to the exact coefficient, and their series expansions for small values of B are given in Table 4.1. Figure 4.4 shows the coefficients for larger values of B. As the number of grids n(= t/Ax) is increased, the cell Biot number B decreases, and all formulations approach the exact one.

The inductive approach generally followed in this chapter, which is based on the five steps of formulation, directly leads to the discrete formulation of a given problem. However, it does not provide information on the accuracy of this formulation. In this section we deal with the error involved with discrete formulations, which is usually called the truncation error. 

Thus, for the particular choice of F — 1/6 the explidt scheme has TE of 0[(Ar)2, (Ax)4], which assures ahigher accuracy. However, the choice At = (Ax)2/(6a) implies time steps too small to be of practical interest in most cases. Irrespective of the size of the time step relative to that of the space step, TE 0, hence FDA —> PDE, as Ai - 0 and Ax —> 0, which shows the discrete formulation to be consistent... 

The second classification category distinguishes between approaches with continuous or discrete formulation for time and pipeline. In a continuous formulation, continuous coordinates are used to describe e.g. the position of a batch or a pipeline branch. The indicator u c, d indicates whether a continuous or discrete formulation is chosen, where the first component refers to the time aspect and the second refers to the pipeline. In general, time-discrete formulations are more intuitive, but to the costs of larger model dimensions in terms of the number of equations and variables. In contrast, time-continuous formulations are more compact but less intuitive. ... 

For the pipeline a discrete formulation impUes that the pipeUne is represented as a sequence of discrete segments. This eases the tracking of product batches along the pipeline and the formulation of product removals at the depots. Table 3.12 summarizes the relevant literature according to the proposed classification scheme. 

Almost all contributions to MIRSP use a time-continuous formulation and consider constant consumption and/or production rates for the products at the sites. One exception is Persson and Gothe-Lundgren (2005) using a time-discrete formulation. Moreover, production planning decisions are incorporated besides routing and inventory management. The second exception propose Christiansen et al. (2011) where a solution procedure for a MIRSP with time-varying consumption rates is described (but no mathematical model). 

When biomacromolecular embedding is considered, such as protein matrices and DNA structures, the discrete formulation is instead to be preferred in those cases a detailed and atomistic description of the macromo-lecular environment is necessary, in order to obtain accurate descriptions of the molecular process of interest. Moreover, for these systems, accurate force fields are generally available. Within this framework, the QM/MM approach is commonly used in combination with MD simulations to both achieve a proper statistical sampling and to account for the effects of fluctuations. Commonly the MD simulations are performed at a fiilly classical level (especially if the systems are large and the time-windows to be explored are long). In the 2010s, however, QM/MM-MD are also becoming feasible for small-medium QM systems for time windows of the order of tens to hundreds ofps. ... 

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