Explicit finite-difference algorithm

Figure 11.20 Schematic representation of the explicit finite difference algorithm. The grey region indicates unknown concentrations, whereas the black regions indicate known concentrations. Starting with = 0, the unknown concentrations cf are calculated from the known values of cf., cf and cf j. Iterating from i = 1 to yields the concentration profile at k = 1. The procedure is repeated to compute all the concentrations profiles until k. ...

Figure 11.22 Convergence plot for the explicit finite difference algorithm.

By adding the forward (explicit) finite-difference approximation to each side of this equation, we can identify both the explicit Euler algorithm and an expression for the local truncation error ... 

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

Shoup D, Szabo A (1986) Explicit hopscotch and implicit finite-difference algorithms for the Cottrell problem exact analytical results. 

Bieniasz LK, 0steiby O, Blitz D (1995) Numerical stability of finite difference algorithms for electrochemical kinetic simulations matrix stability analysis of the classic explicit, fully implicit and Crank-Nicolson methods and typical problems involving mixed boundtiry conditions. Comput Chem 19 121-136... 

In this section we begin with the differential equation, rather than develop the difference equations from the finite control volume as was done in the previous section. Both approaches lead to the same result. We must consider both time and spatial derivatives, and the relationship between them leads to either explicit or implicit algorithms. 

Methods applying reverse differences in time are called implicit. Generally these implicit methods, as e.g. the Crank-Nicholson method, show high numerical stability. On the other side, there are explicit methods, and the methods of iterative solution algorithms. Besides the strong attenuation (numeric dispersion) there is another problem with the finite differences method, and that is the oscillation. 

The numerical solution to the advection-dispersion equation and associated adsorption equations can be performed using finite difference schemes, either in their implicit and/or explicit form. In the one-dimensional MRTM model (Selim et al., 1990), the Crank-Nicholson algorithm was applied to solve the governing equations of the chemical transport and retention in soils. The web-based simulation system for the one-dimensional MRTM model is detailed in Zeng et al. (2002). The alternating direction-implicit (ADI) method is used here to solve the three-dimensional models. 

In many books, radial flow theory is studied superficially and dismissed after cursory derivation of the log r pressure solution. Here we will consider single-phase radial flow in detail. We will examine analytical formulations that are possible in various physical limits, for different types of liquids and gases, and develop efficient models for time and cost-effective solutions. Steady-state flows of constant density liquids and compressible gases can be solved analytically, and these are considered first. In Examples 6-1 to 6-3, different formulations are presented, solved, and discussed the results are useful in formation evaluation and drilling applications. Then, we introduce finite difference methods for steady and transient flows in a natural, informal, hands-on way, and combine the resulting algorithms with analytical results to provide the foundation for a powerful write it yourself radial flow simulator. Concepts such as explicit versus implicit schemes, von Neumann stability, and truncation error are discussed in a self-contained exposition. 

The Verlet algorithm has the numerical disadvantage that the new positions are obtained by adding a term proportional to Af to a difference in positions (2r — r, i). Since At is a small number and (2r, — r, i) is a difference between two large numbers, this may lead to truncation errors due to finite precision. The Verlet furthermore has the disadvantage that velocities do not appear explicitly, which is a problem in connection with generating ensembles with constant temperature, as discussed below. 

The above algorithm can be applied to both lattice and continuum simulations. One important difference, however, must be noted. In the case of a lattice, all possible trial orientations of the molecule can be considered explicitly the sets and are therefore the same (on a cubic lattice, a simple dimer or a bond can adopt an orientation along the coordinate axes, and n = 6). In a continuum, a molecule can adopt one of an infinite number of trial orientations. By restricting the trial orientations to the finite sets and the probabihty of generating these sets also enters the transition matrix along with the probability of choosing a direction from the probability distribution (which itself is a function of these sets [4]). The acceptance criteria are therefore also a function of these two sets ... 

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