Crank—Nicholson finite difference

Denoting the solute species in a binary system by subscripts 1 and 2, the Crank-Nicholson finite difference approximation for... 

The steady-state heat and mass balance equations of the different models were numerically integrated using a fourth-order Runge-Kutta-Gill method for the one-dimensional models, while the Crank-Nicholson finite differences method was used to solve the two-dimensional models. 

FIGURE 8.1 Crank-Nicholson finite difference nomenclature. 

Figure 3.15 Finite difference molecules for explicit (left) and implicit Crank-Nicholson (right) schemes.

When faster reactions are dealt with, it may be profitable to remove the At/Ay2 < 0.5 condition and use an implicit method such as the Crank-Nicholson method.15 17 The finite difference approximation is then applied at the value of t corresponding to the middle of the j to j + 1 interval, leading to... 

Linearizing the kinetic term as before, a set of three unknown linear equations is obtained, which is completed by the finite difference expression of the initial and boundary conditions. Inversion of the ensuing matrix allows the calculation of C at each node of the calculation grid and finally, of the current flowing through the electrode, or of the corresponding dimensionless function, by means of its finite difference expression. Calculation inside thin reaction layers may thus be more efficiently carried out than with explicit methods. The combination of the Crank-Nicholson... 

As initial distribution corresponds to the linear mode (2.11) of the given waveguide, the deviation of T z) with respeet to unity may he eonsidered as a measure of the error in this method. The results presented in Fig.2 allow one to analyze the accuracy of the method depending on the type of finite-difference scheme (Crank-Nicholson" or Douglas" schemes have been applied) and on the method of simulation of conditions at the interface between the core and the cladding for both (2D-FT) and 2D problems. 

In the numerical modeling of optical pulse propagation with account of the SS effect, Eq.(2.8) with = 2 = 3 = 0 has been solved. The Crank-Nicholson scheme was used to transform Eq.(2.8) into a finite-difference equation. 

The Crank-Nicholson scheme was used to transform Eq.(2.8) into a finite-difference equation. 

The equations were transformed into dimensionless form and solved by numerical methods. Solutions of the diffusion equations (7 or 13) were obtained by the Crank-Nicholson method (9) while Equation 2 was solved by a forward finite difference scheme. The theoretical breakthrough curves were obtained in terms of the following dimensionless variables... 

Ehrlich (El) uses the Crank-Nicholson (C16) finite-difference procedure for the integration of the diffusion equation, with a three-point approximation of the space derivatives on either side of the moving... 

Equation (5.62) for the current-potential response in CV has been deduced by assuming that the diffusion coefficients of species O and R fulfill the condition Do = >r = D. If this assumption cannot be fulfilled, this equation is not valid since in this case the surface concentrations are not constant and it has not been possible to obtain an explicit solution. Under these conditions, the CV curves corresponding to Nemstian processes have to be obtained by using numerical procedures to solve the diffusion differential equations (finite differences, Crank-Nicholson methods, etc. see Appendix I and ([28])3. 

Similarly to the above derivation, we can also use the technique to predict transient temperature fields. Again, as with finite elements and boundary elements, the time stepping is done using finite difference techniques. For a Crank-Nicholson transient energy equation formulation given by... 

The time increment was corrected according to a number of iterations necessary for calculation of a new profile. A revision of the space finite difference mesh was performed after five time steps. Frequently, 30-60 mesh points were sufficient also for high values of B. The Eigenberger-Butt method lowers the computer time expenditure in comparison with the classical Crank-Nicholson technique by a factor 4-10. 

In the last section we considered explicit expressions which predict the concentrations in elements at (t + At) from information at time t. An error is introduced due to asymmetry in relation to the simulation time. For this reason implicit methods, which predict what will be the next value and use this in the calculation, were developed. The version most used is the Crank-Nicholson method. Orthogonal collocation, which involves the resolution of a set of simultaneous differential equations, has also been employed. Accuracy is better, but computation time is greater, and the necessity of specifying the conditions can be difficult for a complex electrode mechanism. In this case the finite difference method is preferable7. 

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 this section, a model similar to that used by Cardoso and Luss (1969) is considered, with the exception that the assumption of solid isothermality is relaxed. A finite difference solution for the transient equations with symmetrical boundary conditions is presented using the Crank-Nicholson method (Lapidus, 1962). Another more efficient, method of solution is considered which is based on the orthogonal collocation technique first used by Villadsen and Stewart (1967), Finlayson (1972) and Villadsen and Michelsen (1978). Several assumptions for model reductions are investigated. 

The numerical solution of Eq. (54) as an initial-boundary-value problem, specified to the spatial relaxation problem in uniform electric fields, can be obtained (Sigeneger and Winkler, 1996) by using a finite-difference approach according to the well-known Crank Nicholson scheme for parabolic equations. 

The advantage of Eq. (160) (representing a parabolic partial differential equation from a mathematical viewpoint) is that it can be solved exactly by standard numerical techniques e.g., by the finite-difference Crank-Nicholson scheme under transient (nonstationary) conditions [37,64]. These calculations showed that the duration of the transient regimes is of the order of seconds, as previously estimated. Under the stationary conditions, Eq. (160) is simplified to the ordinary one-dimensional differential equation which can be solved by standard numerical techniques [18,76,118,119]. 

Crank-Nicholson method, 301 collocation method, 303 explicit method, 297 finite-difference formulas, 298 for multiple reactions, 132 for solving fixed-bed conservation equations, 296... 

Traditional finite difference methods [55, 81] for solving time-dependent second-degree partial differential equations (such as modified diffusion equation) include forward time-centered space (ETCS), Crank-Nicholson, and so on. For time-independent second-degree partial differential equations such as Poisson-Boltzmann equation, finite difference equations can be written after discretizing the space and approximating derivatives by their finite difference approximations. For space-independent dielectric constant, that is, E(r) = e, a tridiagonal matrix inversion needs to be carried out in order to obtain a solution for tp for a given/. 

The Reynolds equation and the energy equation in the interfacial film and the conduction equation in the seal rings are solved numerically by the finite difference technique. These equations are coupled by the heat exchange conditions on the boundaries of their domains. They are integrated by using the Crank Nicholson scheme. 

The governing equations are discretized by using the finite difference method. The Reynolds equation solution leads to solving a tri-diagonal system of linear equations. Using the semi-implicit scheme of Crank-Nicholson solves the energy equations in the film and in the rings. 

The third method (called the Crank-Nicholson method) applies some innovation in that the finite difference analog is centered about a fictitious half-way point as shown in Figure 8.1. 

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