Crank Nicholson

The implicit Crank-Nicholson integration method was used to solve the equation. Radial temperature and concentrations were calculated using the Thomas algorithm (Lapidus 1962, Carnahan et al,1969). This program allowed the use of either ideal or non-ideal gas laws. For cases using real gas assumptions, heat capacity and heat of reactions were made temperature dependent. 

The Crank-Nicholson method is a special case of the formula... 

In computer simulations we had chosen the following parameters of the potential h 1, a = 2. With such a choice the coordinates of minima equal Xmin = 1, the barrier height in the absence of driving is A = 1, the critical amplitude Ac is around 1.5, and we have chosen A = 2 to be far enough from Ac. In order to obtain the correlation function K t + x, t] we solved the FPE (2.6) numerically, using the Crank-Nicholson scheme. 

Alternatively, for 0 = 1 /2, i.e., evaluation of the second-order difference as an average of the values at t and t + At, leads to the robust Crank-Nicholson scheme... 

The molecule of the Crank-Nicholson scheme is also shown in Figure 3.15. 

The matrix form of the Crank-Nicholson implicit scheme becomes... 

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

Let us now turn to the implicit Crank-Nicholson method and form the matrix A as... 

Figure 3.17 Same as Figure 3.16 but for the implicit Crank-Nicholson scheme. 

Solve with the Crank-Nicholson scheme the diffusion problem described in the worked example in Section 3.3.1... 

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. 

Crank-Nicholson and Douglas sehemes with improved interfaee conditions have been applied to transform Eq. (2.9) into a finite-differenee equation. 

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. 

So far, relatively little attention has been given to the variational method of solving diffusion problems. Nevertheless, it is a technique which may become of more interest as the nature of problems becomes more complex. Indeed, the variational method is the basis of the finite element method of numerical calculations and so is, in many ways, an equal alternative to the more familiar Crank—Nicholson approach [505a, 505b]. The author hopes that the comments made in this chapter will indicate how useful and versatile this approach can be. 

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... 

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

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. 

Figure 8.21 Crank-Nicholson time marching scheme.

These equations are semi-implicit second order in time typically called Adams-Moulton (AM2) method or Crank-Nicholson (CN), when applied to diffusion problems, and due to the implicit nature of the procedure, the scheme is also unconditionally stable. 

Implicit schemes are unconditionally stable, this is shown in Fig. 8.27 where the evolution of the temperature, in a/ag steps, for values of a/ag higher than 0.5 is shown. Higher values of a/ag mean that we can use higher At, which at the end implies lower computational cost and faster solutions. The results in Fig 8.27 were obtained with the fully implicit Euler scheme, i.e., to = 1. The comparison between the implicit Euler and the Crank-Nicholson, to = 0.5 is illustrated in Fig. 8.28 for the center line temperature evolution. Although there is no apparent significance difference, we expect that the CN scheme is more accurate due to its second order nature. 

Figure 8.28 Comparison of implicit Euler and Crank-Nicholson solutions for a cooling amorphous thermoplastic plate.

There are several methods that we can use to increase the order of approximation of the integral in eqn. (8.72). Two of the most common higher order explicit methods are the Adams-Bashforth (AB2) and the Runge-Kutta of second and fourth order. The Adams-Bashforth is a second order method that uses a combination of the past value of the function, as in the explicit method depicted in Fig. 8.19, and an average of the past two values, similar to the Crank-Nicholson method depicted in Fig. 8.21, and written as... 

What would the constant strain finite element equations look like for the transient heat conduction problem with internal heat generation if you were to use a Crank-Nicholson time stepping scheme ... 

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... 

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