Numerical methods Crank-Nicholson

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... 

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 8-4 Comparison of the numerical solutions obtained by the fully implicit and Crank-Nicholson methods for the Cauchy problem (8-17) to (8-19). The spatial step h = 0.05 and the timestep At = 0.025 have been used.

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 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. 

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. 

The Crank-Nicholson implicit method and the method of lines for numerical solution of these equations do not restrict the radial and axial increments as Eq. (P) does. They are more involved procedures, but the burden is placed on the computer in all cases. 

An unsophisticated numerical method is used here for simplicity. Various implicit and explicit methods of writing dilference equations and their limitations are discussed by Leon Lapidus in Digital Computation for Chemical Engineers, McGraw-Hill Book Company, New York, 1960. The Crank-Nicholson implicit method [G. Crank and P. Nicholson, Proc. Cambridge Phil. Soc., 43, 50 (1947)] is well suited for machine computation. 

Almost always numerical (Crank-Nicholson method, orthogonal collocation, finite element,etc.)... 

Here, a is the thermal boundary conductance. The Crank-Nicholson method can be used to numerically obtain the temperature response of the sample. The temperature response from the model corresponding to a and k value which best fits the experimental data is considered as the appropriate thermophysical property of the sample. 

This again leads to a set of coupled equations that are tri-diagonal in nature. Since this method used a centered time difference approximation, it is accurate to second order in both time and space while the FD and BD methods are only accurate to first order in the time variable. Thus one would expect an improved accuracy with the Crank-Nicholson method. However, numerical stability of the three techniques is perhaps more important than accuracy. 

The question of numerical stability of various time differencing methods as applied to PDEs was discussed in flie previous sections. The method selected to pursue, known as the Crank-Nicholson (or CN) method was shown to have good numerical stability. However, the equally important issue of numerical accuracy was not addressed. It was noted that die developed method was accurate to second order in both the time and space numerical differencing techniques and from previous differencing approaches as applied to single variable differential equations. 

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