Crank-Nicholson algorithm

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 ADI method, first used by Peaceman and Rachford (1955) for solving parabolic PDEs, can also be derived from the Crank-Nicholson algorithm. In the three-dimensional MRTM model, the governing equation can be discretized by the Crank-Nicholson algorithm as... 

The method of zonation was applied to the energy and material conservation equations. Based on centered finite difTerence approximations, this method can transform three partial differential equations in radial distance and time to ordinary differential equations in time only. Following this, the ordinary differential equations were solved by using Crank-Nicholson algorithm. On the basis of this, the volumetric fluxes of those tar-phase and total volatile phase components were integrated with time by using in roved Euler method to evaluate overall pyrolysis product yields, and afterwards the gas yield can be deduced. 

Implicit algorithms that can be used include the globally implicit algorithm of (25.95) or the popular Crank—Nicholson algorithm that can be derived as follows. For the onedimensional problem with constant diffusivity K, we obtain... 

Schematic diagram of the Crank-Nicholson algorithm to solve the diffusion differential equation in combination with the ChemApp program. 

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. 

When p = 0.5, the method is the Crank-Nicholson implicit method. The expansion point should be taken at (i+l/2,j). The truncation error is of the order (Ax)2 plus order (Ay)2. No stability criterion comes out of the von Neumann analysis, but difficulties can come about if diagonal dominance is not kept for the tridiagonal algorithm. 

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 Crank-Nicholson technique is a widely applied method for solving partial diiferential equations such as those for the radial dispersion model. However, it is implicit in approach and thus a little balky sometimes. Use this approach to develop an algorithm for the solution of the equations of Illustration 7.11. You will see that, if the method is developed properly, it will result in equations leading to a tridiagonal matrix similar to those treated in Illustration 6.4. 

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