The Crank-Nicholson implicit method

To obtain an algorithm that is unconditionally stable, we consider an implicit discretization scheme that results from using backward finite-differences for the time derivative. The corresponding difference equation is most conveniently obtained by approximating the diffusion equation at point (Xj,tn+i)  

Analogous to the forward-difference method previously discussed, it is only first-order accurate in At. The only formal difference with respect to the forward-difference equation (8-10) appears to be the fact that the space derivative is evaluated at time tn+i, not at time tn. 

By rearranging the terms in (8-26), the following system of linear equations results  

By using matrix notation, the linear system (8-27) can be written  

It is apparent from the first and last rows of this matrix, that again the simple Dirichlet boundary conditions, Eq. (8-3), have been considered. Since X 0, the matrix A is positive definite and diagonally dominant. For solving system (8-28), the very efficient Crout factorization method for linear systems with tri-diagonal matrix can be applied (see Press et al. 1986, Section 2.4). 

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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. 

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. 

Solution of Mathematical Model for Case 2 and Case 3. For the Case 2 and Case 3 solutions, the technique developed for the Case 1 solution, which was based on the Crank-Nicholson implicit method, was unacceptable when the time derivative was included. 

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 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... 

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. 

Solution of Mathematical Model for Case 1. For the Case 1 solution iterative techniques were ruled unacceptable owing to the excessive time requirements of such methods. Several investigators (27, 28, 29, 30) working with similar noncoupled systems found that the Crank-Nicholson 6-point implicit differencing method (31) provided an excellent solution. For the solution of Equation (8) we decided to apply the Crank-Nicholson method to the second-order partials and corresponding explicit methods to the first-order partials. Nonlinear coefficients were treated in a special manner outlined by Reneau et al (5). 

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. 

Time integration methods more advanced than the one considered in this book (backward implicit, BI) can also be employed. Indeed, the Crank-Nicholson and high-order extrapolation methods [6] have proven to enable the reduction of the number of timesteps (and even improve the accuracy of the simulation) with respect to BI [4]. 

The implicit method, while it requires additional computation, is better than the explicit method as evident from Table 9.3. The implicit method given in Table 9.2 is called the Crank-Nicholson method when the weighting factor 6 is 0.5. As discussed earlier, this method requires simultaneous solution of algebraic equations at each i. When the expressions for the derivatives in Table 9.2 are used in Eq. 9.38, for instance, it is transformed into ... 

This includes as special cases the explicit (forward) Euler method for 0 = 0, the implicit (backward) Euler method for 0 = 1, and the Crank-Nicholson method for 0 = 1/2. 

As discussed in Chapter 4, discretized PDEs yield ODE systems that are very stiff therefore, to avoid a very small time step, an implicit method snch as the Crank-Nicholson method, ode23s, or ode15s should be used. Using ode15s, tubular.reactor.2rxn dyn im.m sim-nlates the reactor start-np dynamics. Initially the reactor is at steady state with an input stream containing only A, and then B is introdnced to start the reaction (Figme 6.13). 

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

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. 

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.

Domain mapping of the WMBVP follows the theory by Joseph. The physical fluid domain shown in Fig. 2.14 for the fully nonlinear WMBVP is mapped to a fixed computational fluid domain, and the discretized coupled free-surface boundary conditions are computed by an implicit Crank Nicholson (C N) method. s each iteration of the C-N method, the potential field is computed by the conjugate gradient method. The wavemaker motion E y/h,t) is assumed to be periodic with period T — 2tt/u , and the WMBVP with the surface tension f is given by... 

This system can be solved with the implicit method of Crank-Nicholson (trapezium rule) this... 

The fractional step methods have become quite popular. To predict an accurate time history of the flow, higher order discretizations must be employed. Kim and Moin [106], for example, used a second order explicit Adams-Bashforth scheme for the convective terms and a second order implicit Crank-Nicholson scheme for the viscous terms. Boundary conditions for the intermediate velocity fields in timesplitting methods are generally a complex issue [3, 106]. There are many variations of the fractional step methods, due to a vast choice of approaches to time and space discretizations, but they are generally based on the principles described above. 

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. 

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