Crank-Nicholson methods

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

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

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... 

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. 

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... 

Among the variety of methods which have been proposed for simulation of packed bed dynamics three techniques have been used with success (1) Crank-Nicholson technique [10], (2) transformation to integral equation [11], (3) orthogonal collocation on finite elements [12]. In the following computation, we have used the Crank-Nicholson method with the nonequidistant space steps in the Eigenberger and Butt version [10]. 

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.

Application of the Crank-Nicholson method based on the spatial difference scheme (5.39) results in the following discretized form of the diffusion equation ... 

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. 

Consider laminar flow of a fluid over a flat plate. Use the Crank-Nicholson method of finite differencing to compute the two dimensionless velocity-component distributions within the boundary layer. 

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. 

This approximation is called the trapezoid rule and is the basis for the popular second order Crank-Nicholson method. 

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. 

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). 

An alternative model uses the Crank-Nicholson method to generate a voltammogram that consists of a layer with a series of microscopic formal potentials, most situated at O.OV and the rest equally spaced 50 mV apart. This also yields a voltanmiogram (Fig. 6.16) similar to the experimental one (Fig. 6.14). The basis for this is the fact that different oligomers of different chain length possess a range of redox potentials. Thus at least qualitatively, two models may account for the electrochemical behavior of a conducting polymer coated on an electrode. 

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

Crank Nicholson method. The reader is advised to consult engineering mathematics texts (3-6). 

In the finite-element process, the left-hand side of Eq. (AID.23) is approximated by Crank-Nicholson method and is given by... 

The zero flux condition is assumed at the other boundary. The Crank-Nicholson method with spatial and temporal mesh Ax = 0.75 and At = 1.0 is used to solve (1).) The dashed curves, which almost perfectly coincide with the dotted curves, are solutions of the kinematic equation (2) (subject to the initial conditions t (0) = (k-l)T-j) based upon the dispersion relation of Fig.1-B. The curves in Fig.2-B show the speeds c (x) E dx/dt (x) of the impulses in the x-c plane. 

The nonstationary transport equations result from the differentiation of the flux with respect to the variation of concentration over time for a given component of the film. To obtain the compositional profiles in the inner layer, the system of equations is solved subject to initial and boundary conditions using a Crank—Nicholson method [86], which takes into account growth mechanisms via oxygen anion and chromium cation vacancies. A system of equations analogous to the inner layer treatment was employed to calculate outer layer compositional profiles. 

Crank-Nicholson method, 301 Crystallite migration kinetics of, 218 mechanism of, 202... 

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... 

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 ... 

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. 

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. 

A VBA program that implements the Crank-Nicholson method for the problem of Equation 8.5 with the initial and boundary conditions of Equation 8.7 is shown below. Equation 8.15 forms the basis for the tridiagonal system to solve at each time step. 

Exercise 8.2 Rework Exercise 8.1 using the Crank-Nicholson method. Exercise 8.3 Rework Exercise 8.1 using the method of lines. 

As (4.115) uses information only about the state values at the beginning, xl l, and end, xl + l, of the current time step, it is said to define a single-step integration method. For example, in the Crank-Nicholson method... 

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. 

Thus, the imphcit Euler and Crank-Nicholson methods, or more generaUy (4.150) whenever 0 > 1/2, are A-stable. Here, we have only shown tiiis to be true when all stable eigenvalues are real. For four values of 9, Figure 4.11 plots the modulus of the growth coefficient tij = as a function of coj in the complex plane. The region of absolute stability,... 

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). 

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