Crank-Nicholson equation

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. 

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... 

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. 

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... 

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. 

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. 

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... 

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]. 

This is the so-called Crank-Nicholson scheme and, formally, it could have been obtained by simply averaging the explicit forward-difference and implicit backward-difference schemes. By conveniently grouping the terms, the following system of linear equations results ... 

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

The equation for the central point (i = 1) actually plays the role of inner boundary condition. The above system should be completed with one more boundary condition for the outer point tm = R. Irrespective of the type of the used time difference scheme (explicit, fully implicit or Crank-Nicholson), the further treatment of the resulting system of difference equations is absolutely analogous to the one developed for Cartesian coordinates. 

Applying the Crank-Nicholson scheme to equation (8-66), relative to a space-time grid characterized by the points ... 

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. 

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 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 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. 

Solution of the differential equations was by Gauss-Seidel iteration (with the fluid property values given in Table II) on an IBM 370 digital computer using implicit difference equations of the Crank-Nicholson type. The program was convergent and stable for all conditions tested. 

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. 

Equations (7) and (8) form a system of six ordinary differential equations in three space dimensions for each individual bubble. This system is integrated in time using a Crank-Nicholson scheme, which provides second order accuracy. Sequential tracking of all bubbles in the system is performed at each time step of the Navier-Stokes solver. 

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. 

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. 

Fine resolution in the normal direction is necessary around the shear layers, and it gives severe limitation on the time step for numerical stability. Thus, it is preferred to compute the derivatives in the normal direction implicitly, while the derivatives in the streamwise direction are treated explicitly. This leads to a hybrid time-integration scheme with a low-storage third-order RK (RK3) scheme for explicitly treated terms and a second-order Crank-Nicholson scheme for implicitly treated terms. The overall accuracy is thus second order in time. The discretized Navier-Stokes equations have the forms ... 

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 algebraic system of equations resulting from the Crank-Nicholson scheme in (25.123) is tridiagonal and can therefore be solved efficiently with specialized routines. 

The numerical solution of Eq. (54) as an initial-boundary-value problem, specified to the spatial relaxation problem in uniform electric fields, can be obtained (Sigeneger and Winkler, 1996) by using a finite-difference approach according to the well-known Crank Nicholson scheme for parabolic equations. 

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. 

The scheme has a single Euler predictor (equation (4,. 20)) and a Crank-Nicholson corrector (equation (. 19)). 

Equations (6) and (7) were solved with two sets of boundary conditions. The first set was source limited , i.e., disassociation rate-controlled and the second was flux limited , i.e., the concentration at the interface S was equal to an equilibrium value. The functions fi and f2 were assumed to be unity, Le., concentration-independent diffusion coefficients were used. The multi-phase Stefan problem was solved numerically [44] using a Crank-Nicholson scheme and the predictions were compared to experimental data for PS dissolution in MEK [45]. Critical angle illumination microscopy was used to measure the positions of the moving boundaries as a function of time and reasonably good agreement was obtained between the data and the model predictions (Fig. 4). 

The movement of the rubbery-solvent interface, S, was governed by the difference between the solvent penetration flux and the dissolution rate, derived earlier. An implicit Crank-Nicholson technique with a fixed grid was used to solve the model equations. A typical concentration profile of the polymer is shown in Fig, 24. Typical Case II behavior was observed. The respective positions of the interfaces R and S are shown in Fig. 25. Typical disentanglement-controlled dissolution was observed. Limited comparisons of the model predictions were made with experimental data for a PMMA-MIBK system. 

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