Crank-Nicolson scheme

There are a number of numerical algorithms to solve the difference equation representation of the partial differential equations. Implicit algorithms such as Crank-Nicolson scheme where the finite difference representations for the spatial derivatives are averaged over two successive times, t = nAt and t = n + l)At, are frequently used because they are usually unconditionally stable algorithms. Most conservation laws lead to equations of the form... 

The Crank-Nicolson scheme improves the accuracy of the implicit scheme to second-order accuracy in time but still requires the usual matrix manipulations of that scheme. 

A finite difference scheme for discretization in time is used at this stage. In order to reduce the set of ordinary differential equations to algebraic equations, a time weighting coefficient is introduced, that allows to use several schemes explicit, implicit or the Crank-Nicolson scheme. 

The physical transport technique can either be the explicit, the implicit difference method, or the Crank-Nicolson-Scheme (see Kinzelbach, 1986, for instance). The explicit difference method, used in this study, shows the following equation for a two-dimensional, unidirectional aquifer ... 

Figures 12.11 shows plots of yj = y x = 0.2) as a function of time. Computations from the forward difference scheme are shown in Fig. 12.11a, while those of the backward difference and the Crank-Nicolson schemes are shown in Figs. 12.11h and c, respectively. Time step sizes of 0.01 and 0.05 are used as parameters in these three figures. The exact solution (Eq. 12.129) is also shown in these figures as dashed lines. It is seen that the backward difference and the Crank-Nicolson methods are stable no matter what step size is used, whereas the forward difference scheme becomes unstable when the stability criterion of Eq. 12.139 is violated. With the grid size of Aa = 0.2, the maximum time step size for stability of the forward difference method is At = (Ajc)V2 = 0.02.

Crank-Nicolson scheme, and the implicit scheme, respectively. The integration with respect to the temporal and the spatial variables eventually results in a system of algebraic equations (one for each control volume), which can be solved by standard numerical techniques. 

Analogous stabihty analyses can be executed for the other time-discretization schemes as well. It is important to note here that although the von Neumann stability analysis yields a limiting time-step estimate to keep the roundoff errors bounded, it does not preclude the occurrence of bounded but unphysical solutions. A classical example is the Crank-Nicolson scheme, which from the von Neumann viewpoint is unconditionally stable but can give rise to bounded unphysical solutirms in case all the coefficients of the discretization equation do not happen to be of the same sign [2]. 

Several different integration methods are available in the literature. Many of them approximate the effect of the Laplace operator by finite differences [32,39]. Among the more elaborate schemes of this kind we mention the Crank-Nicolson scheme [32] and the Dufort-Frankel scheme [39]. 

Difficulties are encountered by both the centered-difference and Crank-Nicolson schemes when both Ui and U2 play a role in the problem formulation. Results are given in Table 8.4. These results also show the flexible and well-behaved characteristics of the state-variable scheme for problems when both convection and diffusion are of consequence. 

Develop a finite-difference solution to the following one-dimensional surfactant flooding problem (Fathi and Ramirez, 1984). You should use the corner scheme for the saturation equation and the Crank-Nicolson scheme for the concentration equation with quasilinearization. 

The procedure with 0 = 0 corresponds to an explicit difference scheme 0 = 1 is an implicit difference scheme and 0 = 0.5 is the Crank-Nicolson scheme. 

The numerical method uses centered finite differences for spatial derivatives and time integrations are performed using the ADI method. The ADI scheme splits each time step into two and a semi-implicit Crank-Nicolson scheme is used treating implicitly the r-direction over half a time step and then the -direction over the second half. In addition, a pseudo-unsteady system is solved which includes a term d tl ldt on the left hand side of (121) and integrating forward to steady state (see Peyret and Taylor [55]). The physical domain is mapped onto a rectangular computational domain by the transformation r = 0 = Try,... 

Runge-Kutta integration to the field. There have been other developments - notably in recent years - but I do not see them becoming very popular, mainly because most of the new techniques require considerable mathematical preparation before the programming step and are difficult to modify when a different reaction is to be simulated (although the contrary is almost always claimed for these methods). Only the traditional explicit and the Runge-Kutta methods are easy to use, to modify and to check for errors, while the Crank-Nicolson schemes are moderately so. 

This removes all the above problems and will be as accurate as Feldberg s (1981) corrected method, again with the X restriction as mentioned above. My feeling here is that, having gone to the trouble of using unequal intervals in order to reduce computing time, it makes sense to combine this with the Crank-Nicolson scheme to get the most out of it. 

Crank-Nicolson scheme. Shoup and Szabo (1982) applied it to electrochemical simulation. It will be fairly briefly dealt with here. 

Similarly, one could attempt to improve the 3C/3T discretisation (other than by using Runge-Kutta integration). In effect, the Crank-Nicolson scheme does this by specifying a central difference approximation at T+J 8T. The same can be done at T by using the Richardson (1911) formula (denoting time steps by the index k) ... 

Britz D, Thomsen KN (1987) Electrochemical digital simulation reevaluation of the Crank-Nicolson scheme. Anal Chim Acta 194 317. 

Heinze J, Storzbach M, Mortensen J (1984) Digital simulation of cyclic voltammetric curves by the implicit Crank-Nicolson scheme. J Electroanal Chem 165 61-70... 

Luskin M, Rannacher R (1982) On the smoothing properties of the Crank-Nicolson scheme. Appl Anal 14 117-135... 

Khaliq AQM, Wade BA (2001) On smoothing of the Crank-Nicolson scheme for nonhomoge-neous parabolic problems. J Comput Methods Sci Eng 1 107-124... 

Gourlay AR, Morris JL (1981) Linear combinations of generalized Crank Nicolson schemes. IMA J Numer Anal 1 347-357... 

The differential equation for head pressure (2) is solved by over-relaxation iterative method [4] filtration rate is calculated from Darcy law by using defined values of hydraulic head. Transport equations of reagent concentration in liquid phase (5), useful element concentration in solid phase (4), and its transition to liquid phase (6) are solved together by the implicit Crank-Nicolson scheme. Crank-Nicolson scheme is implemented in three stages in case of 3D problem by using splitting technique of the alternating direction implicit (ADI) method [4]. 

Fick s second law (Eq. 29.1) was solved by means of the numerical finite-difference technique using the implicit Crank-Nicolson scheme, which is schematically represented in Fig. 29.11. It was assumed that the diffusion coefficients in the x- and y-directions, respectively, are independent of the local concentration and are only affected by temperature. 

The differential equations describing heat transfer through the fabric, air gap and skin are solved by a finite difference model. Due to the non-linear terms of absorption of incident radiation, Gauss-Seidel point-by-point iterative scheme is used to solve these equations. An under-relaxation procedure is utilized to avoid divergence of the iteration method. The Crank-Nicolson scheme [49] is used to solve the resulting ordinary differential equations in time. 

A finite difference Reynolds equation with the central difference formulation of the. first order spatial derivatives and a Crank-Nicolson scheme for incorporating the time dependent term [3] was used for the analyses presented in the current paper, and the results obtained with an alternative finite element formulation were indistinguishable. 

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