Difference scheme Crank-Nicolson

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. 

This then provides a physical derivation of the finite-difference technique and shows how the solution to the differential equations can be propagated forward in time from a knowledge of the concentration profile at a series of mesh points. Algebraic derivations of the finite-difference equations can be found in most textbooks on numerical analysis. There are a variety of finite-difference approximations ranging from the fully explicit method (illustrated above) via Crank-Nicolson and other weighted implicit forward. schemes to the fully implicit backward method, which can be u.sed to solve the equations. The methods tend to increase in stability and accuracy in the order given. The difference scheme for the cylindrical geometry appropriate for a root is... 

Gradient diffusion was assumed in the species-mass-conservation model of Shir and Shieh. Integration was carried out in the space between the ground and the mixing height with zero fluxes assumed at each boundary. A first-order decay of sulfur dioxide was the only chemical reaction, and it was suggested that this reaction is important only under low wind speed. Finite-difference numerical solutions for sulfur dioxide in the St. Louis, Missouri, area were obtained with a second-order central finite-difference scheme for horizontal terms and the Crank-Nicolson technique for the vertical-diffusion terms. The three-dimensional grid had 16,800 points on a 30 x 40 x 14 mesh. 

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... 

So far, we have employed three different numerical schemes (explicit, implicit, Crank-Nicolson) to solve a one-dimensional unsteady conduction problem. Pros and cons for these schemes are ... 

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.

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]. 

The computational way to solve the problem of diffusion by the Crank-Nicolson method is quite different from that followed in the explicit schemes. Once the old concentration profile is known, three unknowns are still present in the equation written above. If n+1 points are required to describe the concentration profile in the diffusion layer, n equations similar to that written above will constitute a system with n+2 unknowns (aq, aj, 2 %1 /+1 n+l)- boundary conditions... 

Looking at Figure 3, considering time on the abscissa (lower scale), f(xQ - /i)= a,- (old value) and f(xQ + /i) = a (new value), the chord AB approximates the slope at A, Le. at a time r, according to a forward difference scheme (classic explicit method) on the other hand, it constitutes a central difference approximation at a time t + 0.5Ar (Crank-Nicolson). We can also use it as a backward difference approximation for the slope at B, Le, at a time t + At (Laasonen, fully implicit method) [4,6] ... 

In the pure diffusion case the centered-difference scheme results were encumbered by severe oscillation, while the Crank-Nicolson performed well. The state-variable scheme followed the Crank-Nicolson results well to U2 values of 10 when oscillations became severe. Table 8.3 presents a summary of the pure diffusion results. 

Solve the model using the Crank-Nicolson finite-difference scheme after quasilinearization. 

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 cases 9 = 0,9 = and 9 = 0.5 correspond to the explicit, implicit and Crank-Nicolson finite difference schemes, respectively. Since (6.64) is satisfied both for the (n- l)th and nth steps, we have... 

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. 

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) ... 

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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