Difference Crank-Nicolson

Errors are proportional to At for small At. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method or Newton-Raphson method. See Ref. 36 for an application to dynamic distillation problems. 

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... 

Transient is a C-program for solving systems of generally non-linear, parabolic partial differential equations in two variables (that is, space and time), in particular, reaction-diffusion equations within the generalized Crank-Nicolson Finite Difference Method. 

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. 

The prior discretization of equation (2.1) uses control volumes with exphcit differences. They are explicit because only the accumulation term contains a concentration at the n -k 1 time step, resulting in an exphcit equation for (equations (E7.1.4), (E7.2.5), (E7.3.4), and (7.25)). Another common option would be fully imphcit (Laasonen) discretization where flux rate terms in equations (7.24) and (7.23) are computed at the n -k 1 time increment, instead of the n increment. Fully implicit is generally preferred over Crank-Nicolson implicit UQ = U Q n + Q.n+i) /P)... 

Most real cases of polymer melting (and solidification) involve complex geometries and shapes, temperature-dependent properties, and a phase change. The rigorous treatment for such problems involve numerical solutions (12-15) using finite difference (FDM) or FEMs. Figure 5.9 presents calculated temperature profiles using the Crank-Nicolson FDM (16) for the solidification of a HDPE melt inside a flat-sheet injection-mold cavity. The HDPE melt that has filled the cavity is considered to be initially isothermal at 300°F, and the mold wall temperature is 100°F. 

The species balance relation Eq. 13.2-8 is transformed to a difference equation using the forward difference on the time derivative and the backward difference on the space derivative. The finite difference form of the x-momentum equation (Eq. 13.2-25) is obtained by using the forward difference on all derivatives, and is solved by the Crank-Nicolson method. The same is true for the energy equation (Eq. 13.2-26). 

It was soon realised that at least unequal intervals, crowded closely around the UMDE edge, might help with accuracy, and Heinze was the first to use these in 1986 [300], as well as Bard and coworkers [71] in the same year. Taylor followed in 1990 [545]. Real Crank-Nicolson was used in 1996 [138], in a brute force manner, meaning that the linear system was simply solved by LU decomposition, ignoring the sparse nature of the system. More on this below. The ultimate unequal intervals technique is adaptive FEM, and this too has been tried, beginning with Nann [407] and Nann and Heinze [408,409], and followed more recently by a series of papers by Harriman et al. [287,288,289, 290,291,292,293], some of which studies concern microband electrodes and recessed UMDEs. One might think that FEM would make possible the use of very few sample points in the simulation space however, as an example, Harriman et al. [292] used up to about 2000 nodes in their work. This is similar to the number of points one needs to use with conformal mapping and multi-point approximations in finite difference methods, for similar accuracy. 

There have thus far been reported only two applications of finite-difference methods to the solution of (7) as they pertain to urban airsheds, both for the Los Angeles Basin. Eschenroeder and Martinez (21) applied the Crank-Nicolson implicit method to the simplified version of, (7),... 

The difference method of Crank-Nicolson is stable for all M. The size of the time steps is limited by the accuracy requirements. Very large values of M lead to finite oscillations in the numerical solution which only slowly decay with increasing k, cf. [2.57],... 

Example 2.7 The cooling problem discussed in Example 2.6 will now be solved using the Crank-Nicolson implicit difference method. The grid divisions will be kept as that in Fig. 2.46. 

To transfer the Crank-Nicolson [2.65] implicit difference method, which is always stable, over to cylindrical coordinates requires the discretisation of the equation... 

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 ... 

Numerical methods of solution Hansen (1971) used the orthogonal collocation method to solve both the steady state and transient equations of six different models of the porous particle of increasing complexity. He found that only 8 collocation points were necessary to obtain accurate results. This leads to a considerable saving in computing time compared to the conventional finite difference methods such as the Crank-Nicolson method. 

Numerical Solution. In the numerical formulation of THCC, Equations (2) and (3) are substituted into Equation (1). The resulting set of Nf, partial differential equations is transcribed into Nb finite-difference equations, using central differencing in space and the Crank-Nicolson method to obtain second-order accuracy in time. The set of unknowns consists of i = 1,..., ATft, and Pjt, k = 1,..., A/p, at each finite-difference node. Residue equations for the basis species are formed by algebraically summing all terms in the finite-difference forms of the transport equations. The finite-difference analogs of Equation (1) provide Ni, residue equations at each node the remaining Np residue equations are provided by the solubility products for the reactive solids. 

Another approach involves using implicit methods (28, 30, 31) for obtaining/(y, k + 1) [e.g., the Crank-Nicolson (32), the //y implicit finite difference (FIFD) (33), and the alternating-direction implicit (ADI) (34) methods] rather than the explicit solution in (B.1.9). In implicit methods, the equations for calculation of new concentrations depend upon knowledge of the new (rather than the old) concentrations. There are a number of examples of the use of such implicit methods in electrochemical problems, such as in cyclic voltammetry (35) and SECM (36). 

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 ... 

Note that the backward difference in time method and the Crank-Nicolson method both yield finite difference equations in the form of tridiagonal matrix, but the latter involves computations of three values of y(yi, y/, and y,+, ) of the previous time tj, whereas the former involves only y,. 

Since the time derivative used in the Crank-Nicolson method is second order correct, its step size can be larger and hence more efficient (see Fig. 12.11c). Moreover, like the backward difference method, the Crank-Nicolson is stable in both space and time. 

Another useful point regarding the Crank-Nicolson method is that the forward and backward differences in time are applied successively. To show this, we evaluate the finite difference equation at the ij + 2)th time by using the backward formula that is,... 

Figure 12.11 Plots of versus time for (a) forward difference ib) backward difference (c) Crank-Nicolson.

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.

Solve Problem 12.9 using the Crank-Nicolson method, and show that the final N finite difference equations have the tridiagonal matrix form. Compute the results and discuss the stability of the simulations. 

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]. 

Carter and Husain<"> Ext. fluid + Pore Diffusion Modified Langmuir Crank Nicolson + Forward finite difference 2 + carrier CO2-H2O on 4A sieve... 

Contrasting the forward difference method is the implicit method of Crank-Nicolson [5,9,18,22,25]. The difference formulation for the homogeneous case of Equation 9.90 with zero end conditions is... 

The Crank-Nicolson method is unconditionally stable. However, it does require the solution of simultaneous equations. Also, this method is more accurate than the forward difference method. 

Different meaning of chord AB in approximating d a/ dr in classic, Crank-Nicolson or Laasonen method (lower abscissa scale). 

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