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. 

The Crank-Nicolson method is popular as a time-step scheme for CFD problems, as it is stable and computationally less expensive than the implicit Euler scheme. 

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... 

More basically, LB with its collision rules is intrinsically simpler than most FV schemes, since the LB equation is a fully explicit first-order discretized scheme (though second-order accurate in space and time), while temporal discretization in FV often exploits the Crank-Nicolson or some other mixed (i.e., implicit) scheme (see, e.g., Patankar, 1980) and the numerical accuracy in FV provided by first-order approximations is usually insufficient (Abbott and Basco, 1989). Note that fully explicit means that the value of any variable at a particular moment in time is calculated from the values of variables at the previous moment in time only this calculation is much simpler than that with any other implicit scheme. 

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. 

Crank-Nicolson implicit method This method is a little more complicated but it offers high precision and unconditional stability. Let... 

Implicit (Crank-Nicolson) An average of conditions in the previous and next time step are used to predict the change in conditions for the next time step. Often requires iteration. 

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). 

The system of Eqs. 13.3-17 and 13.3-18 can be solved for the adiabatic, isothermal, or constant wall flux cases using the Crank-Nicolson method. The thermomechanical and reaction data for such systems were evaluated by Lifsitz, Macosko, and Mussatti (99) at 45°C for a polyester triol and a chain extended 1,6-hexamethylene diisocyanate (HDI) with dibutyltin as a catalyst. Figure 13.46 gives the temperature profiles for the isothermal-wall case. Because of the high heat of polyurethane formation and the low conductivity of... 

Implicit methods have the great advantage of being stable, in contrast with the explicit method. It will be seen (and analysed in detail in Chap. 14) that the Laasonen method, a kind of BI, is very stable and responds to sharp transients with smoothly declining (but relatively large) errors, whereas Crank-Nicolson, also nominally stable, responds with error oscillations of declining amplitude, but is highly accurate. The drawbacks of both methods can be overcome, as will be described below. 

Crank-Nicolson bears the name of its inventors [185], It is interesting to note that in their paper, they cite Hartree and Womersley [296], who describe what amounts to its precursor. 

There are drawbacks, however. It is clear from the above computational molecules, that the second, spatial derivative is approximated in an asymmetric manner, and although these approximations are in fact second-order with respect to the interval H, they are not as good as, say, the Crank-Nicolson ones. Both LR and RL, taken by themselves, do not produce very good results. It was not long after Saul yev s book in 1964, that Larkin (in the same year) published some extensions, as did other workers [223,367,368]. The asymmetry of each of the two variants suggests combining them in some manner. Larkin [352] listed four strategies ... 

Electro chemists first investigated the Saul yev method in 1988 and 1989 [381,382], including GEM, and the incorporation of implicit boundary values was added later [144]. The result of these studies is broadly that the last of Larkin s options above, averaging LR and RL, is the best. This has about the same accuracy as Crank-Nicolson, and could be considered to be easier to program. The third option, alternating LR with RL, produces oscillations. 

The enthusiasm for hopscotch arose from the fact that here was a method with an accuracy thought to be almost comparable with that of Crank-Nicolson, but which was an explicit computation at every step, not requiring the solution of linear systems of equations, as other implicit methods do. It was also stable for all A, thus making it possible to use larger time steps, for example. The convenience of the point-by-point calculation has occasionally led workers to call the method fast [235],... 

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. 

This program does the same work as the earlier one, C0TT EX, but uses Crank-Nicolson (with equal intervals). It also includes the choice of M Pearson substeps within the first step, to damp the oscillations, as discussed in Chap. 8, Sect. 8.5.1. 

Osterby O., The error of the Crank-Nicolson method for linear parabolic equations with a derivative boundary condition, Report PB-534, DAIMI, Aarhus University (1998)... 

Osterby O., Five ways of reducing the Crank-Nicolson oscillations, Tech. Rep. Daimi PB-558, Dept, of Computer Science, Aarhus University (2002)... 

Two-dimensional Model. The same strategy has been used for a two-dimensional model. The mass and energy balances Eqs. 6-8 and Eqs. 10-13,have been integrated by a Crank-Nicolson procedure. After completing the calculation a new distribution of activity is evaluated from Eq. 9 by an explicit Euler integration. 

The Crank-Nicolson method is a mixture of the implicit and explicit schemes which is less stable but more accurate than the fully implicit scheme (see Britz, 1988). A set of simultaneous equations analogous to those necessary for the fully implicit method must be solved the matrix [Af] has exactly the same structure in either case. 

II = parallel to electrode surface 1 = towards electrode surface CN = time-marching Crank-Nicolson MGRID = Multigrid Bfs space marching fully implicit or Crank-Nicolson BI-FIFD = a space-marching FIFO method modelling methods in parentheses have not yet been applied to that particular geometry. 

CN Crank-Nicolson (method for solution of differential equations)... 

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