Crank-Nicolson approximation

The Peaceman-Rachford ADI method is second-order with respect to time, and performs similarly to Crank-Nicolson. Indeed, Lapidus and Finder write [224, p. 246] ... is a variation of the Crank-Nicolson approximation . It is known to be unconditionally stable [225]. As with CN, ADI may show some error oscillations, as also evidenced by the fact that some habitually use expanding time intervals when employing ADI [226-231], although some of these same workers on occasion also use equal time intervals [232,233]. 

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. 

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 ... 

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. 

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]. 

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

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] ... 

To make proper use of the Crank-Nicolson philosophy, according to which the second-order expression (the right-hand side of Eq. 5.85) should be approximated by the mean of the discrete expressions at time T and the new time T+8T (here 0 and need to use the two different X... 

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) ... 

Some comments on the modules provided at the end of this section follow. For VAX/VMS users, the timing routines CPUNUL and CPUOUT may be useful. To compute G from the C-array (Eq. 4.86) with the n-point approximation (point method distribution), use GOFUNC COFUNC does the reverse (Eq. 4.93). For box users, the two equivalent routines GOBOX and COBOX are provided. A standard Crank-Nicolson subroutine is given, for those (rare) cases where good results are obtained with a predicted CJ value, which is one of the transmitted parameters. 

Flanagan and Marcoux [29] were the first to attempt a UMDE time-marching simulation, in order to find the constant in the approximation of Lingane s equation (12.1) they used the explicit method. Crank and Furzeland [40] addressed the steady state for the UMDE and described some of the problems they also briefiy mention time-marching simulations. Their work appears to have come just after that of Evans and Gourlay [101], who used hopscotch. They also found some oscillatory behaviour of the solution, which is not always mentioned. As Gourlay realised [99], hopscotch is mathematically related to ADI, which in turn approximates Crank-Nicolson, known to be oscillatory in response to initial discontinuities such as a potential jump (more on this problem below). 

Bieniasz LK, 0sterby O, Britz D (1995) Numerical stability of finite difference algorithms for electrochemical kinetic simulations. Matrix stability analysis of the classic explicit, firlly implicit and Crank-Nicolson methods, extended to the 3- and 4-point gradient approximation at the electrodes. Comput Chem 19 351-355... 

The basic idea of the finite differences method for solving partial differential equations is to replace spatial and time derivatives by suitable approximations, then to solve numerically the resulting difference equations. In other words, instead of solving for C(x,t) with x and t, it is solved for Cjj = C(Xi, tj), where Xj s iAx, tj = jAt. Thus, the concentrations,, of the diffusing species for the location step, i, and the time step, j -H1, are calculated from the neighbouring concentrations according to the implicit Crank-Nicolson solution for the diffusion differential equation (29.2) ... 

A particularly accurate implicit difference method, which is always stable, has been presented by J. Crank and P. Nicolson [2.65]. In this method the temperatures at the time levels tk and tk+l are used. However the differential equation (2.236) is discretised for the time lying between these two levels tk + At/2. This makes it possible to approximate the derivative (dt>/dt)k+1 2 by means of the accurate central difference quotient... 

The first term omitted in the previous formula contains (At) thus, the formula is of second order correct. This approximation is the famous Crank and Nicolson equation (Crank and Nicolson 1947). 

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