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j D 1; 2:::J using barcode printer for software control to generate, create pdf417 2d barcode image in software applications. .NET CF (20.2.9).

D t (20.2.10) .

x/2 Supplem ented by Dirichlet or Neumann boundary conditions at j D 0 and j D J , equation (20.2.9) is clearly a tridiagonal system, which can easily be solved at each timestep by the method of 2.

4. What is the behavior of (20.2.

8) for very large timesteps The answer is seen most clearly in (20.2.9), in the limit ! 1 ( t ! 1).

Dividing by , we see that the difference equations are just the nite difference form of the equilibrium equation @2 u D0 @x 2 1 1 C 4 sin2 (20.2.11).

where What about stability The a mpli cation factor for equation (20.2.8) is D k x 2 (20.

2.12). Clearly j j < 1 for any pdf417 for None stepsize t. The scheme is unconditionally stable. The details of the small-scale evolution from the initial conditions are obviously inaccurate.

20. Partial Differential Equations for large t. But, as adver Software barcode pdf417 tised, the correct equilibrium solution is obtained. This is the characteristic feature of implicit methods.

Here, on the other hand, is how one gets to the second of our above philosophical answers, combining the stability of an implicit method with the accuracy of a method that is second order in both space and time. Simply form the average of the explicit and implicit FTCS schemes: # " nC1 nC1 nC1 nC1 n n n n uj uj D .uj C1 2uj C uj 1 / C .

uj C1 2uj C uj 1 / D t 2 . x/2 (20.2.

13) Here both the left- and right-hand sides are centered at timestep n C 1 , so the method 2 is second-order accurate in time as claimed. The ampli cation factor is k x 1 2 sin2 2 D (20.2.

14) k x 1 C 2 sin2 2 so the method is stable for any size t. This scheme is called the Crank-Nicolson scheme and is our recommended method for any simple diffusion problem (perhaps supplemented by a few fully implicit steps at the end). (See Figure 20.

2.1.) Now turn to some generalizations of the simple diffusion equation (20.

2.3). Suppose rst that the diffusion coef cient D is not constant, say D D D.

x/. We can adopt either of two strategies. First, we can make an analytic change of variable Z dx (20.

2.15) yD D.x/ Then @ @u @u D D.

x/ @t @x @x becomes @u 1 @2 u D @t D.y/ @y 2 (20.2.

17) (20.2.16).

and we evaluate D at the ap propriate yj . Heuristically, the stability criterion (20.2.

6) in an explicit scheme becomes " # . y/2 (20.2.

18) t min j 2Dj 1 Note that constant spacing y in y does not imply constant spacing in x. An alternative method that does not require analytically tractable forms for D is simply to difference equation (20.2.

16) as it stands, centering everything appropriately. Thus the FTCS method becomes. nC1 uj n uj n Dj C1=2 .uj C1 n uj /. n 1=2 .uj n uj 1 /. . x/2. (20.2.19).

20.2 Diffusive Initial Value Problems t or n FTCS (a) x or j Fully Implicit Crank-Nicolson Figure 20.2.1.

Three differ Software PDF417 encing schemes for diffusive problems (shown as in Figure 20.1.2).

(a) Forward time centered space is rst-order accurate but stable only for suf ciently small timesteps. (b) Fully implicit is stable for arbitrarily large timesteps but is still only rst-order accurate. (c) Crank-Nicolson is second-order accurate and is usually stable for large timesteps.

. where Dj C1=2 D.xj C1=2 / and the heuristic stability criterion is t min (20.2.20).

. x/2 2Dj C1=2 (20.2.21).

The Crank-Nicolson method c an be generalized similarly. The second complication one can consider is a nonlinear diffusion problem, for example where D D D.u/.

Explicit schemes can be generalized in the obvious way. For example, in equation (20.2.

19) write Dj C1=2 D. 1 2 n n D.uj C1 / C D.uj /.

(20.2.22).

Implicit schemes are not as Software PDF-417 2d barcode easy. The replacement (20.2.

22) with n ! n C 1 leaves us with a nasty set of coupled nonlinear equations to solve at each timestep. Often there is an easier way: If the form of D.u/ allows us to integrate dz D D.

u/du (20.2.23).

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