Physical Example of Unstable Calculation

Let us take the heat conduction once again

(8.10)

Applying FTCS discretization scheme depict simple explicit representation as

(8.11)

or

where

(8.12)

This is stable only if

Let us consider a case when For r =1 (which is greater than the stability restriction), we get (which is impossible). The values of u are shown in fig 8.1.

Figure 8.1: Physical Violations Resulting from r =1

Example demonstrating the application of Von Neumann method to multidimensional elliptic problems

Let us take the vorticity transport equation:

(8.13)

We shall extend the Von Neumann stability analysis for this equation, assuming u and v as constant coefficient (within the framework of linear stability analysis). Using FTCS scheme

(8.14)

Let us consider with

(8.15)

where N is the numerical solution obtained from computer, D the exact solution of the FDE and ε is the error.

Substituting Eq. (8.14) into Eq. (8.13) and using the trigonometric identities, we finally obtain

where

where

The obvious stability condition finally leads to

and

(8.16)

when

which means

This is twice as restrictive as the one-dimensional diffusive limitation (compare with Eq. (8.9). Again for the special case and

hence

which is also twice as restrictive as one dimensional convective limitation (compare with Eq. (8.8).

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