Some Non-Trivial Problems with Discretized Equation

The discussion in this section is based upon some ideas indicated by Hirt (1968) which are applied to model Burger's equation as

(13.4)

From this, the modified equation becomes

(13.5)

We define

Courant number

It is interesting to note that the values and C=1 (which are extreme conditions of Von Neumann stability analysis) unfortunately eliminates viscous diffusion completely in Eq. (13.5) and produce a solution from Eq. (13.4) directly as which is unacceptable. From Eq. (13.5) it is clear that in order to obtain a solution for convection diffusion equation, we should have

For meaningful physical result in the case of inviscid flow we require

Combining these two criteria, for a meaningful solution

(13.6)

Here we define the mesh Reynolds-number or cell-peclet number as

So, we get

or

(13.7)

Figure 13.1: Limiting Line ( )

The plot of C vs is shown in Fig. 13.1 to describe the significance of Eq. (13.7).

From the CFL condition, we know that the stability requirement is Under such a restriction, below the calculation is always stable. The interesting information is that it is possible to cross the cell Reynolds number of 2 if C is made less than unity.