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10.11 Model Problem for the MacCormack Method: Quasi 1-D Nozzle 315

MacCormack CFL=0.5 MacCormack CFL=0.9
Theory Theory

1000 1250 250 500 750 1000 1250
X

MacCormack CFL=1.1
Theory

250 500 750 1000 1250
X
Fig. 10.14. Shock tube model problem: gas states at time t — 250.

The numerical results may be compared with the theoretical values for sev-
eral CFL numbers (0.5, 1.0 and 1.1) in Fig. 10.14, which shows dispersion er-
rors associated with the truncation error acting on third derivatives at every
CFL number. The scheme is more accurate at higher CFL number, and the
non-linearities allow solutions to be obtained for a CFL number up to 1.1, as
opposed to the linear stability analysis, which limits the CFL number to unity.
However, dispersion errors are unacceptable at CFL values greater than one
and the code diverges as the MacCormack scheme is unable to damp the high-
frequency errors at values greater than 1.1.

t
10.11 Model Problem for he MacCormack Method:
Quasi 1-D Nozzle

This model problem allows the computation of steady state solutions for a wide
variety of test cases. Since analytical one dimensional nozzle flow solutions can