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312 10. Inviscid Compressible Flow
Lax-Wendroff and MacCormack explicit schemes briefly discussed in Sections
5.2 and 5.3. It then becomes necessary to add explicit artificial dissipation to
the implicit methods of Section 5.4. The amount of dissipation must be enough
to damp the instabilities and, in addition, to prevent oscillations around shock
waves. The model problem of Section 10.13 will show how explicit artificial
dissipation is introduced into the formulation of implicit schemes.
10.9 MacCormack Method for Compressible Euler Equations
In Section 5.3 we discussed the explicit MacCormack method for a one-
dimensional problem and here we discuss its extension to compressible flows.
The Euler equations for one-dimensional flow are considered and from Eq.
(2.1.30),
with Q and E given by Eqs. (2.2.32a) for one-dimensional flow.
/ )
The discussion of Section 5.3 is followed, the predictor values at {xi, £ n + 1 2
is defined by Q™ ' (= Qi) and the convective flux term E represented with
forward differences in the predictor step and backward differences in the cor-
rector step. Thus
predictor:
Qi = Qf - At [ E?+1 Ax E? ) (10-9.2)
corrector:
Q i Q?-At(%-^A (10.9.3)
=
updating
Qi +x = \(Qi + Qi) (10-9-4)
Note that the predictor step could have been obtained with backward differ-
ences, while the corrector step with forward differences as in Eqs. (5.3.3), that
is
predictor:
i 1
' Ax ~ ) (10.9.5)
corrector:
Ei
Qi = Q? - At { Ei+l Ax ) (10-9.6)
Updating is as described by Eq. (10.9.4). It can be shown that the two proce-
dures are identical for linear equations, while for non-linear equations they are