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320 10. Inviscid Compressible Flow

10.12 Beam-Warming Method for Compressible Euler
Equations

The 1-D compressible Euler equation

has been discretized with the implicit procedure in Section 5.4. Here, the case
ip — £ = 0 is considered, and Eq. (5.4.9) is rewritten as
n
[I + 0At6 xA ]AQ n = -At6 xE (10.12.1)
n
/ is the identity matrix of Eq. (5.4.10), A is the jacobian matrix of Eq. (5.4.8)
and 6 X is the central difference operator.
Explicit artificial dissipation must now be added to stabilize the scheme.
Although one would intuitively add a second derivative to the right hand side
of the algorithm to reproduce the artificial viscosity terms of the Lax scheme,
it has been found that a fourth-derivative term reduces the truncation error
while still providing enough stability for the implicit scheme. With a constant
coefficient, 4th derivative term added to Eq. (10.12.3), we obtain
2
n
[I + 6At6 xA ]AQ n = -At6 xE - e e(VA) xQ n (10.12.2)
where e e is the explicit artificial dissipation constant and where A and V are
the forward and backward difference operators, respectively. Stability analysis
of the model scalar equation shows that the bounds on e e are

e e < 0.125
However, in practice this amount of artificial dissipation is too low and a solution
is found by adding an equivalent damping term on the implicit operator
2
[I + 6At8 xA n - e i(VA) x\AQ n = -AtS xE - e e{VA) xQ n (10.12.3)
where Ei is the implicit artificial dissipation constant. Unfortunately, this im-
plicit matrix is now a block pentadiagonal matrix, which is expensive to invert.
Thus, a second-order operator is preferred and maintains the block tridiago-
nal matrix of the original implicit system, while considerably increasing the
damping effects:
2
[I + 6At6 xA n - Si(VA) x}AQ n = -At6 xE - e e(VA) xQ n (10.12.4)
Since the increase in damping is provided on the left hand side of the equation,
it does not degrade the accuracy of the solution upon reaching steady state
values since the AQ term is then close to zero. Linear stability analysis shows
that for stability one must have

Si > 2£ e
In addition, the explicit constants must be scaled by At to retain dimensional
scaling of the equations.