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10.8 Stability Analysis of the Euler Equations 311
Rloo = Uoc ^ (10.7.2)
7 - 1
For a subsonic inflow, we set R\ and s to their freestream values. i?2 repre-
sents the numerical boundary condition extrapolated from the interior of the
computational domain. The velocity u is found by matching R\ and R2 on the
boundary. For a subsonic outflow, R2 is assigned its freestream value, and s
and R\ are extrapolated from the interior of the computational domain. The
velocity is obtained as discussed above. Application of this procedure is given
in Section 10.11.
Along a solid wall, the impermeable wall boundary condition is imposed
V -n = 0 (10.7.3)
and the density and pressure are extrapolated from the interior. Other choices
may be made for the specification of boundary conditions and the reader is
referred to [1] for more information.
10.8 Stability Analysis of the Euler Equations
The stability analysis of Section 5.7 indicates that for the scalar wave equa-
tion, the maximum time step allowable in the explicit Lax method requires the
CFL condition, CFL < 1. The time step restriction disappears in the implicit
schemes. In this section, we address the stability of the Euler equations.
Since the Euler equations can be transformed into a system of three scalar
equations of the form given by Eq. (5.1.19), the stability bounds for each of the
three equations are thus, for the explicit Lax scheme
u At I Ax < 1 for A = u (10.8.1a)
(u + c)At/Ax < 1 for A = u + c (10.8.1b)
{u - c)At/Ax < 1 for A = u - c (10.5.1c)
It is reasonable, although not mathematically strict, to choose the minimum
time step of the three equations, that is
CFL - (\u\ + c)At/Ax < 1 (10.5.3)
Note that this condition must be satisfied at each computational point, and
the time step must be taken as the smallest one in the computational domain.
Such a system is referred to as being numerically stiff, as there is a wide spread
between the smallest and largest eigenvalues as the flow velocity u departs from
c.
The analysis is more complicated for implicit methods. It can be shown that
implicit schemes have a weak instability at sonic lines, which is damped in the