Unformatted text preview: are shown in Figure 10.3. The mesh used for these computations was 38 Hyperbolic Problems Figure 10.3.4: Densities for the RayleighTaylor instability of Example 10.3.1 at t = 1:8
and p = 0 to 3. The mesh used for all computations is shown at the left.
a uniform bisection of each element of the mesh shown in Figure 10.3 into four elements.
Successive frames in Figure 10.3 show the selected values of p and the density at
t = 0:75, 1.2, and 1.5. The computations show the complex series of bifurcations that
occur at the interface between the two uids.)
Example 10.3.2. Flaherty et al. 16] solve a ow problem for the threedimensional Euler equations (10.3.7) in a tube containing a vent (Figure 10.3) using a piecewiseconstant
discontinuous Galerkin method. A van Leer ux vector splitting (10.2.9  10.2.10) 27]
was used to evaluate uxes. No limiting is necessary with a rstorder method. The main
tube initially had a supersonic ow at a Mach number (ratio of the speed of the uid to
the speed of sound) of 1.23. There was no ow in the vent. At time t = 0 a hypothetical
diaphragm between the main and vent cylinders is ruptured and the ow expands into
the vent. Flaherty et al. citeFLS97 solve this problem using an adaptive hre nement
procedure. They used the magnitude of density jumps across element boundaries as a
re nement indicator. Solutions for the Mach number at t = 0 and 10.1 are shown on the
left of Figure 10.3 for a portion of the problem domain. The mesh used in each each case 10.3. Multidimensional Discontinuous Galerkin Methods 39 Figure 10.3.5: Density for the RayleighTaylor instability of Example 10.1.1 at t = 0:75,
1.2, and 1.5 (left to right) obtained by adaptive pre nement. The values of p used on
each element are shown in the rst, third, and fth frames with blue denoting p = 1 and
red denoting p = 3.
is shown on the right of the gure.
A shock forms on the downwind end of the vent tube and expansion forms on the
upwind end. The mesh is largely concentrated in these regions where the rapid solution
change...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, Constant of integration, Partial differential equation, hyperbolic problems

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