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 Rayleigh-Taylor 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 three-dimensional Euler equations (10.3.7) in a tube containing a vent (Figure 10.3) using a piecewise-constant
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 rst-order 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 h-re 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 Rayleigh-Taylor instability of Example 10.1.1 at t = 0:75,
1.2, and 1.5 (left to right) obtained by adaptive p-re 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
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- Spring '14
- Numerical Analysis, Constant of integration, Partial differential equation, hyperbolic problems