Axisymmetric Sedov blast wave calculation run with FLASH using the three solvers. All calculations used FLASH 2.5 with the sedov setup (using this version of init_block.F90 that deals with axisymmetric coords properly). This flash.par was used for both calculations. The code was setup with nxb = nyb = 16.
Here we see the density after 0.01 s using the 3 solvers. Note the odd-even decoupling in the piecewise linear unsplit and split cases -- this is a well-known problem for grid aligned shocks. It is not clear why this is more pronounced in the piecewise linear solutions (note, the FLASH PPM solver has a hybrid Riemann solver that can be used to eliminate this, but it was not enabled for these runs).
| PPM | unsplit piecewise linear | split piecewise linear |
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Here we see that the velocity field looks much better in the piecewise linear solvers. All solvers have some trouble at r=0, but the unsplit/split piecewise linear solvers have far lower errors here than the FLASH PPM calculation. As above, we note the odd-even decoupling noise in the piecewise linear solutions.
| PPM | unsplit piecewise linear | split piecewise linear |
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We can also compare the radial averages to the analytic solution from Sedov (1959). Again, we see that the unsplit/split piecewise linear solvers do much better in the velocity field than the FLASH PPM solver. Comparing the split and unsplit piecewise linear solvers, the velocity field in the unsplit case looks slightly better close to the origin.
| PPM | unsplit piecewise linear | split piecewise linear |
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A collection of 1-d shock tube tests, run in 2-d. Each test is run twice, once with the propagation along the x-direction and once with the propagation along the y-direction. A 128x128 zone grid was used. Again, results from all 3 solvers are presented. Each figure shows the x-direction case as blue diamonds and the y-direction case as red pluses, with the analytic solution plotted as a solid line.
The standard Sod problem run for 0.2 s. The main difference between the PPM and piecewise linear solutions is in the contact discontinuity. The PPM algorithm has a contact steepener, which results in a sharper contact discontinuity.
| PPM | unsplit piecewise linear | split piecewise linear |
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Double rarefaction resulting from initial data: ρL = 1.0; uL = -2.0; pL = 0.4; and ρL = 1.0; uL = 2.0; pL = 0.4, shown after 0.15 s. This test creates a vacuum region at the center (with a stationary contact discontinuity). This is test 2 from Toro's text (Chapter 6).
| PPM | unsplit piecewise linear | split piecewise linear |
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Strong shock resulting from initial data: ρL = 1.0; uL = 0.0; pL = 1000.0; and ρL = 1.0; uL = 0.0; pL = 0.01, shown after 0.012 s. This is test 3 from Toro's text (Chapter 6). Again, the main difference between the PPM and piecewise linear results is in the contact discontinuity, due to the PPM contact steepener.
| PPM | unsplit piecewise linear | split piecewise linear |
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Single mode RT instability comparison between the FLASH PPM solver, a 2nd order unsplit compressible solver (Colella 1990) and a dimensionally split 2nd order Godunov scheme. The initial density ratio is 2:1.
S is the L2 symmetry error about the vertical symmetry axis. We notice that the FLASH PPM solver introduces asymmetry into the solution, which increases with resolution. This is a bug in the FLASH PPM solver.
| 32 | 64 | 128 | 256 |
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| S = 0.0 | S = 1.848e-9 | S = 1.807e-8 | S = 1.560e-6 |
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| S = 0.0 | S = 0.0 | S = 0.0 | S = 0.0 |
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| S = 0.0 | S = 0.0 | S = 0.0 | S = 0.0 |