Implosion

References

"Comparison of several difference schemes on 1D and 2D test problems for the Euler equations" by R. Liska & B. Wendroff (available on the web here). Hui et al., JCP 153, 596 (1999)

Description

This problem is initiated by an overpressured region above the line x+y = 0.5. This overpressure sends a shock wave towards the origin. The shock hits the reflecting boundaries at x=0 and y=0, and the resulting reflections create jets that eventually interact with each other at the origin. This interaction then results in the production of another jet along the diagonal, x=y. This jet interacts with the shocks that are being reflected back and forth in the box. This problem is very good at testing the algorithm's ability to maintain symmetry. Throughout the evolution, the fluid must be symmetric across x=y. If this symmetry is not exactly maintained, there will be no jet along x=y. Furthermore, the less numerical diffusion there is, the longer and narrower this jet will be. As a demonstration of this property, three simulations are presented below, each with a different order of spatial reconstruction.

Set up

Domain 0.0 ≤ x ≤ 1.0, 0.0 ≤ y ≤ 1.0 Boundary conditions Reflecting everywhere Equation of state Adiabatic with γ = 1.4 Initial density ρ = 1 for x+y > 0.5 and ρ = 0.125 for x+y ≤ 0.5 Initial pressure P = 1 for x+y > 0.5 and P = 0.14 for x+y ≤ 0.5 Initial velocity Zero everywhere MHD Components There is no MHD version of this test.

Results

1st-order Reconstruction We ran the simulations on a grid of Nx = 600 and Ny = 600. This particular simulation was run with the 1st-order spatial reconstruction (higher order reconstructions are presented below). Plotted below are linear density maps (higher values are redder). The left image is a frame very near the initial conditions (time 0.04), with density ranging from 0.125 to 1.0. The right image is the final frame (time 5.0), with density ranging from 0.4 to 1.1. Click on the right image to see an animation of the density over the whole simulation (size of animation: 16 MB).

2nd-order Reconstruction This simulation has the exact same parameters as the one described above, but it was run with the 2nd-order spatial reconstruction. The left image is again at time 0.04 and has density ranging from 0.125 to 1.0. The right image is again at the final time, 5.0, and has density ranging from 0.4 to 1.1. Click on the right image to see an animation of the density over the whole simulation (size: 17 MB).

3rd-order Reconstruction This simulation has the exact same parameters as the two described above, but it was run with the 3rd-order spatial reconstruction. The left image is again at time 0.04 and has density ranging from 0.125 to 1.0. The right image is again at the final time, 5.0, and has density ranging from 0.4 to 1.1. Click on the right image to see an animation of the density over the whole simulation (size: 19 MB).

Summary In all three cases presented, the algorithm maintains exact symmetry across x=y. In the simulation run with 1st-order spatial reconstruction, no jet forms along x=y. In the other two cases, a jet forms, but the jet is longer and narrower in the 3rd-order spatial reconstruction than in the 2nd-order reconstruction. The density map at time 0.04 looks roughly the same in all three cases, but even at this early time, one can see slight differences in the sharpness of the contact discontinuity.

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