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Explosion test in 2-D

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The two dimensional Euler equations are solved on a square domain 2x2 in the xy-plane. The initial conditions consists of circular discontinuity of radius 0.4 centered at (1,1). The values inside and outside the circle are given in the table below.

Initial conditions
Variable Inside Outside
Density 1.0 0.125
x velcity 0.0 0.0
y velocity 0.0 0.0
Pressure 1.0 0.1

A typical mesh size that can be used consists of 100x100 cells. It is good to use cell-averaged data as initial conditions. Otherwise the initial data consists of a staircase configuration leading to small amplitude waves at initial times.

The solution exhibits a circular shock wave traveling away from the center, a circular contact surface traveling in the same direction and a circular rarefaction traveling towards the origin (1,1). As time evolves, a complex wave pattern emerges. The circular shock wave travels outwards and becomes weaker. The contact surface follows the shock and also becomes weaker; at some point in time the contact comes to rest and then travels inwards. The rarefaction traveling towards the center reflects, as a rarefaction, and over-expands the flow so as to create an inward traveling shock wave; this circular shock wave implodes into the origin, reflects and travels outwards colliding with the contact surface.

Since the solution has circular symmetry, the 2-D Euler equations can be reduced to 1-D Euler equations in the radial coordinate r,

\frac{\partial U}{\partial t} + \frac{\partial F}{\partial r} = S(U)


\rho \\
\rho u\\
\end{bmatrix}, \quad
F = \begin{bmatrix}
\rho u \\
p + \rho u^2 \\
(E + p)u
\end{bmatrix}, \quad
S = - \frac{1}{r} \begin{bmatrix}
\rho u \\
\rho u^2 \\
(E + p)u

This 1-D problem can be solved using a large number of grid points. The 1-D solution can then be used to assess the accuracy of the 2-D solution.

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