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Noh Problem

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Contact: F.X.Timmes
my one page vitae,
full vitae,
research statement, and
teaching statement.
Code for the analytic solution is released to the public under LA-CC-05-101: noh.f.zip

The Noh problem is a standard verification problem for hydrodynamic and adaptive mesh refinement algorithms. A sphere of gas with a gamma-law equation of state is uniformly compressed, testing the ability to transform kinetic energy into internal energy, and the ability to follow supersonic flows. In the Noh standard problem, a cold gas is initialized with a uniform, radially inward speed of 1 cm s$^{-1}$. A shock forms at the origin and propagates outward as the gas stagnates. For an initial gas density of $\rho_0$=1 g cm$^{-3}$, the analytic solution in spherical geometry for $\gamma$=5/3 predicts a density in the stagnated gas, i.e., after passage of the outward moving shock, of 64 g cm$^{-3}$ .

Most implementations, produce anomalous ``wall-heating'' near the origin, although see Gehmeyr, Cheng, & Mihalas 1997 for a remarkable exception. As the shock forms at the origin the momentum equation tries to establish the correct pressure level. However, numerical dissipation generates entropy. The density near the origin drops below the correct value to compensate for the excess internal energy (e.g., Rider 2000). Thus, the density profile is altered near the origin while the pressure profile remains at the correct constant level in the post-shock region.

In this paper, this paper, and this paper, we discuss the analytic and numerical solutions for the Noh problem.

1D evolutions
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analytical and numerical
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convergence study
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evolution movie


Density uniform mesh in 2D:
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200x200 cells
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400x400 cells
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800x800 cells
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error 200x200 cells
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error 400x400 cells
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error 800x800 cells


Density adaptive mesh in 2D:
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200x200 cells
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400x400 cells
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800x800 cells
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error 200x200 cells
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error 400x400 cells
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error 800x800 cells


Pressure uniform mesh in 2D:
image
200x200 cells
image
400x400 cells
image
800x800 cells
image
error 200x200 cells
image
error 400x400 cells
image
error 800x800 cells


Pressure adaptive mesh in 2D:
image
200x200 cells
image
400x400 cells
image
800x800 cells
image
error 200x200 cells
image
error 400x400 cells
image
error 800x800 cells