Minimum Numerical Viscosity to Care the Carbuncle Instability

1
Minimum Numerical Viscosity to Care the Carbuncle
Instability
before
after

• Tomoyuki Hanawa (Chiba U.)
• Collaborators Hayato Mikami, Tomoaki Matsumoto

2
Carbuncle Instability
It appears only in 2D 3D.
Originally reported by Peery Imlay (1988)
Supersonic flow around a cylinder
Spurious protuberance ahead of the bow shock.
Fig. 3 of Kim et al. (2003)
3
Condition for Carbuncle Ins.

• When the flow is 2D or 3D.
• No carbuncle in 1D simulation.
• When the numerical viscosity is small.
• A Diffusive scheme is stable.
• When the shock is strong.
• When the shock front is parallel to the cell
  surface.
• When the energy equation is solved.
• Stable when the flow is barotropic.

4
Cause of the Carbuncle Ins.

• Physical instability? No
• Inaccuracy of the approximate Riemann solver?
  No Godunov is also unstable.
• Dependence of mass flux on the pressure? (cf.
  Liou 2000) we doubt
• Numerical viscosity is too small.
• Riemman solution is for 1D not for 2D/3D.
• Nonlinear coupling between waves propagating in
  the x-, y- and z-directions.

5
Quirks strategy
A diffusive scheme is stable but the solutions
are dull.

• To supplement numerical viscosity near the shock
  front to the Roe scheme.
• cf. Kim et al. (2003) for hydrodynamics

• How can we identify shock wave?
• How large viscosity do we supplement?

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Carbuncle Care by Kim et al.
MHD shocks? Gravity? How large viscosity?
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Difference in the Characteristics
?? wave compresssion rate Shock index The other
waves will be compressed also at the same
rate. Extra diffusion is needed.
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Maximum Shock Index
Fast 2 Slow 2
8 Adjacent Cell Surfaces
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Supplementary Viscosity (1)
Roe
Average Viscosity
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Supplementary Viscosity (2)
No change
Fast waves
Alfven and slow waves
Entropy wave
otherwise
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Spherical Expansion Test (Roe)
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Spherical Expansion Test- RoeViscosity-
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Detection of Shock Waves
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Detection of Shock Waves
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Supplementary Viscosity
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Supplementary Viscosity
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Odd-Even Decoupling Test
Shock Front
Zigzagged front
Original Roe
Roe Viscosity
Comparison at 200
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Comparison with HLL on B?
HLL
Diffusion of B in HLL
Rotation Axis
19
Twisted Magnetic Field
6.80 ms
5.98 ms
time
P 2 ms
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Minimum Viscosity?

• We need more examples to evaluate the real
  minimum.
• Our scheme might be unstable.
• We can reduce the viscosity more.

Large Viscosity
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Summary

• MHD Carbuncle instability can be removed by
  supplementary viscosity.
• Spatial Difference in the propagation speed is
  good measure for the supplementary viscosity.
• Only one practical problem has been tested.

We would like to ask you to apply this viscosity
to your problem.