Minimum Numerical Viscosity to Care the Carbuncle Instability
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Transcript of Minimum Numerical Viscosity to Care the Carbuncle Instability
Minimum Numerical Viscosity to Care the Carbuncle Instability
Tomoyuki Hanawa (Chiba U.)Collaborators: Hayato Mikami, Tomoaki Matsumoto
before after
Carbuncle Instability
Originally reported by Peery & Imlay (1988)
Fig. 3 of Kim et al. (2003)
Spurious protuberance ahead of the bow shock.
It appears only in 2D & 3D.
Supersonic flow around a cylinder
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.
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 propagatin
g in the x-, y- and z-directions.
Quirk’s strategy
• To supplement numerical viscosity near the shock front to the Roe scheme.– cf. Kim et al. (2003) for hydrodynamics
A diffusive scheme is stable but the solutions are dull.
• How can we identify shock wave?
• How large viscosity do we supplement?
Carbuncle Care by Kim et al.PP /strengthshock
Pj Pj+1
1
1
,max
,min1
jj
jj
PP
PP
P
P
P
P
MHD shocks?
Gravity?
How large viscosity?
Difference in the Characteristics
0,max 1 jj Δλ: wave compresssion rate
Shock index
The other waves will be compressed also at the same rate.
Extra diffusion is needed.
Maximum Shock Index
-slow,,2/1,
slow,,2/1,
fast,,2/1,
fast,,2/1,max
,,2/1,,
,,max
kjixkjix
kjixkjix
kjix
Fast × 2 + Slow × 2
8 Adjacent Cell Surfaces
-slow
2/1,,,2/1,,1,
,2/1,,,2/1,1,max,,2/1
,
,,max
kjizkjiz
kjiykjiy
kji
Supplementary Viscosity (1)
mmmkjixkjixkjix δw mrFFF ||
2
1,,1,,,,,,2/1,
Roe Average Viscosity
kjixkjixm
mδw ,,,,,1, UUrm Urm ||||
mmm δw
mmmkjixkjixkjix δw mrFFF ,,1,,,,,,2/1, 2
1
Supplementary Viscosity (2)
Fast waves || mm No change
max,||max mmAlfven and slow waves
71
714714
Entropy wave 0if max
|| 44 otherwise
viscosity.arysupplementno,0When max
Spherical Expansion Test (Roe)
Spherical Expansion Test- Roe+Viscosity-
Detection of Shock Waves
Detection of Shock Waves
0dx
d
Supplementary Viscosity
Supplementary Viscosity
Odd-Even Decoupling TestShock Front
Original Roe
Roe + Viscosity
....,6,4,2for
...,5,3,1for
0
0sh
jx
jxxx
Zigzagged front
Comparison at #200
Comparison with HLL on B⊥
HLL
Diffusion of B in HLL Rotation Axis
Twisted Magnetic Field
time
6.80 ms 5.98 ms
P = 2 ms
Minimum Viscosity?• We need more examples to evaluate the
real minimum.
• Our scheme might be unstable.
• We can reduce the viscosity more.
Large Viscosity
RoeThis work
Small Viscosity
HLL
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.