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.