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 propagating 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. PjPj P j+1 MHD shocks? Gravity? How large viscosity?

Difference in the Characteristics Δλ: wave compresssion rate Shock index The other waves will be compressed also at the same rate. Extra diffusion is needed.

Maximum Shock Index Fast × 2 + Slow × 2 8 Adjacent Cell Surfaces

Supplementary Viscosity (1) Roe Average Viscosity

Supplementary Viscosity (2) Fast waves No change Alfven and slow waves Entropy wave otherwise

Spherical Expansion Test (Roe)

Spherical Expansion Test - Roe+Viscosity-

Detection of Shock Waves

Supplementary Viscosity

Odd-Even Decoupling Test Shock Front Original Roe Roe + Viscosity Zigzagged front Comparison at #200

Comparison with HLL on B ⊥ HLL Diffusion of B in HLL Rotation Axis

Twisted Magnetic Field time 6.80 ms5.98 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 Roe This 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.