Numerical viscosity and resistivity in MHD turbulence simulations

Abstract

Accurate magnetohydrodynamical (MHD) turbulence simulations require understanding numerical dissipation. We quantify numerical viscosity and resistivity in subsonic (M=0.1) and supersonic (M=10) turbulence regimes. The hydrodynamic (Re) and magnetic Reynolds numbers (Rm) on the turbulence driving scale l_turb in a cubic domain of side length L with N^3 resolution elements are well-described by Re=[2(N/N_Re)(l_turb/L)]^p_Re and Rm=[2(N/N_Rm)(l_turb/L)]^p_Rm. We provide two sets of fit values of (N_Re,p_Re,N_Rm,p_Rm): one with p_Re & p_Rm fixed at their theoretical values, and the other one allowing all 4 parameters to vary. The sets for M=0.1 are (1.57_{-0.12}^{+0.10},4/3,1.55_{-0.14}^{+0.45},4/3) and (0.83_{-0.08}^{+0.09},1.20_{-0.02}^{+0.02},4.19_{-4.05}^{+2.95},1.60_{-0.33}^{+0.18}), respectively. For M=10, they are (3.55_{-0.56}^{+0.78},3/2,1.03_{-0.11}^{+0.12},3/2) and (10.46_{-0.85}^{+0.96},1.90_{-0.04}^{+0.04},0.44_{-0.23}^{+0.61},1.32_{-0.09}^{+0.17}). The resulting magnetic Prandtl numbers (Pm=Rm/Re) are consistent with constant values of 1.0_{-0.2}^{+0.3} for M=0.1, and 6.2_{-4.8}^{+5.6} for M=10. These apply when the magnetic energy (E_mag) is <10% of the kinetic energy (E_kin). When E_mag/E_kin~0.1-1, Rm is reduced by a factor~3 (increase in N_Rm by a factor~2) for M=0.1, while Rm for M=10 and Re (for any M) remain largely unaffected. We compare our Re-N relation with 14 other simulations from the literature, employing various numerical methods (with & without Riemann solvers, different reconstruction schemes & orders, and smoothed particle hydrodynamics), and find agreement within a factor of 3. Additionally, we compare these results to target Re and Rm values from simulations with explicit dissipation. These comparisons and our relations help users determine the Re and Rm achievable at a given N, ensuring physical dissipation dominates over numerical dissipation.