Mitigating numerical dissipation in simulations of subsonic turbulent flows

TL;DR

The study investigates numerical dissipation in subsonic MHD turbulence and demonstrates that the USM-BK relaxation-based scheme substantially reduces energy dissipation and preserves small-scale structures compared with traditional Riemann solvers. Through Balsara vortex tests and high-resolution turbulent-dynamo simulations at Mach numbers $\mathcal{M}=0.1$ and $0.01$, the authors show that USM-BK yields higher effective Reynolds numbers and comparable magnetic Reynolds numbers, with near machine-precision control of $\nabla\cdot\boldsymbol{B}$ when using constrained transport. The results indicate that solver choice strongly affects dynamo growth rates, field structure, and spectral properties, with BK performing best in the deepest low-Mach regimes. However, BK incurs a higher computational cost due to its time-step scaling, motivating exploration of implicit schemes for efficient low-Mach MHD simulations.

Abstract

Magnetohydrodynamic (MHD) simulations of subsonic (Mach number~$<1$) turbulence are crucial to our understanding of several processes including oceanic and atmospheric flows, the amplification of magnetic fields in the early universe, accretion discs, and stratified flows in stars. In this work, we demonstrate that conventional numerical schemes are excessively dissipative in this low-Mach regime. We demonstrate that a new numerical scheme (termed `USM-BK' and implemented in the FLASH MHD code) reduces the dissipation of kinetic and magnetic energy, constrains the divergence of magnetic field to zero close to machine precision, and resolves smaller-scale structure than other, more conventional schemes, and hence, is the most accurate for simulations of low-Mach turbulent flows among the schemes compared in this work. We first compare several numerical schemes/solvers, including Split-Roe, Split-Bouchut, USM-Roe, USM-HLLC, USM-HLLD, and the new USM-BK, on a simple vortex problem. We then compare the schemes/solvers in simulations of the turbulent dynamo and show that the choice of scheme affects the growth rate, saturation level, and viscous and resistive dissipation scale of the dynamo. We also measure the numerical kinematic Reynolds number (Re) and magnetic Reynolds number (Rm) of our otherwise ideal MHD flows, and show that the new USM-BK scheme provides the highest Re and comparable Rm amongst all the schemes compared.

Figures (12)

• Figure 1: Top panel shows the radial profiles of velocity, magnetic field and pressure for the Balsara vortex, following Eqs. (\ref{['eq:BV_vel']})--(\ref{['eq:BV_pres']}) for a sonic Mach number of ${\mathcal{M}}=0.01$ and the ratio of the magnetic to the rotational kinetic energy ${\beta_\mathrm{k}}=1$. Note that the velocity and magnetic pressure profiles have been scaled by a factor of 100 for the sake of clarity. The scaled velocity profile touches the thermal pressure profile ($p_\mathrm{th}\approx1$) at $r=1$ since $\mathcal{M}=0.01$. The bottom panel shows that the centrifugal term $-({\boldsymbol{v}}\cdot\nabla){\boldsymbol{v}}$ is balanced by the magnetic tension $({\boldsymbol{B}}\cdot\nabla){\boldsymbol{B}}$, and the gradients of the thermal pressure ($\nabla p_\mathrm{th}$) and the magnetic pressure ($\nabla p_\mathrm{B}$) balance each other.
• Figure 2: Rotational energy of the vortex after one advection diagonally through the computational domain for the six different numerical solver/scheme combinations (from left to right and top to bottom): Split-Roe, Split-Bouchut, USM-Roe, USM-HLLD, USM-HLLC, and USM-BK. The rotational energy has been normalised by the maximum rotational energy (at $r=1$) at the beginning of the simulation ($t=0$). The value in the top left corner of each panel shows the fraction of the total rotational energy left in the system compared to $t=0$. We see that USM-BK outperforms all other schemes by retaining 88% of the rotational kinetic energy.
• Figure 3: Same as Fig. \ref{['fig:LowMachERot']}, but for the magnetic energy. The value in the top left corner of each panel shows the fraction of the total magnetic energy left in the system after one complete box advection compared to $t=0$. We find that the USM-BK scheme is also the best-performing scheme with respect to the magnetic energy, with only $4\%$ of the initial energy dissipated. The Split schemes dissipate magnetic energy while damping the magnetic monopoles, while the 3-wave USM-HLLC scheme has dissipated almost all of the magnetic energy in the system.
• Figure 4: Same as Fig. \ref{['fig:LowMachERot']}, but for the divergence of the magnetic field, defined in a normalised fashion via Eq. (\ref{['eq:divb']}), such that its magnitude can be compared to order unity. The inset on the top left in each panel shows the root-mean-squared value of $\nabla\cdot{\boldsymbol{\hat{B}}}$. The split schemes keep the value of $\nabla\cdot{\boldsymbol{B}}$ at reasonably low levels while the USM schemes maintain $\nabla\cdot{\boldsymbol{B}}=0$ close to machine precision.
• Figure 5: A slice of the magnetic energy normalised by the mean magnetic energy during the kinematic phase of the dynamo, when $E_\mathrm{mag}/E_\mathrm{kin}=10^{-4}$, emphasising its spatial distribution. The more dissipative Split schemes and USM-HLLC smear features over large-length scales while USM-Roe, USM-HLLD and USM-BK capture finer structures.