Inaccuracy of Spatial Derivatives in Simulations of Supersonic Turbulence

Abstract

We examine the accuracy of spatial derivatives computed from numerical simulations of supersonic turbulence. Two sets of simulations, carried out using a finite-volume code that evolves the hydrodynamic equations with an approximate Riemann solver and a finite-difference code that solves the Navier-Stokes equations, are tested against a number of criteria based on the continuity equation, including exact results at statistically steady state. We find that the spatial derivatives in the Navier-Stokes runs are accurate and satisfy all the criteria. In particular, they satisfy our exact results that the conditional mean velocity divergence, $\langle \nabla \cdot {\bs u}|s\rangle$, where $s$ is the logarithm of density, and the conditional mean of the advection of $s$, $\langle {\bs u} \cdot \nabla s|s\rangle$, vanish at steady state for all density values, $s$. On the other hand, the Riemann solver simulations fail all the tests that require accurate evaluation of spatial derivatives, resulting in apparent violation of the continuity equation, even if the solver enforces mass conservation. In particular, analysis of the Riemann simulations may lead to the incorrect conclusion that the $\pdv$ work tends to preferentially convert kinetic energy into thermal energy, inconsistent with the exact result that the energy exchange by $\pdv$ work is symmetric in barotropic supersonic turbulence at steady state. The inaccuracy of spatial derivatives is a general problem in the post-processing of simulations of supersonic turbulence with Riemann solvers. Solutions from such simulations must be used with caution in post-processing studies concerning the spatial gradients.

arXiv commentSubmitted to ApJ

References in corpus (16)

• The Statistics of Supersonic Isothermal Turbulence
• On the universality of supersonic turbulence
• Exact relation for correlation functions in compressible isothermal turbulence
• Comparing Numerical Methods for Isothermal Magnetized Supersonic Turbulence
• The conservative cascade of kinetic energy in compressible turbulence
• The Temperature of Interstellar Clouds from Turbulent Heating
• Energy cascade and scaling in supersonic isothermal turbulence
• Mixing in Supersonic Turbulence
• Energy transfer in compressible magnetohydrodynamic turbulence for isothermal self-gravitating fluids
• Dissipative Structures in Supersonic Turbulence
• DISPATCH: A Numerical Simulation Framework for the Exa-scale Era. I. Fundamentals
• 3D Solar Null Point Reconnection MHD Simulations
• Exact relations for energy transfer in self-gravitating isothermal turbulence
• Hydrodynamical simulations of the decay of high-speed molecular turbulence. II. Divergence from isothermality
• Detailed Balance and Exact Results for Density Fluctuations in Supersonic Turbulence
• Alignment of the scalar gradient in evolving magnetic fields

Cited by in corpus (4)

• Evolution of the density PDF in star forming clouds: the role of gravity
• The Probability Distribution of Density Fluctuations in Supersonic Turbulence
• Statistical properties and correlation length in star-forming molecular clouds: I. Formalism and application to observations
• Generalised transport equation of the Autocovariance Function of the density field and mass invariant in star-forming clouds