A moving frame algorithm for high Mach number hydrodynamics

Abstract

We present a new approach to Eulerian computational fluid dynamics that is designed to work at high Mach numbers encountered in astrophysical hydrodynamic simulations. Standard Eulerian schemes that strictly conserve total energy suffer from the high Mach number problem and proposed solutions to additionally solve the entropy or thermal energy still have their limitations. In our approach, the Eulerian conservation equations are solved in an adaptive frame moving with the fluid where Mach numbers are minimized. The moving frame approach uses a velocity decomposition technique to define local kinetic variables while storing the bulk kinetic components in a smoothed background velocity field that is associated with the grid velocity. Gravitationally induced accelerations are added to the grid, thereby minimizing the spurious heating problem encountered in cold gas flows. Separately tracking local and bulk flow components allows thermodynamic variables to be accurately calculated in both subsonic and supersonic regions. A main feature of the algorithm, that is not possible in previous Eulerian implementations, is the ability to resolve shocks and prevent spurious heating where both the pre-shock and post-shock fluid are supersonic. The hybrid algorithm combines the high-resolution shock capturing ability of the second-order accurate Eulerian TVD scheme with a low-diffusion Lagrangian advection scheme. We have implemented a cosmological code where the hydrodynamic evolution of the baryons is captured using the moving frame algorithm while the gravitational evolution of the collisionless dark matter is tracked using a particle-mesh N-body algorithm. Hydrodynamic and cosmological tests are described and results presented. The current code is fast, memory-friendly, and parallelized for shared-memory machines.