A primer on Eulerian computational fluid dynamics for astrophysics

We present a pedagogical review of some of the methods employed in Eulerian computational fluid dynamics (CFD). Fluid mechanics is governed by the Euler equations, which are conservation laws for mass, momentum, and energy. The standard approach to Eulerian CFD is to divide space into finite volumes or cells and store the cell-averaged values of conserved hydro quantities. The integral Euler equations are then solved by computing the flux of the mass, momentum, and energy across cell boundaries. We review both first-order and second-order flux assignment schemes. All linear schemes are either dispersive or diffusive. The nonlinear, second-order accurate total variation diminishing (TVD) approach provides high-resolution capturing of shocks and prevents unphysical oscillations. We review the relaxing TVD scheme, a simple and robust method to solve systems of conservation laws such as the Euler equations. A three-dimensional relaxing TVD code is applied to the Sedov-Taylor blast-wave test. The propagation of the blast wave is accurately captured and the shock front is sharply resolved. We apply a three-dimensional self-gravitating hydro code to simulating the formation of blue straggler stars through stellar mergers and present some numerical results. A sample three-dimensional relaxing TVD code is provided in the Appendix.