Toward Higher-order Accuracy in Self-gravitating Hydrodynamics
High-order algorithms have emerged in numerical astrophysics as a promising avenue to reduce truncation error (proportional to a power of the linear resolution Δ x ) with only a moderate increase to computational expense. Significant effort has been placed in the development of finite-volume algorithms for (magneto)hydrodynamics; however, state-of-the-art astrophysical simulations tightly couple a plenitude of physics, additionally including gravity, photon transport, cosmic-ray transport, chemistry, and/or diffusion, to name a few. Algorithms frequently operator-split this additional physics (often a first-order error in time) and/or adopt a model wherein their evaluation is limited to second-order accuracy in space. In this work, we present a fourth-order-accurate finite-volume scheme for self-gravitating hydrodynamics on a uniform Cartesian grid. The method supplies source terms for the gravitational acceleration ( ρ g ) and gravitational energy release ( ρ v · g ) associated with fourth-order-accurate solutions to the Poisson equation. Our scheme (1) guarantees the conservation of total linear momentum while (2) decreasing (in proportion to Δ x 4 ) the effects of spurious heating and/or cooling associated with truncation error in the gravity. We demonstrate expected convergence rates for the algorithm by measuring errors in test problems evolving self-gravity modified linear waves and 3D polytropic equilibria. We test robustness of the algorithm by integrating an induced “inside-out” adiabatic collapse. We also discuss a method to smoothly downgrade the solution to second-order spatial accuracy to avoid spurious overshoots near steep density and/or pressure gradients.