Towards 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 $\Delta x$) 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 ($\rho {\bf g}$) and gravitational energy release ($\rho {\bf v} \cdot {\bf g}$) 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 $\Delta x^4$) 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.