Towards Higher Order Accuracy in Self-Gravitating Hydrodynamics

TL;DR

This paper introduces a fourth order accurate finite volume scheme for self-gravitating hydrodynamics, improving accuracy and conservation properties in astrophysical simulations involving gravity.

Contribution

The work develops a novel fourth order scheme that conserves momentum and reduces truncation errors in gravity calculations, addressing limitations of existing second order methods.

Findings

Demonstrates expected convergence rates in test problems

Ensures conservation of total linear momentum

Reduces spurious heating/cooling effects

Abstract

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.