A purely hyperbolic discontinuous Galerkin approach for self-gravitating gas dynamics

TL;DR

This paper introduces a novel hyperbolic reformulation of the elliptic gravity problem in self-gravitating gas dynamics, enabling the use of explicit hyperbolic solvers for efficient, high-order coupled simulations.

Contribution

It presents a purely hyperbolic discontinuous Galerkin method that reformulates gravity as a hyperbolic diffusion problem, allowing reuse of existing solvers in astrophysical flow simulations.

Findings

Achieves high-order convergence in coupled multi-physics simulations.

Successfully simulates gravitational instability and shock phenomena.

Demonstrates adaptive mesh refinement in self-gravitating systems.

Abstract

One of the challenges when simulating astrophysical flows with self-gravity is to compute the gravitational forces. In contrast to the hyperbolic hydrodynamic equations, the gravity field is described by an elliptic Poisson equation. We present a purely hyperbolic approach by reformulating the elliptic problem into a hyperbolic diffusion problem, which is solved in pseudotime using the same explicit high-order discontinuous Galerkin method we use for the flow solution. The flow and the gravity solvers operate on a joint hierarchical Cartesian mesh and are two-way coupled via the source terms. A key benefit of our approach is that it allows the reuse of existing explicit hyperbolic solvers without modifications, while retaining their advanced features such as non-conforming and solution-adaptive grids. By updating the gravitational field in each Runge-Kutta stage of the hydrodynamics solver, high-order convergence is achieved even in coupled multi-physics simulations. After verifying the expected order of convergence for single-physics and multi-physics setups, we validate our approach by a simulation of the Jeans gravitational instability. Furthermore, we demonstrate the full capabilities of our numerical framework by computing a self-gravitating Sedov blast with shock capturing in the flow solver and adaptive mesh refinement for the entire coupled system.