  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.
