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

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.

A Note on Numerical Fluxes Conserving a Member of Harten's One-Parameter Family of Entropies for the Compressible Euler Equations

“…The semidiscretizations used to compute the spectra above are implemented using Trixi.jl [15,17]. The Jacobian is computed using automatic differentiation [16] in Julia [1].…”

Convergence of Discontinuous Galerkin Schemes for the Euler Equations via Dissipative Weak Solutions

“…With the used SSPRK(10,4), we have not realized any violation of the entropy inequality in the fully discrete setting. All the implementations are done using the Trixi framework [35][36][37]. Trixi is a powerful numerical simulation framework for hyperbolic conservation laws written in Julia and includes all the above mentioned features inside.…”

Adaptive numerical simulations with Trixi.jl: A case study of Julia for scientific computing

“…The package manager Pkg makes it easy to reproduce the exact runtime environment, including binary dependencies, used to generate numerical results. We have used this feature for all of our papers based on Trixi.jl [49,45], including the present manuscript [47]. This facilitates code sharing and reproducible research in computational science, which is arguably important but not yet mainstream [3,12,30].…”

“…Due to Julia's high-level programming approach, dynamic typing, and multiple dispatch, it is easy to combine existing functionality efficiently. For example, the single-physics solvers for hyperbolic PDEs of Trixi.jl were extended to a multi-physics setup for the compressible Euler equations with self-gravity with roughly 350 lines of code [49]. In addition, Trixi.jl can be extended from the outside without modifying the main source code, which makes it easy to set up new simulation approaches and analyze existing ones, e.g., by fluctuation simulations [45].…”