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.

翻译：模拟具有自重力的天体物理流时, 挑战之一是计算引力。 与超偏流流体动力等方程相比, 重力场用椭圆皮质方程描述。 我们提出纯粹的双曲法方法, 将椭圆性问题改造成双曲扩散问题, 以我们用于流解解决方案的相同的高阶不连续 Galerkin 方法在假时解决。 流和重力解解解器通过一个联合等级的Cartesian网格运行, 并通过源术语双向连接。 我们方法的一个重要好处是, 它允许在不修改的情况下重新使用现有的明确的超双曲解决器, 同时保留其先进的特性, 如不相容和解决方案适应电网。 通过在流流解解解的每个运行- Kutta 阶段更新引力场, 即使在同时进行多物理模拟后, 也实现了高度的趋同。 在核实单物理和多物理学的双向精确度组合组合后, 我们通过模拟模拟了我们流压的自动沉积的自我调整方法, 验证了我们的数据模拟了我们的数据震变变的自我。