A Godunov-Type Solver for the Numerical Approximation of Gravitational Flows

Abstract

Abstract. We present a new numerical method to approximate the solutions of an Euler-Poisson model, which is inherent to astrophysical flows where gravity plays an important role. We propose a discretization of gravity which ensures adequate coupling of the Poisson and Euler equations, paying particular attention to the gravity source term involved in the latter equations. In order to approximate this source term, its discretization is introduced into the approximate Riemann solver used for the Euler equations. A relaxation scheme is involved and its robustness is established. The method has been implemented in the software HERACLES [29] and several numerical experiments involving gravitational flows for astrophysics highlight the scheme.

A relaxation scheme for inviscid flows under gravitational influence

“…As mentioned in the previous section, particular attention must be given to the proper discretization of the gravity terms. After the work of Jin-Xin [6] and Bouchut [1], Vides et al [17] proposed to approximate the weak solutions of (8) by the weak solutions of a relaxation system, designed to preserve most of the nonlinearities of the relaxation equilibrium system and in this way, enforce accuracy of the resulting numerical scheme. In the following, we recall some of the main ideas described in [17].…”

“…It is also evident that the choice of a plays an important role in the robustness of the scheme, as will be seen in Section 4. In addition, a relaxation procedure to approximate the potential φ is proposed in [17], but its derivation is not given therein. Here, we present it in more detail.…”

“…Now, in order to approximate equation (5), which is in nonconservative form, Vides et al [17] introduce the discretization of the gravity source term into the approximate Riemann solver for the Euler equations by means of a relaxation technique, a classical numerical approach widely spread across the literature considering fluid flow simulations (see [1,6]). In this paper, we present the complete derivation of their one-dimensional relaxation model.…”

“…In view of numerical simulations, the proper treatment of gravitational effects and the need to preserve the equilibrium states of these flows, present two major challenges. The authors in [17], use an operator splitting approach, and in the context of the first-order decomposition, solve system (2) in two steps. First, being W ∈ Ω d an unknown state vector and φ a given gravitational potential, the following nonlinear hyperbolic system is solved:…”

“…Several gravitational flows reach equilibrium states that are necessary to preserve in the numerical formulation. In this paper, we present the derivation of the relaxation model [17], in which the pressure is a supplementary variable and the Poisson equation is transformed into a hyperbolic equation with a penalty parameter. The corresponding scheme is obtained in the limit as the parameter tends to zero.…”