Astrophysical hydrodynamics with a high-order discontinuous Galerkin scheme and adaptive mesh refinement

Abstract

Solving the Euler equations of ideal hydrodynamics as accurately and efficiently as possible is a key requirement in many astrophysical simulations. It is therefore important to continuously advance the numerical methods implemented in current astrophysical codes, especially also in light of evolving computer technology, which favours certain computational approaches over others. Here we introduce the new adaptive mesh refinement (AMR) code TENET, which employs a high order discontinuous Galerkin (DG) scheme for hydrodynamics. The Euler equations in this method are solved in a weak formulation with a polynomial basis by means of explicit Runge-Kutta time integration and Gauss-Legendre quadrature. This approach offers significant advantages over commonly employed second order finite volume (FV) solvers. In particular, the higher order capability renders it computationally more efficient, in the sense that the same precision can be obtained at significantly less computational cost. Also, the DG scheme inherently conserves angular momentum in regions where no limiting takes place, and it typically produces much smaller numerical diffusion and advection errors than a FV approach. A further advantage lies in a more natural handling of AMR refinement boundaries, where a fall-back to first order can be avoided. Finally, DG requires no wide stencils at high order, and offers an improved data locality and a focus on local computations, which is favourable for current and upcoming highly parallel supercomputers. We describe the formulation and implementation details of our new code, and demonstrate its performance and accuracy with a set of two-and three-dimensional test problems. The results confirm that DG schemes have a high potential for astrophysical applications.

Improved Performances in Subsonic Flows of an SPH Scheme With Gradients Estimated Using an Integral Approach

“…These new schemes are quite numerical complex and, in some cases, their space discretization does not seem to be optimal as required by forthcoming parallel computing systems consisting of several million cores (Schaal et al 2015).…”

High-order magnetohydrodynamics for astrophysics with an adaptive mesh refinement discontinuous Galerkin scheme

“…In the context of decaying isothermal magnetized supersonic turbulence, the code comparison study of Kritsuk et al (2011) has shown that higher-order codes provide a larger turbulence spectral bandwidth and increased effective Reynolds number compared to second-order schemes. Even at second order, recent studies by Mocz et al (2014) and Schaal et al (2015) show that DG methods demonstrate overall superior accuracy compared to finite volume methods, in particular due to lower advection errors and reduced angular momentum diffusion. Beyond second order, Schaal et al (2015) find that, on smooth problems at least, increasing the DG order reduces advection errors, and decreases the total time-to-solution for a given target accuracy.…”

“…Even at second order, recent studies by Mocz et al (2014) and Schaal et al (2015) show that DG methods demonstrate overall superior accuracy compared to finite volume methods, in particular due to lower advection errors and reduced angular momentum diffusion. Beyond second order, Schaal et al (2015) find that, on smooth problems at least, increasing the DG order reduces advection errors, and decreases the total time-to-solution for a given target accuracy.…”

“…These properties make DG schemes attractive for fixed grid Eulerian astrophysical codes, where large structures such as galaxies may travel across the grid with large bulk advection velocities. As a result, there has been significant recent interest in DG methods in astrophysics, for hydrodynamics (Schaal et al 2015;Velasco Romero et al 2018), ideal magnetohydrodynamics (Mocz et al 2014;Dumbser & Loubère 2016;Karami Halashi & Luo 2016), non-ideal MHD (Boscheri & Dumbser 2017), as well as special-and general-relativistic MHD (Zanotti et al 2015;Kidder et al 2017;Anninos et al 2017;Zhao & Tang 2017;Fambri et al 2018).…”

CosmosDG: An hp-adaptive Discontinuous Galerkin Code for Hyper-resolved Relativistic MHD

“…We also implement a bounded version of this limiter in the spirit of and Schaal et al (2015), where the central difference slope is first evaluated against a threshold parameter before applying the minmod operator…”