Higher-order accurate space-time schemes for computational astrophysics—Part I: finite volume methods

Abstract

As computational astrophysics comes under pressure to become a precision science, there is an increasing need to move to high accuracy schemes for computational astrophysics. The algorithmic needs of computational astrophysics are indeed very special. The methods need to be robust and preserve the positivity of density and pressure. Relativistic flows should remain sub-luminal. These requirements place additional pressures on a computational astrophysics code, which are usually not felt by a traditional fluid dynamics code. Hence the need for a specialized review. The focus here is on weighted essentially non-oscillatory (WENO) schemes, discontinuous Galerkin (DG) schemes and PNPM schemes. WENO schemes are higher order extensions of traditional second order finite volume schemes. At third order, they are most similar to piecewise parabolic method schemes, which are also included. DG schemes evolve all the moments of the solution, with the result that they are more accurate than WENO schemes. PNPM schemes occupy a compromise position between WENO and DG schemes. They evolve an Nth order spatial polynomial, while reconstructing higher order terms up to Mth order. As a result, the timestep can be larger. Time-dependent astrophysical codes need to be accurate in space and time with the result that the spatial and temporal accuracies must be matched. This is realized with the help of strong stability preserving Runge-Kutta schemes and ADER (Arbitrary DERivative in space and time) schemes, both of which are also described. The emphasis of this review is on computer-implementable ideas, not necessarily on the underlying theory.

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

“…A typical workaround to this problem is to introduce in the scheme some non-linear procedure, whose role is to detect and attempt to control oscillations by locally modifying the solution. This so-called limiting process is recognized as a major challenge for high-order schemes (Qiu & Shu 2004;Balsara 2017). Many limiters have been proposed in the literature, first in the context of finite volume methods, but also specifically for discontinuous Galerkin schemes.…”

“…As a result, DG methods have been the focus of intense research over the last three decades (see e.g. Cockburn et al 2004;Li & Shu 2005;Hesthaven & Warburton 2008;Li et al 2011;Shu 2013;Balsara 2017).…”

Magnetic, thermal and rotational evolution of isolated neutron stars

“…These techniques have been successfully applied to a range of problems, from a simple 1D Burgers equation to complex ideal MHD problems (Antón et al 2006;Giacomazzo and Rezzolla 2007;Cerdá-Durán et al 2008), avoiding the appearance of spurious oscillations near discontinuities. We refer to Martí and Müller (2015) for a general review on grid-based methods and to Balsara (2017) for a review on finite-volume methods, applied to other astrophysical scenarios. Let us review some of the main characteristics of these methods, of particular interest in our problem.…”

“…Other popular higher order reconstructions, are PPM (Colella and Woodward 1984), PHM (Donat and Marquina 1996), MP5 (Suresh and Huynh 1997), the FDOC families (Bona et al 2009), or the Weighted-Essentially-Non-Oscillatory (WENO) reconstructions (Jiang and Shu 1996;Shu 1998;Yamaleev and Carpenter 2009;Balsara 2017). In Viganò et al (2019) they presented and thoroughly tested a two-step method consisting of the reconstruction with WENO methods of a combination of fluxes and fields at each node, known as flux-splitting (Shu 1998).…”

Efficient, divergence-free, high-order MHD on 3D spherical meshes with optimal geodesic meshing

“…Unlike the ADER formulation in Boscheri & Dumbser (2013, 2017 this ADER formulation achieves its simplification because we exploit the time-independence of the Jacobian.…”