ECHO: a Eulerian conservative high-order scheme for general relativistic magnetohydrodynamics and magnetodynamics

Abstract

Aims. We present a new numerical code, ECHO, based on a Eulerian conservative high-order scheme for time dependent threedimensional general relativistic magnetohydrodynamics (GRMHD) and magnetodynamics (GRMD). ECHO is aimed at providing a shock-capturing conservative method able to work at an arbitrary level of formal accuracy (for smooth flows), where the other existing GRMHD and GRMD schemes yield an overall second order at most. Moreover, our goal is to present a general framework based on the 3 + 1 Eulerian formalism, allowing for different sets of equations and different algorithms and working in a generic space-time metric, so that ECHO may be easily coupled to any solver for Einstein's equations. Methods. Our finite-difference conservative scheme previously developed for special relativistic hydrodynamics and MHD is extended here to the general relativistic case. Various high-order reconstruction methods are implemented and a two-wave approximate Riemann solver is used. The induction equation is treated by adopting the upwind constrained transport (UCT) procedures, appropriate to preserving the divergence-free condition of the magnetic field in shock-capturing methods. The limiting case of magnetodynamics (also known as force-free degenerate electrodynamics) is implemented by simply replacing the fluid velocity with the electromagnetic drift velocity and by neglecting the contribution of matter to the stress tensor. Results. ECHO is particularly accurate, efficient, versatile, and robust. It has been tested against several astrophysical applications, like magnetized accretion onto black holes and constant angular momentum thick disks threaded by toroidal fields. A novel test of the propagation of large-amplitude, circularly polarized Alfvén waves is proposed, and this allows us to prove the spatial and temporal high-order properties of ECHO very accurately. In particular, we show that reconstruction based on a monotonicity-preserving (MP) filter applied to a fixed 5-point stencil gives highly accurate results for smooth solutions, both in flat and curved metric (up to the nominal fifth order), while at the same time providing sharp profiles in tests involving discontinuities.

Electromagnetic counterparts of recoiling black holes: general relativistic simulations of non-Keplerian discs

“…We solve the corresponding equations of general relativistic non-dissipative hydrodynamics through the ECHO code (Del Zanna et al 2007). Because the dynamics of the EM emission takes place on a timescale which is of the order of the orbital one and because the latter is much shorter than the viscous timescale 1 , the use of inviscid hydrodynamics is indeed a very good approximation.…”

“…The set of hydrodynamics equations is discretised in time with the method of lines and the evolution is performed with a second-order modified Euler scheme. A fifth-order finite-difference algorithm based on an upwind monotonicity preserving filter is employed for spatial reconstruction of primitive variables, whereas a two-wave HLL Riemann solver is used to ensure the shock-capturing properties (see Del Zanna et al 2007, for further details). The timestep is generically chosen to be sufficiently small so that the second-order truncation error in time is comparable with the fifth-order one in space.…”

General relativistic magnetohydrodynamics in axisymmetric dynamical spacetimes: the X-ECHO code

“…Finally, in Sect. 2.3 we review the GRMHD equations in 3 + 1 conservative form, as implemented in the original ECHO scheme (Del Zanna et al 2007). …”

“…In the present paper we describe a novel code for GRMHD in dynamical spacetimes, named X-ECHO, aimed at studying the evolution of magnetized relativistic stars and the gravitational collapse of the magnetized rotating cores of massive stars. X-ECHO is built on top of the Eulerian conservative high-order code (Del Zanna et al 2007) for GRMHD in a given and stationary background metric (Cowling approximation), which in turn has upgraded the previous version for a Minkowskian spacetime (Del Zanna & Bucciantini 2002;Del Zanna et al 2003). ECHO relies on robust shock-capturing methods within a finite-difference discretization scheme (two-wave Riemann solvers and limited high-order reconstruction routines), with a staggered constrained-transport method used to preserve the divergence-free condition for the magnetic field (to machine accuracy for second order of spatial accuracy), as proposed by Londrillo & Del Zanna (2000, 2004.…”

Three-dimensional simulations of compressible tearing instability

“…A related version of the code has been designed for special relativistic 33 and general relativistic ideal MHD equations. 34 With respect to previous versions, in the code an explicit plasma resistivity has been introduced and space resolution has been improved by using high-order compact ͑or implicit͒ difference schemes 30 to approximate flux derivatives. Here we briefly summarize the main procedures adopted in the code by focusing on the specific aspects related to the present physical application.…”