ECHO: a Eulerian conservative high-order scheme for general relativistic magnetohydrodynamics and magnetodynamics. We present a new numerical code, ECHO, based on an Eulerian Conservative High Order scheme for time dependent three-dimensional 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, different algorithms, and working in a generic space-time metric, so that ECHO may be easily coupled to any solver for Einstein’s equations. 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 preserve 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 matter contribution to the stress tensor. ECHO is particularly accurate, efficient, versatile, and robust. It has been tested against several astrophysical applications, including a novel test on the propagation of large amplitude circularly polarized Alfven waves. In particular, we show that reconstruction based on a Monotonicity Preserving 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.

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  1. Cipolletta, F.; Kalinani, J. V.; Giangrandi, E.; Giacomazzo, B.; Ciolfi, R.; Sala, L.; Giudici, B.: Spritz: general relativistic magnetohydrodynamics with neutrinos (2021)
  2. Gaburro, Elena; Dumbser, Michael: A posteriori subcell finite volume limiter for general (P_NP_M) schemes: applications from gasdynamics to relativistic magnetohydrodynamics (2021)
  3. Wu, Kailiang; Shu, Chi-Wang: Provably physical-constraint-preserving discontinuous Galerkin methods for multidimensional relativistic MHD equations (2021)
  4. Cipolletta, F.; Kalinani, J. V.; Giacomazzo, B.; Ciolfi, R.: Spritz: a new fully general-relativistic magnetohydrodynamic code (2020)
  5. Ha, Cong-Tu; Lee, Jae Hwa: A modified monotonicity-preserving high-order scheme with application to computation of multi-phase flows (2020)
  6. Reyes, Adam; Lee, Dongwook; Graziani, Carlo; Tzeferacos, Petros: A variable high-order shock-capturing finite difference method with GP-WENO (2019)
  7. Shahbazi, Khosro: High-order finite difference scheme for compressible multi-component flow computations (2019)
  8. Dumbser, Michael; Fambri, Francesco; Tavelli, Maurizio; Bader, Michael; Weinzierl, Tobias: Efficient implementation of ADER discontinuous Galerkin schemes for a scalable hyperbolic PDE engine (2018)
  9. Felker, Kyle Gerard; Stone, James M.: A fourth-order accurate finite volume method for ideal MHD via upwind constrained transport (2018)
  10. Kidder, Lawrence E.; Field, Scott E.; Foucart, Francois; Schnetter, Erik; Teukolsky, Saul A.; Bohn, Andy; Deppe, Nils; Diener, Peter; Hébert, François; Lippuner, Jonas; Miller, Jonah; Ott, Christian D.; Scheel, Mark A.; Vincent, Trevor: SpECTRE: A task-based discontinuous Galerkin code for relativistic astrophysics (2017)
  11. Wu, Kailiang; Tang, Huazhong: Admissible states and physical-constraints-preserving schemes for relativistic magnetohydrodynamic equations (2017)
  12. Balsara, Dinshaw S.; Amano, Takanobu; Garain, Sudip; Kim, Jinho: A high-order relativistic two-fluid electrodynamic scheme with consistent reconstruction of electromagnetic fields and a multidimensional Riemann solver for electromagnetism (2016)
  13. Balsara, Dinshaw S.; Kim, Jinho: A subluminal relativistic magnetohydrodynamics scheme with ADER-WENO predictor and multidimensional Riemann solver-based corrector (2016)
  14. Chen, Yuxi; Tóth, Gábor; Gombosi, Tamas I.: A fifth-order finite difference scheme for hyperbolic equations on block-adaptive curvilinear grids (2016)
  15. Dumbser, Michael; Balsara, Dinshaw S.: A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems (2016)
  16. Goetz, Claus R.; Dumbser, Michael: A novel solver for the generalized Riemann problem based on a simplified LeFloch-Raviart expansion and a local space-time discontinuous Galerkin formulation (2016)
  17. Meliani, Z.; Grandclément, P.; Casse, F.; Vincent, F. H.; Straub, O.; Dauvergne, F.: GR-AMRVAC code applications: accretion onto compact objects, boson stars versus black holes (2016)
  18. Amano, Takanobu: Divergence-free approximate Riemann solver for the quasi-neutral two-fluid plasma model (2015)
  19. Kim, Jinho; Balsara, Dinshaw S.: A stable HLLC Riemann solver for relativistic magnetohydrodynamics (2014)
  20. Núñez, Manuel: On the one-dimensional relativistic induction equation (2014)

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