Primitive Variable Determination in Conservative Relativistic Magnetohydrodynamic Simulations

Abstract

Abstract. In nonrelativistic hydrodynamics and magnetohydrodynamics, conservative integration schemes for the fluid equations of motion are generally employed. The computed quantities, namely, the mass density, (vector) momentum density, and energy density, can readily be converted back into the primitive variables that define the problem, namely, the mass density, (vector) velocity, and thermal pressure. In practical terms, the primitive variables can be "peeled away" from the computed variables. In relativistic problems, however, the appearance of the Lorentz factor in the computed quantities dramatically complicates the problem owing to its near-singular dependence upon relativistic velocities. Conservative integration schemes for the hyperbolic partial differential equations of special relativistic magnetohydrodynamics (RMHD) yield estimates of the five conserved quantities that are related in a highly nonlinear way to the five primitive variables. We also observe that an equivalent set of five nonlinear equations emerges in describing the general relativistic magnetohydrodynamics problem. Conversion from the conserved quantities to primitive variables is the most computationally intensive part of the simulation, consuming almost all the computational effort. This paper presents a new algorithm for accurately and rapidly addressing this problem. We provide an analytic representation for this nonlinear system as a single equation in a single unknown. This equation lends itself to an iterative approach, emerging from its underlying physical and analytic properties, one whose convergence is rapid and sufficiently close to geometric that the Aitken acceleration scheme renders the iterations quadratically convergent. The new algorithm enjoys robust convergence properties that render it well-suited to parallel computing architectures. We show how our new scheme facilitates the rapid and accurate computation of solutions to the RMHD equations for moderate Lorentz factors as well as extreme relativistic situations that are observed in high-energy astrophysical objects, where the Lorentz factor can exceed 10 4 and magnetic fields can exceed 10 13 Gauss, and in laboratory plasmas associated with fusion research where very high velocities and magnetic fields are introduced.

On physical-constraints-preserving schemes for special relativistic magnetohydrodynamics with a general equation of state

“…In practice, the values of (ρ, p, v) should be derived from the given value of U by solving some nonlinear algebraic equation, see e.g., [2,12,22,29,31,32]. The present paper considers the following nonlinear algebraic equation (consistent with the one used in Ref.…”

Spectral and imaging properties of Sgr A* from high-resolution 3D GRMHD simulations with radiative cooling

“…HARM2D (Gammie et al 2003;Noble et al 2006) in its early days. It is accelerated by Graphical Processing Units (GPUs) and improved with a staggered grid for constrained transport of magnetic fields (Gardiner & Stone 2005) to preserve ∇ • B = 0, more robust inversion (Newman & Hamlin 2014) adaptive mesh refinement (AMR, not utilised in this work), static mesh refinement (SMR), and a locally adaptive time step (LAT; see Chatterjee et al 2019, Appendix A). It adopts a piece-wise parabolic method (PPM;Colella & Woodward 1984) for reconstruction of cell-centred quantities at cell faces, which is third-order accurate, for the spatial reconstruction at cell faces from cell centres, and a second-order time-stepping.…”

Simulating magnetized neutron stars with discontinuous Galerkin methods

“…We use the recently proposed scheme of Kastaun et al [58]. If this scheme fails to recover the primitives, we try the Newman-Hamlin scheme [59]. If the Newman-Hamlin scheme fails, we use the scheme of Palenzuela et al [60], and if that fails we terminate the simulation.…”

“…A standard test problem for GRMHD codes is the cylindrical blast wave [67,68], where a magnetized fluid initially at rest in a constant magnetic field along the xaxis is evolved. The fluid obeys the ideal fluid equation of state (59) In the region 0.8 ≤ r ≤ 1, the solution transitions continuously and exponentially (i.e., transitions such that the logarithms of the pressure and density are linear functions of r). The fluid begins threaded with a uniform magnetic field with Cartesian components…”