Primitive Variable Determination in Conservative Relativistic Magnetohydrodynamic Simulations

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.