Characteristic Decomposition for Relativistic Numerical Simulations: I. Hydrodynamics

TL;DR

This work addresses the long-standing challenge of obtaining a practical characteristic decomposition for relativistic hydrodynamics in a form usable by conservation-based numerical codes. It introduces quasi-invertible transformations to move from a comoving, Lagrangian frame where the decomposition is simple to Eulerian and then to the conserved-variable formulation used in simulations. The authors derive closed-form characteristic speeds and complete sets of right and left eigenvectors in the comoving, Eulerian, and conserved representations, including extensions to fluid composition (Y_e). The approach enables the use of full-wave Riemann solvers for relativistic hydrodynamics and sets the stage for the GRMHD decomposition in Paper II, with significant implications for accuracy in shock treatment and boundary conditions in numerical relativity and astrophysics.

Abstract

The characteristic decomposition for GRMHD is not known in a form useful for current numerical simulations. This prevents us from using the most accurate known computational methods, such as full-wave Riemann solvers. In this paper, we present a new method of finding decompositions. The method is based on transformations from the comoving frame, where the fluid flow is simplest and the decomposition has been known for a long time. The key innovation we introduce is that of quasi-invertible transformations. In this first paper, we introduce these transformations using the simpler example of relativistic hydrodynamics. We recover the known decomposition for relativistic hydrodynamics in somewhat simpler form than previously derived, and without the need for computer algebra. A new result in this paper is the characteristic decomposition when the the evolution tracks the composition of a fluid in nuclear statistical equilibrium. In Paper II of this series, we apply a quasi-invertible transformation to derive the complete characteristic decomposition for GRMHD in the conserved variables used in simulations.