[Paper Review] General Relativistic Hydrodynamics with Special Relativistic Riemann Solvers

J. A. Pons, José A. Font+3 more|arXiv (Cornell University)|Jul 21, 1998

Scientific Research and Discoveries8 references46 citations

TL;DR

This paper presents a novel, computationally efficient method to solve general relativistic hydrodynamics (GRH) by locally transforming spacetime coordinates to a Minkowski-like frame at each numerical interface, enabling the direct use of existing special relativistic Riemann solvers (SRRS). The approach achieves numerical accuracy comparable to previous GRH schemes while allowing researchers to leverage advanced SRRS without developing new solvers, with validation through multidimensional simulations of black hole accretion.

ABSTRACT

We present a general and practical procedure to solve the general relativistic hydrodynamic equations by using any of the special relativistic Riemann solvers recently developed for describing the evolution of special relativistic flows. Our proposal relies on a local change of coordinates in terms of which the spacetime metric is locally Minkowskian and permits accurate numerical calculations of general relativistic hydrodynamics problems using the numerical tools developed for the special relativistic case with negligible computational cost. The feasibility of the method has been confirmed by a number of numerical experiments.

Motivation & Objective

• To develop a general and practical method for solving general relativistic hydrodynamics (GRH) using existing special relativistic Riemann solvers (SRRS).
• To overcome the limitation of prior GRH schemes that relied only on linearized Riemann solvers, which are less accurate and robust.
• To enable researchers in special relativistic hydrodynamics to easily transition into GRH by reusing their existing numerical tools.
• To validate the method through multidimensional simulations of complex astrophysical flows, such as non-spherical black hole accretion.
• To provide a framework applicable beyond hydrodynamics, including to general relativistic magnetohydrodynamics (GRMHD).

Proposed method

• The method performs a local change of coordinates at each numerical interface such that the spacetime metric becomes locally Minkowskian, effectively transforming the GRH equations into a form solvable by SRRS.
• At each interface, the fluid states are transformed into the local inertial frame, where the Riemann problem is solved using any available SRRS (e.g., Roe-like, HLL, Marquina’s solver).
• The resulting numerical fluxes from the SRRS are then transformed back into the original coordinate system for updating the conservative variables in the GRH scheme.
• The coordinate transformation is linear and involves only a few arithmetic operations per interface, ensuring negligible computational cost.
• The approach preserves the conservative and high-resolution shock-capturing (HRSC) properties of the original SRRS, ensuring stability and accuracy in discontinuous flow regions.
• The method is fully compatible with the 3+1 formalism of general relativity and can be implemented in existing HRSC GRH codes with minimal modifications.

Experimental results

Research questions

• RQ1 Can existing special relativistic Riemann solvers be effectively applied to general relativistic hydrodynamics without significant modification?
• RQ2 Does a local coordinate transformation to a Minkowski-like frame preserve the accuracy and stability of the Riemann solver in curved spacetime?
• RQ3 How does the performance of this method compare to previous GRH schemes that use linearized Riemann solvers?
• RQ4 Can this approach be generalized to other systems of conservation laws, such as general relativistic magnetohydrodynamics?
• RQ5 What is the computational cost of the local transformation, and is it negligible in practice?

Key findings

• The method successfully enables the use of any special relativistic Riemann solver—whether approximate (e.g., Roe-like, HLL) or exact—for general relativistic hydrodynamics with negligible computational overhead.
• Numerical simulations of non-spherical black hole accretion show that results obtained with the new method are indistinguishable from those produced by previous, more complex GRH schemes, confirming consistency and accuracy.
• The results obtained with different SRRS (Roe-like, HLL, Marquina’s) are remarkably similar, with minor differences attributable only to the intrinsic numerical properties of each solver, such as diffusion and oscillation levels.
• The method is analytically equivalent to the previous approach using linearized Riemann solvers, validating its theoretical soundness.
• The approach is robust and stable, maintaining high accuracy in regions with strong shocks and high Lorentz factors, as demonstrated in 2D simulations of accretion flows.
• The computational cost of the local coordinate transformation is negligible, making the method highly efficient and practical for real astrophysical simulations.