A direct Eulerian GRP scheme for spherically symmetric general relativistic hydrodynamics

TL;DR

This paper introduces a second-order accurate Eulerian GRP scheme for spherically symmetric general relativistic hydrodynamics and Einstein equations, demonstrating high accuracy and effectiveness through numerical experiments.

Contribution

It presents a novel second-order Eulerian GRP scheme that directly solves spherically symmetric general relativistic hydrodynamics and Einstein equations.

Findings

Achieves second-order accuracy in numerical tests.

Provides high-resolution solutions for spherically symmetric RHD problems.

Effective in simulating general relativistic hydrodynamics with Einstein equations.

Abstract

The paper proposes a second-order accurate direct Eulerian generalized Riemann problem (GRP) scheme for the spherically symmetric general relativistic hydrodynamical (RHD) equations and a second-order accurate discretization for the spherically symmetric Einstein (SSE) equations. The former is directly using the Riemann invariants and the Runkine-Hugoniot jump conditions to analytically resolve the left and right nonlinear waves of the local GRP in the Eulerian formulation together with the local change of the metrics to obtain the limiting values of the time derivatives of the conservative variables along the cell interface and the numerical flux for the GRP scheme. While the latter utilizes the energy-momentum tensor obtained in the GRP solver to evaluate the fluid variables in the SSE equations and keeps the continuity of the metrics at the cell interfaces. Several numerical experiments show that the GRP scheme can achieve second-order accuracy and high resolution, and is effective for spherically symmetric general RHD problems.