The geometric finite volume method for compressible fluid flows on Schwarzschild spacetime

TL;DR

This paper introduces a geometrically formulated, well-balanced finite volume method for relativistic Euler equations on Schwarzschild spacetime, accurately capturing steady states and shock waves.

Contribution

It develops a second-order accurate, coordinate-free finite volume scheme that preserves steady solutions on curved spacetime, with demonstrated robustness and convergence.

Findings

Method accurately preserves steady solutions in simulations.

Numerical experiments show robustness with shock waves.

Late-time asymptotics confirm convergence to steady states.

Abstract

We consider the relativistic Euler equations governing spherically symmetric, perfect fluid flows on the outer domain of communication of Schwarzschild spacetime, and we introduce a version of the finite volume method which is formulated geometrically (without choosing coordinates a priori) and is well--balanced, in the sense that it preserves steady solutions to the Euler equations on the curved geometry under consideration. In order to formulate our method, we first derive a closed formula describing all steady and spherically symmetric solutions to the Euler equations posed on Schwarzschild spacetime. Second, we describe a finite volume method which is formulated geometrically from the family of steady solutions to the Euler system. Our scheme is second--order accurate and, as required, preserves the family of steady solutions at the discrete level. Numerical experiments are presented which demonstrate the efficiency and robustness of the proposed method even for solutions containing shock waves and nonlinear interacting wave patterns. As an application, we investigate the late--time asymptotics of perturbed steady solutions and demonstrate its convergence for late time toward another steady solution, taking the overall effect of the perturbation into account.