A Geometry-Preserving Finite Volume Method for Compressible Fluids on Schwarzschild Spacetime (Communications in Computational Physics, 2014)
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 formulatedfrom the geometric formulation (and thus takes the geometry into account at the discretization level) and is well-balanced, in the sense that it preserves steady solutions tothe Euler equations on the curved geometry under consideration. In order to formulate our method, we first derive a closed formula describing all steady and sphericallysymmetric solutions to the Euler equations posed on Schwarzschild spacetime. Second, we describe a geometry-preserving, finite volume method which is based on 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 nonlinearinteracting wave patterns. As an application, we investigate the late-time asymptoticsof perturbed steady solutions and demonstrate its convergence for late time towardanother steady solution, taking the overall effect of the perturbation into account.