The finite volume method on a Schwarzschild background
Shijie Dong, Philippe G. LeFloch
TL;DR
The paper develops a rigorous finite-volume framework for nonlinear hyperbolic balance laws posed on a Schwarzschild black hole background. It derives a scalar velocity model with flux $f$ and source $h$ from relativistic fluid dynamics, presents equivalent non-singular formulations, and analyzes characteristics and steady states in both exterior and interior (horizon-penetrating) coordinates. A key contribution is a provably convergent finite-volume scheme that respects curved geometry and requires no boundary data at the horizon; the authors establish discrete entropy inequalities and prove convergence to an entropy solution via measure-valued (Young measure) techniques and DiPerna’s theory. This work provides a solid numerical methodology and a theoretical existence/stability framework for conservation laws in curved spacetime, with potential applications to relativistic fluid dynamics in black-hole spacetimes and numerical relativity.
Abstract
We introduce a class of nonlinear hyperbolic conservation laws on a Schwarzschild black hole background and derive several properties satisfied by (possibly weak) solutions. Next, we formulate a numerical approximation scheme which is based on the finite volume methodology and takes the curved geometry into account. An interesting feature of our model is that no boundary conditions is required at the black hole horizon boundary. We establish that this scheme converges to an entropy weak solution to the initial value problem and, in turn, our analysis also provides us with a theory of existence and stability for a new class of conservation laws.