# The finite volume method on a Schwarzschild background

## TL;DR

This paper develops a finite volume numerical scheme for nonlinear hyperbolic conservation laws on a Schwarzschild black hole background, proving convergence and stability without boundary conditions at the horizon.

## Contribution

It introduces a novel finite volume method tailored for curved spacetime, establishing convergence, existence, and stability results for solutions.

## Key findings

- Scheme converges to an entropy weak solution
- No boundary conditions needed at the black hole horizon
- Provides a new theory for conservation laws on curved backgrounds

## 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.