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.
Highlights
We design and study a finite volume scheme for a class of nonlinear hyperbolic equations posed on a Schwarzschild black hole background
As is common in the mathematical theory of hyperbolic balance laws, we consider a simplified version of the compressible Euler equations and we describe the fluid evolution by a single scalar unknown function, typically representing the velocity of the fluid
Choosing M to be a Schwarzschild black hole, we introduce a class of hyperbolic balance laws (1.2) and formulate the associated initial value problem
Summary
We design and study a finite volume scheme for a class of nonlinear hyperbolic equations posed on a Schwarzschild black hole background. Defined on a “spacetime” M (explicitly described below in a global coordinate chart) and satisfying the following hyperbolic balance law. Hyperbolic conservation law, Schwarzschild black hole, weak solution, finite volume scheme, convergence analysis. Choosing M to be (the outer domain of communication of) a Schwarzschild black hole, we introduce a class of hyperbolic balance laws (1.2) and formulate the associated initial value problem. The entropy inequalities satisfied by the weak solutions and their discrete version are derived, and the proof of convergence is completed
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