A geometric approach to error estimates for conservation laws posed on a spacetime

TL;DR

This paper introduces a geometric method for deriving L1 error estimates for hyperbolic conservation laws on (N+1)-dimensional spacetimes, extending classical techniques to more general geometric settings.

Contribution

It generalizes Kuznetsov's method to a geometric framework applicable to conservation laws on Lorentzian manifolds and related spacetime structures.

Findings

Derived a new L1 error estimate for conservation laws on spacetime

Extended the framework to handle multiple entropy solutions with different flux fields

Applicable to equations on globally hyperbolic Lorentzian manifolds

Abstract

We consider a hyperbolic conservation law posed on an (N+1)-dimensional spacetime, whose flux is a field of differential forms of degree N. Generalizing the classical Kuznetsov's method, we derive an L1 error estimate which applies to a large class of approximate solutions. In particular, we apply our main theorem and deal with two entropy solutions associated with distinct flux fields, as well as with an entropy solution and an approximate solution. Our framework encompasses, for instance, equations posed on a globally hyperbolic Lorentzian manifold.