Symmetric hyperbolic form of systems of second-order evolution equations subject to constraints

TL;DR

This paper introduces a symmetric hyperbolic formulation for second-order evolution systems, inspired by Einstein equations, enabling constraint-preserving boundary conditions, with detailed illustration using electromagnetism as a model.

Contribution

It defines symmetric hyperbolicity for second-order in space systems and demonstrates its application to constraint-preserving boundary conditions, extending existing theory.

Findings

Provides a new framework for second-order systems in hyperbolic form

Enables constraint-preserving boundary conditions in complex systems

Illustrates the approach with electromagnetism as a practical example

Abstract

Motivated by the initial-boundary value problem for the Einstein equations, we propose a definition of symmetric hyperbolicity for systems of evolution equations that are first order in time but second order in space. This can be used to impose constraint-preserving boundary conditions. The general methods are illustrated in detail in the toy model of electromagnetism.