On the Theoretical Foundation of Overset Grid Methods for Hyperbolic Problems: Well-Posedness and Conservation

TL;DR

This paper establishes the mathematical foundations of overset grid methods for hyperbolic problems, proving well-posedness and conservation properties using energy methods and proposing a penalty approach for complex systems.

Contribution

It provides a rigorous analysis of overset grid methods' well-posedness and introduces a novel penalty technique for systems that cannot be diagonalized.

Findings

Overset domain problems are well-posed with characteristic coupling conditions in 1D.

The penalty approach ensures energy boundedness and conservation in multi-dimensional systems.

Solutions of penalized problems are equivalent to single domain problems under certain conditions.

Abstract

We use the energy method to study the well-posedness of initial-boundary value problems approximated by overset mesh methods in one and two space dimensions for linear constant-coefficient hyperbolic systems. We show that in one space dimension, for both scalar equations and systems of equations, the problem where one domain partially oversets another is well-posed when characteristic coupling conditions are used. If a system cannot be diagonalized, as is ususally the case in multiple space dimensions, then the energy method does not give proper bounds in terms of initial and boundary data. For those problems, we propose a novel penalty approach. We show, by using a global energy that accounts for the energy in the overlap region of the domains, that under well-defined conditions on the coupling matrices the penalized overset domain problems are energy bounded, conservative, well-posed and have solutions equivalent to the original single domain problem.