On the theoretical foundation of overset grid methods for hyperbolic problems: Well-posedness and conservation

• Published in: Journal of computational physics Vol. 448; p. 110732
• Main Authors: Kopriva, David A., Nordström, Jan, Gassner, Gregor J.
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01-01-2022; Elsevier Science Ltd
• Subjects: Boundary value problems; Chimera method; Computational physics; Conservation; Coupling; Domains; Energy methods; Hyperbolic systems; Mathematical analysis; Overset grids; Penalty methods; Stability; Well posed problems; Well-posedness; Overset grids; Penalty methods; Chimera method; Stability; Conservation; Well-posedness
• ISSN: 0021-9991; 1090-2716; 1090-2716
• DOI: 10.1016/j.jcp.2021.110732

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 usually 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.
•Analyzes well-posedness of multi-dimensional IBVPs approximated by overset grid methods.•Shows the 1D characteristic formulation for overset domains is well-posed in one space dimension.•Shows the multiple space extension is not.•Proposes a well-posed problem for systems in multiple dimensions that uses a penalty technique.•Extends the formulation with a new idea to penalize solutions within the overlap region.