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Wave-heat coupling in one-dimensional unbounded domains: artificial boundary conditions and an optimized Schwarz method

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Abstract

This paper deals with the coupling between one-dimensional heat and wave equations in unbounded subdomains, as a simplified prototype of fluid-structure interaction problems. First we apply appropriate artificial boundary conditions that yield an equivalent problem, but with bounded subdomains, and we carry out the stability analysis for this coupled problem in truncated domains. Then we devise an optimized Schwarz-in-time (or Schwarz Waveform Relaxation) method for the numerical solving of the coupled equations. Particular emphasis is made on the design of optimized transmission conditions. Notably, for this setting, the optimal transmission conditions can be expressed analytically in a very simple manner. This result is illustrated by some numerical experiments.

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Acknowledgements

The two authors thank the two anonymous referees for their comments that allowed to improve the manuscript. They also thank Miguel A. Fernandez, Yvon Maday, Martin Gander, Véronique Martin, Christophe Besse and Xavier Antoine for stimulating discussions and helpful comments. This work was granted for access to the resources of the Mesocentre of Franche-Comté.

Funding

F.C. and P.K.’s work is partially supported by Université de Franche-Comté and Région Bourgogne Franche-Comté. F.C.’s work is partially supported by the I-Site BFC project NAANoD and the EIPHI Graduate School (contract ANR-17-EURE-0002).

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Authors and Affiliations

  1. Université Bourgogne Franche-Comté, Institut de Mathématiques de Bourgogne, 21078, Dijon, France

    Franz Chouly

  2. Center for Mathematical Modeling and Department of Mathematical Engineering, University of Chile and IRL 2807 – CNRS, Santiago, Chile

    Franz Chouly

  3. Departamento de Ingenería Matématica, CI2MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Franz Chouly

  4. Laboratoire de Mathématiques de Besançon, Université Bourgogne Franche-Comté, CNRS, UMR-6623, 25030, Besançon, France

    Pauline Klein

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Correspondence to Franz Chouly.

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Chouly, F., Klein, P. Wave-heat coupling in one-dimensional unbounded domains: artificial boundary conditions and an optimized Schwarz method. Numer Algor 90, 631–668 (2022). https://doi.org/10.1007/s11075-021-01201-x

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