## Coupling conditions for linear hyperbolic relaxation systems in two-scale problems

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Juntao Huang, Ruo Li and Yizhou Zhou
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## Abstract:

This work is concerned with coupling conditions for linear hyperbolic relaxation systems with multiple relaxation times. In the region with a small relaxation time, an equilibrium system can be used for computational efficiency. The key assumption is that the relaxation system satisfies Yong’s structural stability condition [J. Differential Equations, 155 (1999), pp. 89–132]. For the non-characteristic case, we derive a coupling condition at the interface to couple two systems in a domain decomposition setting. We prove the validity by the energy estimate and Laplace transform, which shows how the error of the domain decomposition method depends on the smaller relaxation time and the boundary-layer effects. In addition, we propose a discontinuous Galerkin (DG) numerical scheme for solving the interface problem with the derived coupling condition and prove the $L^2$ stability. We validate our analysis on the linearized Carleman model and the linearized Grad’s moment system and show the effectiveness of the DG scheme.## References

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## Additional Information

**Juntao Huang**- Affiliation: Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409
- MR Author ID: 1121918
- ORCID: 0000-0003-0527-7431
- Email: juntao.huang@ttu.edu
**Ruo Li**- Affiliation: School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic ofChina
- Email: rli@math.pku.edu.cn
**Yizhou Zhou**- Affiliation: School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
- Email: zhouyz@math.pku.edu.cn
- Received by editor(s): July 11, 2022
- Received by editor(s) in revised form: December 8, 2022, and February 28, 2023
- Published electronically: May 8, 2023
- Additional Notes: This work was supported by the National Key R&D Program of China, Project Number 2020YFA0712000 and the China Postdoctoral Science Foundation, Project Number 2021M700002. The third author is the corresponding author.
- © Copyright 2023 American Mathematical Society
- Journal: Math. Comp.
**92**(2023), 2133-2165 - MSC (2020): Primary 35L50, 65M55
- DOI: https://doi.org/10.1090/mcom/3845