A. Arnold, M. Ehrhardt, and I. Sofronov, Discrete transparent boundary conditions for the Schrödinger equation: fast calculation, approximation, and stability, Communications in Mathematical Sciences, vol.1, issue.3, pp.501-556, 2003.

S. Benzoni-gavage and D. Serre, Multi-dimensional hyperbolic partial differential equations, First-order systems and applications, 2006.

C. Besse, M. Ehrhardt, and I. Lacroix-violet, Discrete artificial boundary conditions for the linearized Korteweg-de Vries equation. Numerical Methods for Partial Differential Equations, An International Journal, vol.32, issue.5, pp.1455-1484, 2016.

C. Besse, P. Noble, and D. Sanchez, Discrete transparent boundary conditions for the mixed KDV-BBM equation, Journal of Computational Physics, vol.345, pp.484-509, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01373388

B. Boutin, T. H. Nguyen, and N. Seguin, A stiffly stable semi-discrete scheme for the characteristic linear hyperbolic relaxation with boundary, ESAIM: Mathematical Modelling and Numerical Analysis, 2020.
URL : https://hal.archives-ouvertes.fr/hal-02136776

C. Cercignani, The Boltzmann Equation. Applied Mathematical Sciences, 1988.

G. Q. Chen, C. D. Levermore, and T. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Communications on Pure and Applied Mathematics, vol.47, issue.6, pp.787-830, 1994.

P. Colella, A. Majda, and V. Roytburd, Theoretical and Numerical Structure for Reacting Shock Waves, SIAM Journal on Scientific and Statistical Computing, vol.7, issue.4, pp.1059-1080, 1986.

J. Coulombel, Transparent numerical boundary conditions for evolution equations: derivation and stability analysis, Toulouse. Mathématiques. Série, vol.6, issue.2, pp.259-327, 2019.
URL : https://hal.archives-ouvertes.fr/hal-01369975

M. Goldberg, On a boundary extrapolation theorem by Kreiss. Mathematics of Computation, vol.31, pp.469-477, 1977.

M. J. Grote and J. B. Keller, Exact Nonreflecting Boundary Conditions for the Time Dependent Wave Equation, SIAM Journal on Applied Mathematics, vol.55, issue.2, pp.280-297, 1995.

B. Gustafsson, H. Kreiss, and J. Oliger, Time dependent problems and difference methods, 2013.

B. Hanouzet and R. Natalini, Global Existence of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy. Archive for Rational Mechanics and, Analysis, vol.169, issue.2, pp.89-117, 2003.

S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Communications on Pure and Applied Mathematics, vol.48, issue.3, pp.235-276, 1995.

E. I. Jury, Theory and Application of the Z-Transform Method, 1964.

M. Kazakova and P. Noble, Discrete Transparent Boundary Conditions for the Linearized Green-Naghdi System of Equations, SIAM Journal on Numerical Analysis, vol.58, issue.1, pp.657-683, 2020.
URL : https://hal.archives-ouvertes.fr/hal-01613071

E. M. , Landau. Statistical Physics, 1980.

T. Liu, Hyperbolic conservation laws with relaxation, Communications in Mathematical Physics, vol.108, issue.1, pp.153-175, 1987.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems (2Nd Ed, 1996.

J. J. Stoker, Water waves. The mathematical theory with applications, 1992.

W. G. Vincenti and C. H. Kruger, Introduction to Physical Gas Dynamics, vol.70, 1966.

G. B. Whitham, Linear and nonlinear waves, 1974.

Z. Xin and W. Xu, Stiff Well-Posedness and Asymptotic Convergence for a Class of Linear Relaxation Systems in a Quarter Plane, Journal of Differential Equations, vol.167, issue.2, pp.388-437, 2000.

W. Yong, Boundary conditions for hyperbolic systems with stiff source terms, Indiana University Mathematics Journal, vol.48, issue.1, pp.115-137, 1999.

W. Yong, Singular Perturbations of First-Order Hyperbolic Systems with Stiff Source Terms, Journal of Differential Equations, vol.155, issue.1, pp.89-132, 1999.

W. Yong, Entropy and Global Existence for Hyperbolic Balance Laws. Archive for Rational Mechanics and, Analysis, vol.172, issue.2, pp.247-266, 2004.

, 35042 Rennes Cedex, France. Email address: benjamin.boutin@univ-rennes1.fr Email address: thi-hoai-thuong.nguyen@univ-rennes1, CNRS 6625)