Stability of stationary solutions of singular systems of balance laws

Nicolas Seguin

doi:10.5802/cml.52

Nicolas Seguin ¹

¹ Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

Confluentes Mathematici, Tome 10 (2018) no. 2, pp. 93-112.

Résumé

The stability of stationary solutions of first-order systems of PDE’s is considered. The systems under investigation may include singular geometric terms, leading to discontinuous flux and non-conservative products. Based on several examples in Fluid Mechanics, we assume that these systems are endowed with a partially convex entropy. We first construct an associated relative entropy which allows to compare two states which share the same geometric data. This way, we are able to prove the stability of some stationary states within entropy weak solutions. Let us stress that these solutions are only required to have a bounded total variation, i.e. they can be discontinuous. This result applies for instance to the shallow-water equations with bathymetry. Besides, this relative entropy can be used to study finite volume schemes which are entropy-stable and well-balanced, and due to the numerical dissipation inherent to these methods, asymptotic stability of discrete stationary solutions is obtained. This analysis does not make us of any specific definition of the non-conservative products, applies to non-strictly hyperbolic systems, and is fully multidimensional with unstructured meshes for the numerical methods.

Reçu le : 2017-09-13
Accepté le : 2017-12-17
Publié le : 2019-03-03

DOI : 10.5802/cml.52

Classification : 35L60, 35B35, 35B25, 65M08
Mots clés : Hyperbolic systems, stationary state, stability, relative entropy, non-conservative systems, finite volume schemes, well-balanced schemes.

Affiliations des auteurs :

Nicolas Seguin ¹

¹ Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

Licence :

CC-BY-NC-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Nicolas Seguin. Stability of stationary solutions of singular systems of balance laws. Confluentes Mathematici, Tome 10 (2018) no. 2, pp. 93-112. doi : 10.5802/cml.52. https://cml.centre-mersenne.org/articles/10.5802/cml.52/

Bibliographie
Cité par

[1] D. Amadori and L. Gosse. Error estimates for well-balanced schemes on simple balance laws. SpringerBriefs in Mathematics. Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao, 2015. | DOI | Zbl

[2] B. Andreianov, K. H. Karlsen, and N. H. Risebro. A theory of $L^{1}$ -dissipative solvers for scalar conservation laws with discontinuous flux. Arch. Ration. Mech. Anal., 201(1):27–86, 2011. | DOI | MR | Zbl

[3] B. Andreianov and N. Seguin. Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes. Discrete Contin. Dyn. Syst., 32(6):1939–1964, 2012. | DOI | Zbl

[4] E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein, and B. Perthame. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput., 25(6):2050–2065, 2004. | DOI | MR | Zbl

[5] C. Berthon and C. Chalons. A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations. Math. Comp., 85(299):1281–1307, 2016. | DOI | MR | Zbl

[6] F. Bouchut. Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2004. | Zbl

[7] Y. Brenier, C. De Lellis, and L. Székelyhidi, Jr. Weak-strong uniqueness for measure-valued solutions. Comm. Math. Phys., 305(2):351–361, 2011. | DOI | MR | Zbl

[8] C. Cancès, H. Mathis, and N. Seguin. Error estimate for time-explicit finite volume approximation of strong solutions to systems of conservation laws. SIAM J. Numer. Anal., 54(2):1263–1287, 2016. | DOI | MR | Zbl

[9] P. Cargo and A.-Y. LeRoux. Un schéma équilibre adapté au modèle d’atmosphère avec termes de gravité. C. R. Acad. Sci. Paris Sér. I Math., 318(1):73–76, 1994. | Zbl

[10] C. Chalons and F. Coquel. A new comment on the computation of non conservative products using Roe-type path conservative scheme. J. Comput. Phys., 335:592–604, 2017. | DOI | MR | Zbl

[11] C. Chalons, F. Coquel, E. Godlewski, P.-A. Raviart, and N. Seguin. Godunov-type schemes for hyperbolic systems with parameter-dependent source. The case of Euler system with friction. Math. Models Methods Appl. Sci., 20(11):2109–2166, 2010. | DOI | MR | Zbl

[12] G. Chen and S. Noelle. A new hydrostatic reconstruction scheme based on subcell reconstructions. SIAM J. Numer. Anal., 55(2):758–784, 2016. | DOI | MR | Zbl

[13] A. Chinnayya, A.-Y. LeRoux, and N. Seguin. A well-balanced numerical scheme for the approximation of the shallow-water equations with topography: the resonance phenomenon. Int. J. Finite Volumes, pages 1–33, 2004.

[14] J.-F. Colombeau. Multiplication of distributions. Springer Verlag, 1992. | DOI

[15] F. Coquel, K. Saleh, and N. Seguin. A robust and entropy-satisfying numerical scheme for fluid flows in discontinuous nozzles. Math. Models Methods Appl. Sci., 24(10):2043–2083, 2014. | DOI | MR | Zbl

[16] C. M. Dafermos. The second law of thermodynamics and stability. Arch. Ration. Mech. Anal., 70:167–179, 1979. | DOI | MR | Zbl

[17] C. M. Dafermos. Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, troisième edition, 2010. | DOI | Zbl

[18] G. Dal Maso, P.G. LeFloch, and F. Murat. Definition and weak stability of non conservative products. J. Math. Pures Appl., 74:483–548, 1995. | Zbl

[19] R. J. DiPerna. Uniqueness of solutions to hyperbolic conservation laws. Indiana U. Math. J., 28:137–188, 1979. | Zbl

[20] U. S. Fjordholm, S. Mishra, and E. Tadmor. Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography. J. Comput. Phys., 230(14):5587–5609, 2011. | DOI | MR | Zbl

[21] G. Gallice. Solveurs simples positifs et entropiques pour les systèmes hyperboliques avec terme source. C. R. Math. Acad. Sci. Paris, 334(8):713–716, 2002. | DOI | Zbl

[22] J. Glimm. Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math., 18:697–715, 1965. | DOI | MR | Zbl

[23] J. Glimm and P. D. Lax. Decay of solutions of systems of nonlinear hyperbolic conservation laws. Memoirs of the American Mathematical Society, No. 101. American Mathematical Society, Providence, R.I., 1970. | DOI | Zbl

[24] P. Goatin and P. G. LeFloch. The Riemann problem for a class of resonant hyperbolic systems of balance laws. Ann. Inst. H. Poincaré Anal. Non Linéaire, 21(6):881–902, 2004. | DOI | Numdam | MR | Zbl

[25] E. Godlewski and P.-A. Raviart. Numerical approximation of hyperbolic systems of conservation laws, volume 118 of Applied Mathematical Sciences. Springer-Verlag, New York, 1996. | DOI | Zbl

[26] S. K. Godunov. Finite difference method for numerical computation of discontinous solution of the equations of fluid dynamics. Mat. Sb., 47:271–300, 1959.

[27] L. Gosse. A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms. Math. Models Methods Appl. Sci., 11(2):339–365, 2001. | DOI | MR | Zbl

[28] L. Gosse. Localization effects and measure source terms in numerical schemes for balance laws. Math. Comp., 71(238):553–582 (electronic), 2002. | DOI | MR | Zbl

[29] L. Gosse. Computing qualitatively correct approximations of balance laws, volume 2 of SIMAI Springer Series. Springer, Milan, 2013. Exponential-fit, well-balanced and asymptotic-preserving. | DOI | Zbl

[30] J. M. Greenberg and A.-Y. LeRoux. A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal., 33(1):1–16, 1996. | DOI | MR | Zbl

[31] A. Harten, P. D. Lax, and B. van Leer. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev., 25(1):35–61, 1983. | DOI | MR | Zbl

[32] E. Isaacson and B. Temple. Nonlinear resonance in systems of conservation laws. SIAM J. Applied Math., 52(5):1260–1278, 1992. | DOI | MR | Zbl

[33] E. Isaacson and B. Temple. Convergence of the $2 \times 2$ Godunov method for a general resonant nonlinear balance law. SIAM J. Applied Math., 55(3):625–640, 1995. | DOI | MR | Zbl

[34] V. Jovanović and C. Rohde. Error estimates for finite volume approximations of classical solutions for nonlinear systems of hyperbolic balance laws. SIAM J. Numer. Anal., 43(6):2423–2449, 2006. | DOI | MR | Zbl

[35] M.-J. Kang and A. F. Vasseur. Criteria on contractions for entropic discontinuities of systems of conservation laws. Arch. Ration. Mech. Anal., 222(1):343–391, 2016. | DOI | MR | Zbl

[36] C. Klingenberg and N. H. Risebro. Convex conservation laws with discontinuous coefficients. Existence, uniqueness and asymptotic behavior. Comm. Partial Differential Equations, 20(11-12):1959–1990, 1995. | DOI | MR | Zbl

[37] B. Perthame and C. Simeoni. A kinetic scheme for the Saint-Venant system with a source term. Calcolo, 38(4):201–231, 2001. | DOI | MR | Zbl

[38] E. Tadmor. The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Math. Comp., 49(179):91–103, 1987. | DOI | MR | Zbl

[39] B. Temple and R. Young. A Nash-Moser framework for finding periodic solutions of the compressible Euler equations. J. Sci. Comput., 64(3):761–772, 2015. | DOI | MR | Zbl

[40] E. F. Toro. Riemann solvers and numerical methods for fluid dynamics. Springer-Verlag, Berlin, third edition, 2009. A practical introduction. | DOI | Zbl

[41] A. Vasseur, and Y. Wang. The inviscid limit to a contact discontinuity for the compressible Navier–Stokes–Fourier system using the relative entropy method. SIAM J. Math. Anal., 47(6):4350–4359, 2015. | DOI | MR | Zbl

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