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Large-time asymptotics, vanishing viscosity and numerics for 1-D scalar conservation laws

Authors: Liviu I. Ignat, Alejandro Pozo and Enrique Zuazua
Journal: Math. Comp. 84 (2015), 1633-1662
MSC (2010): Primary 35B40, 35L65, 65M12
Published electronically: December 11, 2014
MathSciNet review: 3335886
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Abstract: In this paper we analyze the large time asymptotic behavior of the discrete solutions of numerical approximation schemes for scalar hyperbolic conservation laws. We consider three monotone conservative schemes that are consistent with the one-sided Lipschitz condition (OSLC): Lax-Friedrichs, Engquist-Osher and Godunov. We mainly focus on the inviscid Burgers equation, for which we know that the large time behavior is of a self-similar nature, described by a two-parameter family of N-waves. We prove that, at the numerical level, the large time dynamics depends on the amount of numerical viscosity introduced by the scheme: while Engquist-Osher and Godunov yield the same N-wave asymptotic behavior, the Lax-Friedrichs scheme leads to viscous self-similar profiles, corresponding to the asymptotic behavior of the solutions of the continuous viscous Burgers equation. The same problem is analyzed in the context of self-similar variables that lead to a better numerical performance but to the same dichotomy on the asymptotic behavior: N-waves versus viscous waves. We also give some hints to extend the results to more general fluxes. Some numerical experiments illustrating the accuracy of the results of the paper are also presented.

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  • [1] Juan J. Alonso and Michael R. Colonno, Multidisciplinary optimization with applications to sonic-boom minimization, Annual review of fluid mechanics. Volume 44, 2012, Annu. Rev. Fluid Mech., vol. 44, Annual Reviews, Palo Alto, CA, 2012, pp. 505-526. MR 2882607 (2012m:76076),
  • [2] C. Bardos, A. Y. le Roux, and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations 4 (1979), no. 9, 1017-1034. MR 542510 (81b:35052),
  • [3] Yann Brenier and Stanley Osher, The discrete one-sided Lipschitz condition for convex scalar conservation laws, SIAM J. Numer. Anal. 25 (1988), no. 1, 8-23. MR 923922 (89a:65134),
  • [4] Miguel Escobedo, Juan Luis Vázquez, and Enrike Zuazua, Asymptotic behaviour and source-type solutions for a diffusion-convection equation, Arch. Rational Mech. Anal. 124 (1993), no. 1, 43-65. MR 1233647 (94j:35077),
  • [5] Miguel Escobedo and Enrike Zuazua, Large time behavior for convection-diffusion equations in $ {\bf R}^N$, J. Funct. Anal. 100 (1991), no. 1, 119-161. MR 1124296 (92i:35063),
  • [6] Lawrence C. Evans, Partial Differential Equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943 (2011c:35002)
  • [7] Edwige Godlewski and Pierre-Arnaud Raviart, Hyperbolic Systems of Conservation Laws, Mathématiques & Applications (Paris) [Mathematics and Applications], vol. 3/4, Ellipses, Paris, 1991. MR 1304494 (95i:65146)
  • [8] Eduard Harabetian, Rarefactions and large time behavior for parabolic equations and monotone schemes, Comm. Math. Phys. 114 (1988), no. 4, 527-536. MR 929127 (89d:35084)
  • [9] Eberhard Hopf, The partial differential equation $ u_t+uu_x=\mu u_{xx}$, Comm. Pure Appl. Math. 3 (1950), 201-230. MR 0047234 (13,846c)
  • [10] Yong-Jung Kim and Wei-Ming Ni, On the rate of convergence and asymptotic profile of solutions to the viscous Burgers equation, Indiana Univ. Math. J. 51 (2002), no. 3, 727-752. MR 1911052 (2003g:35184),
  • [11] Yong Jung Kim and Athanasios E. Tzavaras, Diffusive $ N$-waves and metastability in the Burgers equation, SIAM J. Math. Anal. 33 (2001), no. 3, 607-633 (electronic). MR 1871412 (2002i:35121),
  • [12] Corrado Lattanzio and Pierangelo Marcati, Global well-posedness and relaxation limits of a model for radiating gas, J. Differential Equations 190 (2003), no. 2, 439-465. MR 1970037 (2004c:35259),
  • [13] Philippe Laurençot, Asymptotic self-similarity for a simplified model for radiating gases, Asymptot. Anal. 42 (2005), no. 3-4, 251-262. MR 2138795 (2006d:35230)
  • [14] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537-566. MR 0093653 (20 #176)
  • [15] Tai-Ping Liu and Michel Pierre, Source-solutions and asymptotic behavior in conservation laws, J. Differential Equations 51 (1984), no. 3, 419-441. MR 735207 (85i:35094),

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Additional Information

Liviu I. Ignat
Affiliation: Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania — and — Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei Street, 010014 Bucharest, Romania — and —BCAM - Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Basque Country, Spain

Alejandro Pozo
Affiliation: BCAM, Basque Center for Applied Mathematics, Alameda de Mazarredo 14, E-48009 Bilbao, Basque Country, Spain

Enrique Zuazua
Affiliation: BCAM, Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Basque Country, Spain — and — Ikerbasque, Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, 48011 Bilbao, Basque Country, Spain

Received by editor(s): June 19, 2013
Received by editor(s) in revised form: November 8, 2013
Published electronically: December 11, 2014
Additional Notes: The first author was partially supported by Grant PN-II-ID-PCE-2012-4-0021 of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI
The second author was supported by the Grant BFI-2010-339
This work was supported by the Grant MTM2011-29306-C02-00 of the MICINN (Spain), the Advanced Grant FP7-246775 of the European Research Council Executive Agency and the Grant PI2010-04 of the Basque Government
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