Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Entropic sub-cell shock capturing schemes via Jin-Xin relaxation and Glimm front sampling for scalar conservation laws


Authors: Frédéric Coquel, Shi Jin, Jian-Guo Liu and Li Wang
Journal: Math. Comp. 87 (2018), 1083-1126
MSC (2010): Primary 35L65, 35L67, 65M12, 76M12, 76M45
DOI: https://doi.org/10.1090/mcom/3253
Published electronically: September 7, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce a sub-cell shock capturing method for scalar conservation laws built upon the Jin-Xin relaxation framework. Here, sub-cell shock capturing is achieved using the original defect measure correction technique. The proposed method exactly restores entropy shock solutions of the exact Riemann problem and, moreover, it produces monotone and entropy satisfying approximate self-similar solutions. These solutions are then sampled using Glimm's random choice method to advance in time. The resulting scheme combines the simplicity of the Jin-Xin relaxation method with the resolution of the Glimm's scheme to achieve the sharp (no smearing) capturing of discontinuities. The benefit of using defect measure corrections over usual sub-cell shock capturing methods is that the scheme can be easily made entropy satisfying with respect to infinitely many entropy pairs. Consequently, under a classical CFL condition, the method is proved to converge to the unique entropy weak solution of the Cauchy problem for general non-linear flux functions. Numerical results show that the proposed method indeed captures shocks--including interacting shocks--sharply without any smearing.


References [Enhancements On Off] (What's this?)

  • [1] Paolo Baiti, Alberto Bressan, and Helge Kristian Jenssen, Instability of travelling wave profiles for the Lax-Friedrichs scheme, Discrete Contin. Dyn. Syst. 13 (2005), no. 4, 877-899. MR 2166709, https://doi.org/10.3934/dcds.2005.13.877
  • [2] Alberto Bressan, Helge Kristian Jenssen, and Paolo Baiti, An instability of the Godunov scheme, Comm. Pure Appl. Math. 59 (2006), no. 11, 1604-1638. MR 2254446, https://doi.org/10.1002/cpa.20141
  • [3] Weizhu Bao and Shi Jin, The random projection method for hyperbolic conservation laws with stiff reaction terms, J. Comput. Phys. 163 (2000), no. 1, 216-248. MR 1777727, https://doi.org/10.1006/jcph.2000.6572
  • [4] J. Boris and D. Book, Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works, J. Comput. Phys. 11 (1973), no. 1, 38-69.
  • [5] Christophe Chalons and Frederic Coquel, Modified Suliciu relaxation system and exact resolution of isolated shock waves, Math. Models Methods Appl. Sci. 24 (2014), no. 5, 937-971. MR 3187187, https://doi.org/10.1142/S0218202513500723
  • [6] C. Chalons, F. Coquel, P. Engel, and C. Rohde, Fast relaxation solvers for hyperbolic-elliptic phase transition problems, SIAM J. Sci. Comput. 34 (2012), no. 3, A1753-A1776. MR 2970272, https://doi.org/10.1137/110848815
  • [7] Gui Qiang Chen, C. David Levermore, and Tai-Ping Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math. 47 (1994), no. 6, 787-830. MR 1280989, https://doi.org/10.1002/cpa.3160470602
  • [8] Juan Cheng and C.-W. Shu, Second order symmetry-preserving conservative Lagrangian scheme for compressible Euler equations in two-dimensional cylindrical coordinates, J. Comput. Phys. 272 (2014), 245-265. MR 3212271, https://doi.org/10.1016/j.jcp.2014.04.031
  • [9] Phillip Colella, Andrew Majda, and Victor Roytburd, Theoretical and numerical structure for reacting shock waves, SIAM J. Sci. Statist. Comput. 7 (1986), no. 4, 1059-1080. MR 857783, https://doi.org/10.1137/0907073
  • [10] James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697-715. MR 0194770, https://doi.org/10.1002/cpa.3160180408
  • [11] Edwige Godlewski and Pierre-Arnaud Raviart, Hyperbolic systems of conservation laws, Mathématiques and Applications (Paris) [Mathematics and Applications], vol. 3/4, Ellipses, Paris, 1991. MR 1304494
  • [12] Ami Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1983), no. 3, 357-393. MR 701178, https://doi.org/10.1016/0021-9991(83)90136-5
  • [13] Ami Harten, ENO schemes with subcell resolution, J. Comput. Phys. 83 (1989), no. 1, 148-184. MR 1010163, https://doi.org/10.1016/0021-9991(89)90226-X
  • [14] Ami Harten and James M. Hyman, Self-adjusting grid methods for one-dimensional hyperbolic conservation laws, J. Comput. Phys. 50 (1983), no. 2, 235-269. MR 707200, https://doi.org/10.1016/0021-9991(83)90066-9
  • [15] Amiram Harten and Peter D. Lax, A random choice finite difference scheme for hyperbolic conservation laws, SIAM J. Numer. Anal. 18 (1981), no. 2, 289-315. MR 612144, https://doi.org/10.1137/0718021
  • [16] Amiram Harten, Peter D. Lax, and Bram van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev. 25 (1983), no. 1, 35-61. MR 693713, https://doi.org/10.1137/1025002
  • [17] Shi Jin and Jian-Guo Liu, The effects of numerical viscosities. I. Slowly moving shocks, J. Comput. Phys. 126 (1996), no. 2, 373-389. MR 1404378, https://doi.org/10.1006/jcph.1996.0144
  • [18] Shi Jin and Zhou Ping Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995), no. 3, 235-276. MR 1322811, https://doi.org/10.1002/cpa.3160480303
  • [19] S. N. Kružkov, First order quasilinear equations with several independent variables., Mat. Sb. (N.S.) 81 (123) (1970), 228-255 (Russian). MR 0267257
  • [20] Philippe G. LeFloch and Siddhartha Mishra, Numerical methods with controlled dissipation for small-scale dependent shocks, Acta Numer. 23 (2014), 743-816. MR 3202244, https://doi.org/10.1017/S0962492914000099
  • [21] Roberto Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws, Comm. Pure Appl. Math. 49 (1996), no. 8, 795-823. MR 1391756, https://doi.org/10.1002/(SICI)1097-0312(199608)49:8$ \langle$795::AID-CPA2$ \rangle$3.0.CO;2-3
  • [22] W. F. Noh, Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux, J. Comp. Phys. 72 (1987), 78-120.
  • [23] E. Yu. Panov, Uniqueness of the solution of the Cauchy problem for a first-order quasilinear equation with an admissible strictly convex entropy, Mat. Zametki 55 (1994), no. 5, 116-129, 159 (Russian, with Russian summary); English transl., Math. Notes 55 (1994), no. 5-6, 517-525. MR 1296003, https://doi.org/10.1007/BF02110380
  • [24] James J. Quirk, A contribution to the great Riemann solver debate, Internat. J. Numer. Methods Fluids 18 (1994), no. 6, 555-574. MR 1270805, https://doi.org/10.1002/fld.1650180603
  • [25] Denis Serre and Systems of conservation laws. 2, Cambridge University Press, Cambridge, 2000. Geometric structures, oscillations, and initial-boundary value problems; Translated from the 1996 French original by I. N. Sneddon. MR 1775057
  • [26] Chi-Wang Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Advanced numerical approximation of nonlinear hyperbolic equations (Cetraro, 1997) Lecture Notes in Math., vol. 1697, Springer, Berlin, 1998, pp. 325-432. MR 1728856, https://doi.org/10.1007/BFb0096355
  • [27] C.-W. Shu, A brief survey on discontinuous Galerkin methods in computational fluid dynamics, Advances in Mechanics 43 (2013), 541-554.
  • [28] M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal. 81 (1983), no. 4, 301-315. MR 683192, https://doi.org/10.1007/BF00250857
  • [29] B. van Leer, Towards the ultimate conservative difference scheme. v. a second-order sequel to godunov's method, J. Comput. Phys. 32 (1979), no. 1, 101-136.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 35L65, 35L67, 65M12, 76M12, 76M45

Retrieve articles in all journals with MSC (2010): 35L65, 35L67, 65M12, 76M12, 76M45


Additional Information

Frédéric Coquel
Affiliation: CNRS and Centre de Mathématiques Appliquées UMR 7641, École Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France
Email: frederic.coquel@cmap.polytechnique.fr

Shi Jin
Affiliation: Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, Wisconsin 53706
Email: jin@math.wisc.edu

Jian-Guo Liu
Affiliation: Department of Physics and Department of Mathematics, Duke University, Durham, North Carolina 27708
Email: Jian-Guo.Liu@duke.edu

Li Wang
Affiliation: Department of Mathematics and Computational Data-Enabled Science and Engineering Program, State University of New York at Buffalo, 244 Mathematics Building, Buffalo, New York 14260
Email: lwang46@buffalo.edu

DOI: https://doi.org/10.1090/mcom/3253
Keywords: Conservation laws, relaxation, defect measure, Riemann problem, shock capturing methods, cell entropy condition, Glimm's random choice
Received by editor(s): June 1, 2015
Received by editor(s) in revised form: November 2, 2016
Published electronically: September 7, 2017
Additional Notes: The second author was partly supported by NSF grants DMS-1522184 and DMS-1107291 (RNMS:KI-Net), and NSFC grant No. 91330203. The research of the third author was partially supported by KI-Net NSF RNMS grant No. 1107444 and NSF grant DMS 0514826. The fourth author was partially supported by NSF grant DMS-1620135. Support for this research was also provided by the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin-Madison with funding from the Wisconsin Alumni Research Foundation.
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society