Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
권호 표지
DOI QR코드

DOI QR Code

NUMERICAL COUPLING OF TWO SCALAR CONSERVATION LAWS BY A RKDG METHOD

  • OKHOVATI, NASRIN (DEPARTMENT OF MATHEMATICS, KERMAN BRANCH, ISLAMIC AZAD UIVERSITY) ;
  • IZADI, MOHAMMAD (DEPARTMENT OF APPLIED MATHEMATICS, FACULTY OF MATHEMATICS AND COMPUTER, SHAHID BAHONAR UNIVERSITY OF KERMAN)
  • Received : 2019.07.08
  • Accepted : 2019.09.14
  • Published : 2019.09.25
Download PDF
⟨ Previous Next ⟩

Abstract

This paper is devoted to the study and investigation of the Runge-Kutta discontinuous Galerkin method for a system of differential equations consisting of two hyperbolic conservation laws. The numerical coupling flux which is used at a given interface (x = 0) is the upwind flux. Moreover, in the linear case, we derive optimal convergence rates in the $L_2$-norm, showing an error estimate of order ${\mathcal{O}}(h^{k+1})$ in domains where the exact solution is smooth; here h is the mesh width and k is the degree of the (orthogonal Legendre) polynomial functions spanning the finite element subspace. The underlying temporal discretization scheme in time is the third-order total variation diminishing Runge-Kutta scheme. We justify the advantages of the Runge-Kutta discontinuous Galerkin method in a series of numerical examples.

Keywords

References

  1. E. Godlewski and P.-A. Raviart, The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: I. the scalar case, Numer. Math., 97 (2004), 81-130. https://doi.org/10.1007/s00211-002-0438-5
  2. E. Godlewski, K.-C. Le Thanh, P.-A. Raviart, The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: II. The system case, M2AN Math. Model. Numer. Anal., 39 (2005), 649-692. https://doi.org/10.1051/m2an:2005029
  3. A. Ambroso, Ch. Chalons, F. Coquel, E. Godlewski, J.-M. Herard, F. Lagoutiere, P.-A. Raviart, and N. Seguin, The coupling of multiphase flow models, Proceedings of Nureth-11, Avignon, France, 2005.
  4. J.-M. Herard, O. Hurisse, Coupling two and one-dimensional unsteady Euler equations through a thin interface, Computer and Fluids, 36 (2007), 651-666. https://doi.org/10.1016/j.compfluid.2006.03.007
  5. A. Ambroso, Ch. Chalons, F. Coquel, E. Godlewski, J.-M. Herard, F. Lagoutiere, P.-A. Raviart, and N. Seguin, The coupling of homogeneous models for two-phase flows, Int. Journal for Finite Volume, 4 (2007), 1-39.
  6. A. Ambroso, Ch. Chalons, F. Coquel, E. Godlewski, J.-M. Herard, F. Lagoutiere, P.-A. Raviart, N. Seguin, and J.-M. Herard, Coupling of multiphase flow models, Proceedings of the 11th international meeting on nuclear thermohydraulics, Nureth, 2005.
  7. F. Coquel, Coupling of nonlinear hyperbolic systems: A journey from mathematical to numerical issues, in Vazquez-Cendon et al. (Eds.), Numerical Methods for Hyperbolic Equations, Taylor & Francis Group, London, (2013), 21-35.
  8. B. Andreianov, K. H. Karlsen, and N. H. Risebro, A theory of $L_1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Rational Mech. Anal., 201 (2011), 27-86. https://doi.org/10.1007/s00205-010-0389-4
  9. R. Burger, K. H. Karlsen, Conservation laws with discontinuous flux: a short introduction, J. Engrg. Math., 60 (2008), 241-247. https://doi.org/10.1007/s10665-008-9213-7
  10. R. Burger, K.H. Karlsen, J. Towers, On Enquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections, SIAM J. Numer. Anal., 3 (2009), 1684-1712.
  11. S. Diehl, On scalar conservation laws with point source and discontinuous flux function, SIAM J. Math. Anal., 26 (1995), 1425-1451. https://doi.org/10.1137/S0036141093242533
  12. K. H. Karlsen, N. H. Risebro, and J. D. Towers, Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient, IMA J. Numer. Anal., 22(4) (2004), 623-664. https://doi.org/10.1093/imanum/22.4.623
  13. K.H. Karlsen, N.H. Risebro, J. D. Towers, $L_1$-stability for entropy solutions of nonlinear degenerate parabolic connection-diffusion equations with discontinuous coefficients, Skr.-K. Nor. Vidensk. Selsk. 3 (2003) 1-49.
  14. W.H. Reed and T.R. Hill, Triangular mesh methods for the neutron transport equation, Tech. Report LA-UR- 73-479, Los Alamos Scientific Laboratory, Los Alamos, 1973.
  15. P. LeSaint, and P. A. Raviart, On a finite element method for solving the neutron transport equation, In de Boor, C. (Eds.), Mathematical Aspects of Finite Elements in Partial Differential Equations, Academic Press, (1974), 89-145.
  16. C. Johnson and J. Pitkaranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp., 46 (1986), 1-26. https://doi.org/10.1090/S0025-5718-1986-0815828-4
  17. T. Peterson, A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM. J. Numer. Anal., 28 (1991), 133-140. https://doi.org/10.1137/0728006
  18. G.R. Richter, An optimal-order error estimate for the discontinuous Galerkin method, Math. Comp., 50 (1988), 75-88. https://doi.org/10.1090/S0025-5718-1988-0917819-3
  19. B. Cockburn, S. Hou, and C.W. Shu, The Runge-Kutta local projection discontinuous Galerkin method for conservation laws IV: the multidimensional case, Math. Comp., 54 (1990), 545-581. https://doi.org/10.1090/S0025-5718-1990-1010597-0
  20. C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), 439-471. https://doi.org/10.1016/0021-9991(88)90177-5
  21. B. Cockburn, C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: General framework, Math. Comp., 52 (1989), 411-435. https://doi.org/10.1090/S0025-5718-1989-0983311-4
  22. B. Cockburn, C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws V: Multidimensional systems, J. Comput. Phys.,141 (1998), 199-224. https://doi.org/10.1006/jcph.1998.5892
  23. B. Cockburn, G.E. Karniadakis, and C. W. Shu (Eds.), Discontinuous Galerkin methods theory, computation and applications, Lecture Notes in Computational Science and Engineering, vol. 11, Springer, Berlin, 2000.
  24. J.S. Hesthaven, T.Warburton, Nodal Discontinuous Galerkin Methods: algorithms, analysis, and applications, Texts in Applied Mathematics, vol. 54, Springer Verlag, New York, USA, 2008.
  25. M. Izadi, Streamline diffusion methods for treating the coupling equations of two hyperbolic conservation laws, Math. Comput. Model., 45 (2007), 201-214. https://doi.org/10.1016/j.mcm.2006.05.004
  26. A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws. Arch. Ration. Mech. Anal., 160 (2001), 181-193. https://doi.org/10.1007/s002050100157
  27. S.N. Kruzkov, First order quasilinear equations in several independent variables, USSR Math. Sbornik. 10 (2) (1970) 217-243. https://doi.org/10.1070/SM1970v010n02ABEH002156
  28. Adimurthi and G. D. V. Gowda, Conservation laws with discontinuous flux, J. Math. Kyoto Univ., 43 (2003), 27-70. https://doi.org/10.1215/kjm/1250283740
  29. Y. Cheng, C.-W. Shu, Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension, SIAM J. Numer. Anal., 47 (2010), 4044-4072. https://doi.org/10.1137/090747701
  30. S. Osher, Riemann solvers, the entropy condition, and difference approximations, SIAM. J. Numer. Anal., 21 (1984), 217-235. https://doi.org/10.1137/0721016
  31. G. Jiang, C.-W. Shu, On a cell entropy inequality for discontinuous Galerkin methods, Math. Comput., 206 (1994), 531-538. https://doi.org/10.1090/S0025-5718-1994-1223232-7
  32. M. Izadi, A posteriori error estimates for the coupling equations of scalar conservation laws, BIT Numer. Math., 49(4) (2009), 697-720. https://doi.org/10.1007/s10543-009-0243-y
  33. M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, Dover, New York, 1965.
  34. S. Bertoluzza, S. Falletta, G. Russo, and C.-W. Shu, Numerical Solution of Partial Differential Equations, in: Advanced Courses in Mathematics, CRM, Barcelona, 2008.
  35. P.G. Ciarlet, The Finite Element Method for Elliptic Problems, Amesterdam, North Holland, 1987.
  36. B. Cockburn, C.-W, Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16 (2001), 173-261. https://doi.org/10.1023/A:1012873910884
  37. J.-M. Herard, Schemes to couple flows between free and porous medium, Proceedings of AIAA (2005), 2005-4861.