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Journal of Numerical Mathematics

Editor-in-Chief: Hoppe, Ronald H. W. / Kuznetsov, Yuri

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A priori error estimates of Adams–Bashforth discontinuous Galerkin methods for scalar nonlinear conservation laws

Charles Puelz
  • Rice University, Department of Computational and Applied Mathematics, 6100 Main MS–134, Houston, Texas, 77005, USA
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/ Béatrice Rivière
  • Rice University, Department of Computational and Applied Mathematics, 6100 Main MS–134, Houston, Texas, 77005, USA
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Published Online: 2017-05-19 | DOI: https://doi.org/10.1515/jnma-2017-0011

Abstract

In this paper we show theoretical convergence of a second–order Adams–Bashforth discontinuous Galerkin method for approximating smooth solutions to scalar nonlinear conservation laws with E-fluxes. A priori error estimates are also derived for a first–order forward Euler discontinuous Galerkin method. Rates are optimal in time and suboptimal in space; they are valid under a CFL condition.

Keywords: discontinuous Galerkin; error estimates; hyperbolic conservation law

MSC 2010: 65M12; 65M15; 65M60

Bibliography

  • [1]

    J. Alastruey, S.M. Moore, K.H. Parker, T. David, J. Peiró and S.J. Sherwin, Reduced modelling of blood flow in the cerebral circulation: coupling 1-D, 0-D and cerebral auto-regulation models, Internat. J. Numer. Methods Fluids 56 (2008), 1061.Google Scholar

  • [2]

    E. Boileau, P. Nithiarasu, P.J. Blanco, L.O. Müller, F.E. Fossan, L.R. Hellevik, W.P. Donders, W. Huberts, M. Willemet and J. Alastruey, A benchmark study of numerical schemes for one-dimensional arterial blood flow modelling, Internat. J. Numer. Methods Biomed. Eng. (2015).CrossrefGoogle Scholar

  • [3]

    E. Bollache, N. Kachenoura, A. Redheuil, F. Frouin, E. Mousseaux, P. Recho and D. Lucor, Descending aorta subject-specific one-dimensional model validated against in vivo data, Journal of Biomechanics 47 (2014), 424–431.CrossrefWeb of ScienceGoogle Scholar

  • [4]

    S. Čanić and E.H. Kim, Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi-symmetric vessels, Math. Methods Appl. Sci. 26 (2003), 1161–1186.CrossrefGoogle Scholar

  • [5]

    R.C. Cascaval, C. D’Apice, M.P. D’Arienzo and R. Manzo, Boundary Control for an Arterial System, Journal of Fluid Flow 3 (2016).Google Scholar

  • [6]

    B. Cockburn, S. Hou and C.W. Shu, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case, Math. Comp. 54 (1990), 545–581.Google Scholar

  • [7]

    B. Cockburn, S.Y. Lin and C.W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems, J. Comput. Phys. 84 (1989), 90–113.CrossrefGoogle Scholar

  • [8]

    B. Cockburn and C.W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework, Math. Comp. 52 (1989), 411–435.Google Scholar

  • [9]

    B. Cockburn and C.W. Shu, The Runge-Kutta local projection P1 discontinuous- Galerkin finite element method for scalar conservation laws, RAIRO-Modélisation Mathématique et Analyse Numérique 25 (1991), 337–361.Google Scholar

  • [10]

    B. Cockburn and C.W. Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys. 141 (1998), 199–224.CrossrefGoogle Scholar

  • [11]

    C.M. Dafermos, Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2010.Google Scholar

  • [12]

    E. Deriaz, Stability conditions for the numerical solution of convection-dominated problems with skew-symmetric discretizations, SIAM J. Numer. Anal. 50 (2012), 1058–1085.CrossrefWeb of ScienceGoogle Scholar

  • [13]

    D.A. Di Pietro and A. Ern, Mathematical aspects of discontinuous Galerkin methods, 69, Springer Science & Business Media, 2011.Google Scholar

  • [14]

    L. Dumas, T. El Bouti and D. Lucor, A Robust and Subject-Specific Hemodynamic Model of the Lower Limb Based on Noninvasive Arterial Measurements, Journal of Biomechanical Engineering 139 (2017), 011002.Google Scholar

  • [15]

    J. Luo, C.W. Shu and Q. Zhang, A priori error estimates to smooth solutions of the third order Runge–Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws, ESAIM: Math. Model. Numer. Anal. 49 (2015), 991–1018.Web of ScienceCrossrefGoogle Scholar

  • [16]

    K.S. Matthys, J. Alastruey, J. Peiró, A.W. Khir, P. Segers, P.R. Verdonck, K.H. Parker and S.J. Sherwin, Pulse wave propagation in a model human arterial network: assessment of 1-D numerical simulations against in vitro measurements, Journal of Biomechanics 40 (2007), 3476–3486.CrossrefWeb of ScienceGoogle Scholar

  • [17]

    A. Mikelic, G. Guidoboni and S. Čanić, Fluid-structure interaction in a pre-stressed tube with thick elastic walls I: the stationary Stokes problem, Netw. Heterog. Media 2 (2007), 397.Google Scholar

  • [18]

    C. Puelz, B. Rivière, S. Čanić and C.G. Rusin, Comparison of reduced blood flow models using Runge–Kutta discontinuous Galerkin methods, Appl. Numer. Math. 115 (2017), 114–141.CrossrefWeb of ScienceGoogle Scholar

  • [19]

    S.J. Sherwin, L. Formaggia, J. Peiró and V. Franke, Computational modelling of 1D blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system, Internat. J. Numer. Methods Fluids 43 (2003), 673–700.Google Scholar

  • [20]

    S.J. Sherwin, V. Franke, J. Peiró and K. Parker, One-dimensional modelling of a vascular network in space-time variables, J. Engrg. Math. 47 (2003), 217–250.Google Scholar

  • [21]

    H. Wang, C.W. Shu and Q. Zhang, Stability analysis and error estimates of local discontinuous Galerkin methods with implicit–explicit time–marching for nonlinear convection-diffusion problems, Appl. Math. Comput. 272 (2016), 237–258.Web of ScienceGoogle Scholar

  • [22]

    X. Wang, J.M. Fullana and P.Y. Lagrée, Verification and comparison of four numerical schemes for a 1D viscoelastic blood flow model, Computer Methods in Biomechanics and Biomedical Engineering 18 (2015), 1704–1725.Google Scholar

  • [23]

    M. Zakerzadeh and G. May, On the convergence of a shock capturing discontinuous Galerkin method for nonlinear hyperbolic systems of conservation laws, SIAM J. Numer. Anal. 54 (2016), 874–898.CrossrefWeb of ScienceGoogle Scholar

  • [24]

    Q. Zhang and C.W. Shu, Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws, SIAM J. Numer. Anal. 42 (2004), 641–666.CrossrefGoogle Scholar

  • [25]

    Q. Zhang and C.W. Shu, Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws, SIAM J. Numer. Anal. 44 (2006), 1703–1720.CrossrefGoogle Scholar

  • [26]

    Q. Zhang and C.W. Shu, Stability analysis and a priori error estimates of the third order explicit Runge-Kutta discontinuous Galerkin method for scalar conservation laws, SIAM J. Numer. Anal. 48 (2Web of ScienceGoogle Scholar

About the article

The authors are funded in part by the grants NSF-DMS 1312391 and NSF 1318348 and by a training fellowship from the Keck Center of the Gulf Coast Consortia, on the Training Program in Biomedical Informatics, National Library of Medicine (NLM) T15LM007093.


Published Online: 2017-05-19


Citation Information: Journal of Numerical Mathematics, ISSN (Online) 1569-3953, ISSN (Print) 1570-2820, DOI: https://doi.org/10.1515/jnma-2017-0011.

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