\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes

Abstract Related Papers Cited by
  • We define entropy weak solutions and establish well-posedness for the Cauchy problem for the formal equation $$\partial_t u(t,x) + \partial_x \frac{u^2}2(t,x) = - \lambda \, u(t,x)\,\delta_0(x),$$ which can be seen as two Burgers equations coupled in a non-conservative way through the interface located at $x=0$. This problem appears as an important auxiliary step in the theoretical and numerical study of the one-dimensional particle-in-fluid model developed by Lagoutière, Seguin and Takahashi [30].
        The interpretation of the non-conservative product "$ u(t,x) \, \delta_0(x)$" follows the analysis of [30]; we can describe the associated interface coupling in terms of one-sided traces on the interface. Well-posedness is established using the tools of the theory of conservation laws with discontinuous flux ([4]).
        For proving existence and for practical computation of solutions, we construct a finite volume scheme, which turns out to be a well-balanced scheme and which allows a simple and efficient treatment of the interface coupling. Numerical illustrations are given.
    Mathematics Subject Classification: Primary: 35L65; Secondary: 35L81, 35R06, 65M12.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    D. Amadori, L. Gosse and G. Guerra, Godunov-type approximation for a general resonant balance law with large data, J. Differ. Equ., 198 (2004), 233-274.doi: 10.1016/j.jde.2003.10.004.

    [2]

    B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws, Numer. Math., 115 (2010), 609-645.doi: 10.1007/s00211-009-0286-7.

    [3]

    B. Andreianov, K. H. Karlsen and N. H. Risebro, On vanishing viscosity approximation of conservation laws with discontinuous flux, Netw. Heterog. Media, 5 (2010), 617-633.

    [4]

    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 (2011), 27-86.doi: 10.1007/s00205-010-0389-4.

    [5]

    B. Andreianov, F. Lagoutière, N. Seguin and T. Takahashi, Small solids in an inviscid fluid, Netw. Heter. Media, 5 (2010), 385-404.

    [6]

    B. Andreianov, F. Lagoutière, N. Seguin and T. TakahashiWell-posedness for a one-dimensional fluid-particle interaction model, In preparation.

    [7]

    Adimurthi, S. Mishra and G. D. Veerappa Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ., 2 (2005), 783-837.doi: 10.1142/S0219891605000622.

    [8]

    E. Audusse and B. Perthame, Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 253-265.

    [9]

    F. Bachmann, "Équations Hyperboliques Scalaires à Flux Discontinu,'' Ph.D thesis, Université de Provence, France, 2005.

    [10]

    B. Boutin, F. Coquel and E. Godlewski, Dafermos regularization for interface coupling of conservation laws, in "Hyperbolic Problems: Theory, Numerics, Applications,'' Springer, Berlin, (2008), 567-575.doi: 10.1007/978-3-540-75712-2_55.

    [11]

    R. Bürger, A. García, K. H. Karlsen and J. D. Towers, A family of numerical schemes for kinematic flows with discontinuous flux, J. Engrg. Math., 60 (2008), 387-425.

    [12]

    P. Baiti and H. K. Jenssen, Well-posedness for a class of $2\times2$ conservation laws with $L^\infty$ data, J. Differ. Equ., 140 (1997), 161-185.doi: 10.1006/jdeq.1997.3308.

    [13]

    R. Bürger, K. H. Karlsen and J. D. Towers, An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections, SIAM J. Numer. Anal., 47 (2009), 1684-1712.

    [14]

    B. Boutin, "Couplage de Lois de Conservation Scalaires par une Régularisation à la Dafermos,'' Master's thesis, Université Pierre et Marie Curie-Paris 6, 2006.

    [15]

    R. Botchorishvili, B. Perthame and A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources, Math. Comp., 72 (2003), 131-157.doi: 10.1090/S0025-5718-01-01371-0.

    [16]

    R. M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint, J. Differ. Equ., 234 (2007), 654-675.doi: 10.1016/j.jde.2006.10.014.

    [17]

    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, 1 (2004), 33 pp.

    [18]

    M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings, Proc. AMS, 78 (1980), 385-390.

    [19]

    R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in "Handbook of Numerical Analysis,'' Vol. VII, North-Holland, Amsterdam, (2000), 713-1020.

    [20]

    L. Gosse and A.-Y. LeRoux, Un schéma-équilibre adapté aux lois de conservation scalaires non-homogènes, C. R. Acad. Sci. Paris Sér. I Math., 323 (1996), 543-546.

    [21]

    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 (1996), 1-16.doi: 10.1137/0733001.

    [22]

    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 (2004), 881-902.

    [23]

    L. Gosse, Localization effects and measure source terms in numerical schemes for balance laws, Math. Comp., 71 (2002), 553-582.doi: 10.1090/S0025-5718-01-01354-0.

    [24]

    E. Godlewski and P.-A. Raviart, "Hyperbolic Systems of Conservation Laws,'' Mathématiques & Applications (Paris) [Mathematics and Applications], 3/4, Ellipses, Paris, 1991.

    [25]

    L. Gosse and G. Toscani, Space localization and well-balanced schemes for discrete kinetic models in diffusive regimes, SIAM J. Numer. Anal., 41 (2003), 641-658.doi: 10.1137/S0036142901399392.

    [26]

    G. Guerra, Well-posedness for a scalar conservation law with singular nonconservative source, J. Differ. Equ., 206 (2004), 438-469.doi: 10.1016/j.jde.2004.04.008.

    [27]

    E. Isaacson and B. Temple, Convergence of the $2\times 2$ Godunov method for a general resonant nonlinear balance law, SIAM J. Appl. Math., 55 (1995), 625-640.doi: 10.1137/S0036139992240711.

    [28]

    S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (123) (1970), 228-255.

    [29]

    A.-Y. LeRoux, Riemann solvers for some hyperbolic problems with a source term, In "Actes du 30ème Congrès d'Analyse Numérique: CANum '98'' (Arles, 1998), ESAIM Proc., 6, Soc. Math. Appl. Indust., Paris, (1999), 75-90.

    [30]

    F. Lagoutière, N. Seguin and T. Takahashi, A simple 1D model of inviscid fluid-solid interaction, J. Differ. Equ., 245 (2008), 3503-3544.

    [31]

    R. J. LeVeque and B. Temple, Stability of Godunov's method for a class of $2\times 2$ systems of conservation laws, Trans. Amer. Math. Soc., 288 (1985), 115-123.doi: 10.2307/2000429.

    [32]

    P. G. LeFloch and A. E. Tzavaras, Representation of weak limits and definition of nonconservative products, SIAM J. Math. Anal., 30 (1999), 1309-1342.doi: 10.1137/S0036141098341794.

    [33]

    E. Yu. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyperbolic Differ. Equ., 4 (2007), 729-770.doi: 10.1142/S0219891607001343.

    [34]

    N. Seguin and J. Vovelle, Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients, Math. Models Methods Appl. Sci., 13 (2003), 221-257.doi: 10.1142/S0218202503002477.

    [35]

    B. Temple, Global solution of the Cauchy problem for a class of $2\times 2$ nonstrictly hyperbolic conservation laws, Adv. in Appl. Math., 3 (1982), 335-375.doi: 10.1016/S0196-8858(82)80010-9.

    [36]

    A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., 160 (2001), 181-193.doi: 10.1007/s002050100157.

    [37]

    A. Vasseur, Well-posedness of scalar conservation laws with singular sources, Methods Appl. Anal., 9 (2002), 291-312.

    [38]

    J. Vovelle, Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains, Numer. Math., 90 (2002), 563-596.doi: 10.1007/s002110100307.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(114) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return