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On the convergence of a second order approximation of conservation laws with discontinuous flux

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Abstract

A second order scheme is constructed for the scalar conservation laws with flux function allowed to be discontinuous in the space variable. The corresponding numerical solutionis shown to converge to the (A,B) entropysolution. Numerical results are presented to validate the convergence analysis. In addition, it is inferred that these numerical solutions are comparable with those obtained using the standard minmod limiter.

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References

  1. Adimurthi R. Dutta, J. Jaffré and G.D. Veerappa Gowda. Monotone (A,B) entropy stable numerical scheme for scalar conservation laws with discontinuous flux. ESAIM Math. Model. Numer. Anal., 48(6) (2014), 1725–1755.

    Article  MathSciNet  MATH  Google Scholar 

  2. Adimurthi R. Dutta, S.S. Ghoshal and G.D. Veerappa Gowda. Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux. Comm. Pure. Appl. Math., LXIV (2011), 0084–0115.

    MathSciNet  MATH  Google Scholar 

  3. Adimurthi J. Jaffré and G.D. Veerappa Gowda. Godunov-type methods for conservation laws with a flux function discontinuous in space. SIAM Journal in Numerical Analysis, 42 (2004), 179–208.

    Article  MathSciNet  MATH  Google Scholar 

  4. Adimurthi S. Mishra and G.D. Veerappa Gowda. Optimal entropy solutions for conservation laws with discontinuous flux-functions. J. Hyperbolic Differ. Equ., 2(4) (2005), 783–837.

    Article  MathSciNet  MATH  Google Scholar 

  5. Adimurthi S. Mishra and G.D. Veerappa Gowda. Convergence of Godunov type methods for a conservation law with a spatially varying discontinuous flux function. Math. Comp., 76(259) (2007), 1219–1242.

    Article  MathSciNet  MATH  Google Scholar 

  6. Adimurthi S. Mishra and Gowda G.D. Veerappa. Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes. Netw. Heterog. Media, 2(1) (2007), 127–157.

    MathSciNet  MATH  Google Scholar 

  7. Adimurthi S. Mishra and G.D Veerappa Gowda. Conservation laws with discontinuous flux. J. Math. Kyoto. Univ., 43(1) (2003), 27–70.

    MathSciNet  MATH  Google Scholar 

  8. R. Burger, K.H. Karlsen, N.H. Risebro and J.D. Towers. Well-posedness in BVt and convergence of a difference scheme for continuous sedimentation in ideal clarifier thickener units. Numer. Math., 97 (1) (2004), 25–65.

    Article  MathSciNet  MATH  Google Scholar 

  9. R. Burger, K.H. Karlsen, H. Torres and J.D. Towers. Second-order schemes for conservation laws with discontinuous flux modelling clarifier-thickener units. Numer. Math., 116(4) (2010), 579–617.

    Article  MathSciNet  MATH  Google Scholar 

  10. R. Burger, 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(3) (2009), 1684–1712.

    Article  MathSciNet  MATH  Google Scholar 

  11. T. Gimse and N.H. Risebro. Solution of Cauchy problem for a conservation law with discontinuous flux function. SIAM J. Math. Anal., 23(3) (1992), 635–648.

    Article  MathSciNet  MATH  Google Scholar 

  12. K.H. Karlsen, N.H. Risebro and J.D. Towers. On a nonlinear degenerate parabolic transport-diffusion equation with discontinuous coefficient. Electron. J. Differential Equations, (2002), no. 93, 1–23.

    MathSciNet  MATH  Google Scholar 

  13. K.H. Karlsen, N.H. Risebro and J.D. Towers. Upwind difference approximationsfor degenerate parabolic convection-diffusionequationswith a discontinuous coefficient. SIAMJ. Numer. Anal., 22(4) (2002), 623–664.

    Article  MathSciNet  MATH  Google Scholar 

  14. K.H. Karlsen and J.D. Towers. Convergence of the Lax-Friedrichs scheme and stability of conservation laws with a discontinuous time-dependent flux. Chinese Ann. Math. Ser. B., 25(3) (2004), 287–318.

    Article  MathSciNet  MATH  Google Scholar 

  15. C. Klingenberg and N.H. Risebro. Convex conservation laws with discontinuous coefficients: Existence, uniqueness and Asymptotic behavior. Comm. Part. Diff. Eqns., 20(11-12) (1995), 1959–1990.

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Mishra. Convergence of upwind finite difference schemes for a scalar conservation law with indefinite discontinuities in the flux function. SIAM J. Num. Anal., 43(2) (2005), 559–577.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Mishra and J. Jaffré, On the upstream mobility scheme for two-phase flow in porous media. Comput. Geosci., 14(1) (2010), 105–124.

    Article  MathSciNet  MATH  Google Scholar 

  18. J.D. Towers. Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAMJ. Numer. Anal., 38(2) (2000), 681–698.

    Article  MathSciNet  MATH  Google Scholar 

  19. J.D. Towers. A difference scheme for conservation laws with a discontinuous flux: the nonconvex case. SIAMJ. Numer. Anal., 39(4) (2001), 1197–1218.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. TIFR Centre for Applicable Mathematics, Bangalore, India

    Adimurthi & G. D. Veerappa Gowda

  2. Centro de Investigación en Ingeniería Matemática, UDEC, Casilla-160, Concepcion, 4030000, Chile

    Sudarshan Kumar Kenettinkara

Authors
  1. Adimurthi
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  2. Sudarshan Kumar Kenettinkara
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  3. G. D. Veerappa Gowda
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Correspondence to Sudarshan Kumar Kenettinkara.

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Adimurthi, Kumar Kenettinkara, S. & Veerappa Gowda, G.D. On the convergence of a second order approximation of conservation laws with discontinuous flux. Bull Braz Math Soc, New Series 47, 21–35 (2016). https://doi.org/10.1007/s00574-016-0119-0

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  • DOI: https://doi.org/10.1007/s00574-016-0119-0

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