## Convergence of Godunov type methods for a conservation law with a spatially varying discontinuous flux function

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- by Adimurthi, Siddhartha Mishra and G. D. Veerappa Gowda PDF
- Math. Comp.
**76**(2007), 1219-1242 Request permission

## Abstract:

We deal with single conservation laws with a spatially varying and possibly discontinuous coefficient. This equation includes as a special case single conservation laws with conservative and possibly singular source terms. We extend the framework of optimal entropy solutions for these classes of equations based on a two-step approach. In the first step, an interface connection vector is used to define infinite classes of entropy solutions. We show that each of these classes of solutions is stable in $L^1$. This allows for the possibility of choosing one of these classes of solutions based on the physics of the problem. In the second step, we define optimal entropy solutions based on the solution of a certain optimization problem at the discontinuities of the coefficient. This method leads to optimal entropy solutions that are consistent with physically observed solutions in two-phase flows in heterogeneous porous media. Another central aim of this paper is to develop suitable numerical schemes for these equations. We develop and analyze a set of Godunov type finite volume methods that are based on exact solutions of the corresponding Riemann problem. Numerical experiments are shown comparing the performance of these schemes on a set of test problems.## References

- Adimurthi and G. D. Veerappa Gowda,
*Conservation law with discontinuous flux*, J. Math. Kyoto Univ.**43**(2003), no. 1, 27–70. MR**2028700**, DOI 10.1215/kjm/1250283740 - Adimurthi, Jérôme Jaffré, and G. D. Veerappa Gowda,
*Godunov-type methods for conservation laws with a flux function discontinuous in space*, SIAM J. Numer. Anal.**42**(2004), no. 1, 179–208. MR**2051062**, DOI 10.1137/S003614290139562X - Adimurthi, Siddhartha Mishra, and G. D. Veerappa Gowda,
*Optimal entropy solutions for conservation laws with discontinuous flux-functions*, J. Hyperbolic Differ. Equ.**2**(2005), no. 4, 783–837. MR**2195983**, DOI 10.1142/S0219891605000622 - Adimurthi, Siddhartha Mishra and G. D. Veerappa Gowda,
*Conservation laws with flux function discontinuous in the space variable - II, Discontinuous convex-concave type fluxes and generalised entropy solutions*, To appear in J. Comp. Appl. Math. - Adimurthi, Siddhartha Mishra and G. D. Veerappa Gowda,
*Conservation laws with flux function discontinuous in the space variable - III, The general case*, Preprint. - R. Bürger, K. H. Karlsen, N. H. Risebro, and J. D. Towers,
*Well-posedness in $BV_t$ and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units*, Numer. Math.**97**(2004), no. 1, 25–65. MR**2045458**, DOI 10.1007/s00211-003-0503-8 - Michael G. Crandall and Andrew Majda,
*Monotone difference approximations for scalar conservation laws*, Math. Comp.**34**(1980), no. 149, 1–21. MR**551288**, DOI 10.1090/S0025-5718-1980-0551288-3 - Stefan Diehl,
*On scalar conservation laws with point source and discontinuous flux function*, SIAM J. Math. Anal.**26**(1995), no. 6, 1425–1451. MR**1356452**, DOI 10.1137/S0036141093242533 - Stefan Diehl,
*A conservation law with point source and discontinuous flux function modelling continuous sedimentation*, SIAM J. Appl. Math.**56**(1996), no. 2, 388–419. MR**1381652**, DOI 10.1137/S0036139994242425 - J. M. Greenberg, A. Y. Leroux, R. Baraille, and A. Noussair,
*Analysis and approximation of conservation laws with source terms*, SIAM J. Numer. Anal.**34**(1997), no. 5, 1980–2007. MR**1472206**, DOI 10.1137/S0036142995286751 - Tore Gimse and Nils Henrik Risebro,
*Riemann problems with a discontinuous flux function*, Third International Conference on Hyperbolic Problems, Vol. I, II (Uppsala, 1990) Studentlitteratur, Lund, 1991, pp. 488–502. MR**1109304** - Tore Gimse and Nils Henrik Risebro,
*Solution of the Cauchy problem for a conservation law with a discontinuous flux function*, SIAM J. Math. Anal.**23**(1992), no. 3, 635–648. MR**1158825**, DOI 10.1137/0523032 - S. Godunov,
*Finite difference methods for numerical computation of discontinuous solutions of the equations of fluid dynamics*, Math.Sbornik, 47 (1959), 271-306. - Edwige Godlewski and Pierre-Arnaud Raviart,
*Hyperbolic systems of conservation laws*, Mathématiques & Applications (Paris) [Mathematics and Applications], vol. 3/4, Ellipses, Paris, 1991. MR**1304494** - Jerome Jaffre and Siddhartha Mishra,
*On the upstream mobility flux scheme for simulating two phase flow in heterogeneous porous media*, Preprint. - S. N. Kruzkhov,
*First order quasilinear equations in several independent variables*, Mat. Sb, 10(1970), 217-243. - Christian Klingenberg and Nils Henrik Risebro,
*Convex conservation laws with discontinuous coefficients. Existence, uniqueness and asymptotic behavior*, Comm. Partial Differential Equations**20**(1995), no. 11-12, 1959–1990. MR**1361727**, DOI 10.1080/03605309508821159 - Kenneth H. Karlsen, Nils H. Risebro, and John D. Towers,
*On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient*, Electron. J. Differential Equations (2002), No. 93, 23. MR**1938389** - 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**(2002), no. 4, 623–664. MR**1937244**, DOI 10.1093/imanum/22.4.623 - K. H. Karlsen, N. H. Risebro, and J. D. Towers,
*$L^1$ stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients*, Skr. K. Nor. Vidensk. Selsk.**3**(2003), 1–49. MR**2024741** - K. H. Karlsen and J. D. Towers,
*Convergence of the Lax-Friedrichs scheme and stability for conservation laws with a discontinous space-time dependent flux*, Chinese Ann. Math. Ser. B**25**(2004), no. 3, 287–318. MR**2086124**, DOI 10.1142/S0252959904000299 - E. F. Kaasschieter,
*Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium*, Comput. Geosci.**3**(1999), no. 1, 23–48. MR**1696184**, DOI 10.1023/A:1011574824970 - S. Mochon,
*An analysis of the traffic on highways with changing surface conditions*, Math. Modelling**9**(1987), no. 1, 1–11. MR**898784**, DOI 10.1016/0270-0255(87)90068-6 - David Stewart Ross,
*Two new moving boundary problems for scalar conservation laws*, Comm. Pure Appl. Math.**41**(1988), no. 5, 725–737. MR**948078**, DOI 10.1002/cpa.3160410511 - Siddhartha Mishra,
*Convergence of upwind finite difference schemes for a scalar conservation law with indefinite discontinuities in the flux function*, SIAM J. Numer. Anal.**43**(2005), no. 2, 559–577. MR**2177880**, DOI 10.1137/030602745 - Siddhartha Mishra,
*Analysis and Numerical approximation of conservation laws with discontinuous coefficients*, Ph.D. thesis, Indian Institute of Science, 2005. - John D. Towers,
*Convergence of a difference scheme for conservation laws with a discontinuous flux*, SIAM J. Numer. Anal.**38**(2000), no. 2, 681–698. MR**1770068**, DOI 10.1137/S0036142999363668 - John D. Towers,
*A difference scheme for conservation laws with a discontinuous flux: the nonconvex case*, SIAM J. Numer. Anal.**39**(2001), no. 4, 1197–1218. MR**1870839**, DOI 10.1137/S0036142900374974

## Additional Information

**Adimurthi**- Affiliation: TIFR center, P.O. Box 1234, Bangalore 560012, India
- Email: aditi@math.tifrbng.res.in
**Siddhartha Mishra**- Affiliation: Center of Mathematics for Applications, University of Oslo, P.O. Box 1053, Oslo–0316, Norway
- Email: siddharm@cma.uio.no
**G. D. Veerappa Gowda**- Affiliation: TIFR center, P.O.Box 1234, Bangalore 560012, India
- Email: gowda@math.tifrbng.res.in
- Received by editor(s): September 19, 2005
- Received by editor(s) in revised form: June 23, 2006
- Published electronically: January 25, 2007
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**76**(2007), 1219-1242 - MSC (2000): Primary 35L65, 65M06, 65M12
- DOI: https://doi.org/10.1090/S0025-5718-07-01960-6
- MathSciNet review: 2299772