# Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients

Kenneth Hvistendahl Karlsen; Nils Henrik Risebro

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 35, Issue: 2, page 239-269
- ISSN: 0764-583X

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topKarlsen, Kenneth Hvistendahl, and Risebro, Nils Henrik. "Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients." ESAIM: Mathematical Modelling and Numerical Analysis 35.2 (2010): 239-269. <http://eudml.org/doc/197398>.

@article{Karlsen2010,

abstract = {
We consider the initial value problem for degenerate
viscous and inviscid scalar conservation laws where the
flux function depends on the spatial location through a
"rough"coefficient function k(x).
We show that the Engquist-Osher
(and hence all monotone)
finite difference approximations converge
to the unique entropy solution
of the governing equation
if, among other demands, k' is in BV, thereby providing
alternative (new) existence proofs for entropy solutions of
degenerate convection-diffusion equations as
well as new convergence results for their finite difference
approximations. In the inviscid case, we also provide
a rate of convergence. Our convergence proofs are based on
deriving a series of a priori estimates
and using a general Lp compactness criterion.
},

author = {Karlsen, Kenneth Hvistendahl, Risebro, Nils Henrik},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Conservation law; degenerate convection-diffusion equation;
entropy solution; finite difference scheme; convergence; error estimate.; initial value problem; degenerate scalar conservation laws; entropy solutions; Engquist-Osher finite difference approximations; flux function; degenerate convection-diffusion equations; rate of convergence; a priori estimates; compactness criterion},

language = {eng},

month = {3},

number = {2},

pages = {239-269},

publisher = {EDP Sciences},

title = {Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients},

url = {http://eudml.org/doc/197398},

volume = {35},

year = {2010},

}

TY - JOUR

AU - Karlsen, Kenneth Hvistendahl

AU - Risebro, Nils Henrik

TI - Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 35

IS - 2

SP - 239

EP - 269

AB -
We consider the initial value problem for degenerate
viscous and inviscid scalar conservation laws where the
flux function depends on the spatial location through a
"rough"coefficient function k(x).
We show that the Engquist-Osher
(and hence all monotone)
finite difference approximations converge
to the unique entropy solution
of the governing equation
if, among other demands, k' is in BV, thereby providing
alternative (new) existence proofs for entropy solutions of
degenerate convection-diffusion equations as
well as new convergence results for their finite difference
approximations. In the inviscid case, we also provide
a rate of convergence. Our convergence proofs are based on
deriving a series of a priori estimates
and using a general Lp compactness criterion.

LA - eng

KW - Conservation law; degenerate convection-diffusion equation;
entropy solution; finite difference scheme; convergence; error estimate.; initial value problem; degenerate scalar conservation laws; entropy solutions; Engquist-Osher finite difference approximations; flux function; degenerate convection-diffusion equations; rate of convergence; a priori estimates; compactness criterion

UR - http://eudml.org/doc/197398

ER -

## References

top- M. Afif and B. Amaziane, Convergence of finite volume schemes for a degenerate convection-diffusion equation arising in two-phase flow in porous media. Preprint (1999). Zbl1052.65516
- F. Bouchut, F.R. Guarguaglini and R. Natalini, Diffusive BGK approximations for nonlinear multidimensional parabolic equations. Indiana Univ. Math. J.49 (2000) 723-749. Zbl0964.35011
- R. Bürger, S. Evje and K.H. Karlsen, On strongly degenerate convection-diffusion problems modeling sedimentation-consolidation processes. J. Math. Anal. Appl.247 (2000) 517-556. Zbl0961.35078
- M.C. Bustos, F. Concha, R. Bürger and E.M. Tory, Sedimentation and thickening: Phenomenological foundation and mathematical theory. Kluwer Academic Publishers, Dordrecht (1999). Zbl0936.76001
- J. Carrillo, Entropy solutions for nonlinear degenerate problems. Arch. Rational Mech. Anal.147 (1999) 269-361. Zbl0935.35056
- C. Chainais-Hillairet, Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate. RAIRO-Modél. Math. Anal. Numér.33 (1999) 129-156. Zbl0921.65071
- S. Champier, T. Gallouët and R. Herbin, Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh. Numer. Math.66 (1993) 139-157. Zbl0801.65089
- B. Cockburn, F. Coquel and P. Le Floch, An error estimate for finite volume methods for multidimensional conservation laws. Math. Comp.63 (1994) 77-103. Zbl0855.65103
- B. Cockburn, F. Coquel and P.G. LeFloch, Convergence of the finite volume method for multidimensional conservation laws. SIAM J. Numer. Anal.32 (1995) 687-705. Zbl0845.65051
- B. Cockburn and P.-A. Gremaud, A priori error estimates for numerical methods for scalar conservation laws. I. The general approach. Math. Comp.65 (1996) 533-573. Zbl0848.65067
- B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal.35 (1998) 2440-2463 (electronic). Zbl0927.65118
- M.G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws. Math. Comp.34 (1980) 1-21. Zbl0423.65052
- M.G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings. Proc. Amer. Math. Soc.78 (1980) 385-390. Zbl0449.47059
- B. Engquist and S. Osher, One-sided difference approximations for nonlinear conservation laws. Math. Comp.36 (1981) 321-351. Zbl0469.65067
- M.S. Espedal and K.H. Karlsen, Numerical solution of reservoir flow models based on large time step operator splitting algorithms, in Filtration in Porous media and industrial applications. Lect. Notes Math.1734, Springer, Berlin (2000) 9-77. Zbl1077.76546
- S. Evje and K.H. Karlsen, Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations. Numer. Math.86 (2000) 377-417. Zbl0963.65094
- S. Evje and K.H. Karlsen, Degenerate convection-diffusion equations and implicit monotone difference schemes, in Hyperbolic problems: Theory, numerics, applications, Vol. I (Zürich, 1998). Birkhäuser, Basel (1999) 285-294. Zbl0931.65094
- S. Evje and K.H. Karlsen, Viscous splitting approximation of mixed hyperbolic-parabolic convection-diffusion equations. Numer. Math.83 (1999) 107-137. Zbl0961.65084
- S. Evje and K.H. Karlsen, Monotone difference approximations of BV solutions to degenerate convection-diffusion equations. SIAM J. Numer. Anal.37 (2000) 1838-1860 (electronic). Zbl0985.65100
- S. Evje and K.H. Karlsen, Second order difference schemes for degenerate convection-diffusion equations. Preprint (in preparation). Zbl0931.65094
- R. Eymard, T. Gallouët, M. Ghilani and R. Herbin, Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes. IMA J. Numer. Anal.18 (1998) 563-594. Zbl0973.65078
- R. Eymard, T. Gallouët, D. Hilhorst and Y. Naït Slimane, Finite volumes and nonlinear diffusion equations. RAIRO-Modél. Math. Anal. Numér.32 (1998) 747-761. Zbl0914.65101
- T. Gimse and N.H. Risebro, Solution of the Cauchy problem for a conservation law with a discontinuous flux function. SIAM J. Math. Anal.23 (1992) 635-648. Zbl0776.35034
- A. Harten, J.M. Hyman and P.D. Lax, On finite-difference approximations and entropy conditions for shocks. Comm. Pure Appl. Math.XXIX (1976) 297-322. Zbl0351.76070
- H. Holden, K.H. Karlsen and K.-A. Lie, Operator splitting methods for degenerate convection-diffusion equations I: Convergence and entropy estimates, in Stochastic processes, physics and geometry: New interplays. A volume in honor of Sergio Albeverio. Amer. Math. Soc. (to appear).
- H. Holden, K.H. Karlsen, K.-A. Lie and N.H. Risebro, Operator splitting for nonlinear partial differential equations: An L1 convergence theory. Preprint (in preparation). Zbl0989.65093
- E. Isaacson and B. Temple, Convergence of the 2 x 2 Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math.55 (1995) 625-640. Zbl0838.35075
- K.H. Karlsen and N.H. Risebro, On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Preprint, Department of Mathematics, University of Bergen (2000). Zbl1027.35057
- C. Klingenberg and N.H. Risebro, Stability of a resonant system of conservation laws modeling polymer flow with gravitation. J. Differential Equations March (2000). Zbl0977.35083
- C. Klingenberg and N.H. Risebro, Convex conservation laws with discontinuous coefficients. Existence, uniqueness and asymptotic behavior. Comm. Partial Differential Equations20 (1995) 1959-1990. Zbl0836.35090
- D. Kröner, S. Noelle and M. Rokyta, Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions. Numer. Math.71 (1995) 527-560. Zbl0841.65079
- D. Kröner and M. Rokyta, Convergence of upwind finite volume schemes for scalar conservation laws in two dimensions. SIAM J. Numer. Anal.31 (1994) 324-343. Zbl0856.65104
- S.N. Kruzkov, Results on the nature of the continuity of solutions of parabolic equations, and certain applications thereof. Mat. Zametki6 (1969) 97-108.
- S.N. Kruzkov, First order quasi-linear equations in several independent variables. Math. USSR Sbornik10 (1970) 217-243.
- A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys.160 (2000) 241-282. Zbl0987.65085
- N.N. Kuznetsov, Accuracy of some approximative methods for computing the weak solutions of a first-order quasi-linear equation. USSR Comput. Math. Math. Phys. Dokl.16 (1976) 105-119. Zbl0381.35015
- B.J. Lucier, Error bounds for the methods of Glimm, Godunov and LeVeque. SIAM J. Numer. Anal.22 (1985) 1074-1081. Zbl0584.65059
- S. Noelle, Convergence of higher order finite volume schemes on irregular grids. Adv. Comput. Math.3 (1995) 197-218. Zbl0834.65088
- M. Ohlberger, A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations. Preprint, Mathematische Fakultät, Albert-Ludwigs-Universität Freiburg (2000). Zbl0992.65100
- O.A. Oleĭnik, Discontinuous solutions of non-linear differential equations. Amer. Math. Soc Transl. Ser. 226 (1963) 95-172.
- S. Osher and E. Tadmor, On the convergence of difference approximations to scalar conservation laws. Math. Comp.50 (1988) 19-51. Zbl0637.65091
- É. Rouvre and G. Gagneux, Solution forte entropique de lois scalaires hyperboliques-paraboliques dégénérées. C. R. Acad. Sci. Paris Sér. I Math.329 (1999) 599-602.
- A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov and A.P. Mikhailov, Blow-up in quasilinear parabolic equations. Walter de Gruyter & Co., Berlin (1995). Translated from the 1987 Russian original by Michael Grinfeld and revised by the authors. Zbl1020.35001
- R. Sanders, On convergence of monotone finite difference schemes with variable spatial differencing. Math. Comp.40 (1983) 91-106. Zbl0533.65061
- B. Temple, Global solution of the Cauchy problem for a class of 2 x 2 nonstrictly hyperbolic conservation laws. Adv. in Appl. Math.3 (1982) 335-375. Zbl0508.76107
- J. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux. Preprint, Available at the URL Zbl0972.65060URIhttp://www.math.ntnu.no/conservation/
- J. Towers, A difference scheme for conservation laws with a discontinuous flux - the nonconvex case. Preprint, Available at the URL Zbl1055.65104URIhttp://www.math.ntnu.no/conservation/
- J.-P. Vila, Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws. I. Explicit monotone schemes. RAIRO-Modél. Math. Anal. Numér.28 (1994) 267-295. Zbl0823.65087
- A.I. Vol'pert, The spaces BV and quasi-linear equations. Math. USSR Sbornik2 (1967) 225-267.
- A.I. Vol'pert and S.I. Hudjaev, Cauchy's problem for degenerate second order quasilinear parabolic equations. Math. USSR Sbornik7 (1969) 365-387.

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