𝐿¹ error estimates for difference approximations of degenerate convection-diffusion equations

Karlsen, K.; Risebro, N.; Storrøsten, E.

doi:10.1090/S0025-5718-2014-02818-4

$L^1$ error estimates for difference approximations of degenerate convection-diffusion equations
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by K. H. Karlsen, N. H. Risebro and E. B. Storrøsten;

Math. Comp. 83 (2014), 2717-2762

DOI: https://doi.org/10.1090/S0025-5718-2014-02818-4

Published electronically: March 28, 2014

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Abstract:

We analyze monotone finite difference schemes for strongly degenerate convection-diffusion equations in one spatial dimension. These nonlinear equations are well-posed within a class of (discontinuous) entropy solutions. We prove that the $L^1$ error between the approximate and exact solutions is $\mathcal {O}(\Delta x^{1/3})$, where $\Delta x$ is the spatial grid parameter. This result should be compared with the classical $\mathcal {O}(\Delta x^{1/2})$ error estimate for conservation laws (Kuznecov, 1976), and a recent estimate of $\mathcal {O}(\Delta x^{1/11})$ for degenerate convection-diffusion equations (Karlsen, Koley, Risebro 2012).

References

Boris Andreianov and Noureddine Igbida, On uniqueness techniques for degenerate convection-diffusion problems, Int. J. Dyn. Syst. Differ. Equ. 4 (2012), no. 1-2, 3–34. MR 2908633, DOI 10.1504/IJDSDE.2012.045992
B. Andreianov, M. Bendahmane, and K. H. Karlsen, Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations, J. Hyperbolic Differ. Equ. 7 (2010), no. 1, 1–67. MR 2646796, DOI 10.1142/S0219891610002062
D. Aregba-Driollet, R. Natalini, and S. Tang, Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems, Math. Comp. 73 (2004), no. 245, 63–94. MR 2034111, DOI 10.1090/S0025-5718-03-01549-7
F. Bouchut, F. R. Guarguaglini, and R. Natalini, Diffusive BGK approximations for nonlinear multidimensional parabolic equations, Indiana Univ. Math. J. 49 (2000), no. 2, 723–749. MR 1793689, DOI 10.1512/iumj.2000.49.1811
José Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal. 147 (1999), no. 4, 269–361. MR 1709116, DOI 10.1007/s002050050152
Gui-Qiang Chen and Kenneth H. Karlsen, Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients, Commun. Pure Appl. Anal. 4 (2005), no. 2, 241–266. MR 2149515, DOI 10.3934/cpaa.2005.4.241
Bernardo Cockburn, Continuous dependence and error estimation for viscosity methods, Acta Numer. 12 (2003), 127–180. MR 2249155, DOI 10.1017/S0962492902000107
Bernardo Cockburn and Pierre-Alain Gremaud, A priori error estimates for numerical methods for scalar conservation laws. I. The general approach, Math. Comp. 65 (1996), no. 214, 533–573. MR 1333308, DOI 10.1090/S0025-5718-96-00701-6
B. Cockburn and G. Gripenberg, Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations, J. Differential Equations 151 (1999), no. 2, 231–251. MR 1669570, DOI 10.1006/jdeq.1998.3499
Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2010. MR 2574377, DOI 10.1007/978-3-642-04048-1
Steinar Evje and Kenneth Hvistendahl Karlsen, Degenerate convection-diffusion equations and implicit monotone difference schemes, Hyperbolic problems: theory, numerics, applications, Vol. I (Zürich, 1998) Internat. Ser. Numer. Math., vol. 129, Birkhäuser, Basel, 1999, pp. 285–294. MR 1717198
Steinar Evje and Kenneth Hvistendahl Karlsen, Monotone difference approximations of BV solutions to degenerate convection-diffusion equations, SIAM J. Numer. Anal. 37 (2000), no. 6, 1838–1860. MR 1766850, DOI 10.1137/S0036142998336138
Steinar Evje and Kenneth Hvistendahl Karlsen, Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations, Numer. Math. 86 (2000), no. 3, 377–417. MR 1785415, DOI 10.1007/s002110000156
Steinar Evje and Kenneth H. Karlsen, An error estimate for viscous approximate solutions of degenerate parabolic equations, J. Nonlinear Math. Phys. 9 (2002), no. 3, 262–281. MR 1916386, DOI 10.2991/jnmp.2002.9.3.3
Robert Eymard, Thierry Gallouët, and Raphaèle Herbin, Error estimate for approximate solutions of a nonlinear convection-diffusion problem, Adv. Differential Equations 7 (2002), no. 4, 419–440. MR 1869118
Robert Eymard, Thierry Gallouët, Raphaèle Herbin, and Anthony Michel, Convergence of a finite volume scheme for nonlinear degenerate parabolic equations, Numer. Math. 92 (2002), no. 1, 41–82. MR 1917365, DOI 10.1007/s002110100342
Helge Holden, Kenneth H. Karlsen, Knut-Andreas Lie, and Nils Henrik Risebro, Splitting methods for partial differential equations with rough solutions, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2010. Analysis and MATLAB programs. MR 2662342, DOI 10.4171/078
Kenneth Hvistendahl Karlsen and Nils Henrik Risebro, On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients, Discrete Contin. Dyn. Syst. 9 (2003), no. 5, 1081–1104. MR 1974417, DOI 10.3934/dcds.2003.9.1081
Kenneth Hvistendahl Karlsen and Nils Henrik Risebro, Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients, M2AN Math. Model. Numer. Anal. 35 (2001), no. 2, 239–269. MR 1825698, DOI 10.1051/m2an:2001114
K. H. Karlsen, U. Koley, and N. H. Risebro, An error estimate for the finite difference approximation to degenerate convection-diffusion equations, Numer. Math. 121 (2012), no. 2, 367–395. MR 2917166, DOI 10.1007/s00211-011-0433-9
S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.) 81(123) (1970), 228–255 (Russian). MR 267257
N. N. Kuznecov, The accuracy of certain approximate methods for the computation of weak solutions of a first order quasilinear equation, Ž. Vyčisl. Mat i Mat. Fiz. 16 (1976), no. 6, 1489–1502, 1627 (Russian). MR 483509
G. E. Ladas and V. Lakshmikantham, Differential equations in abstract spaces, Mathematics in Science and Engineering, Vol. 85, Academic Press, New York-London, 1972. MR 460832
Mario Ohlberger, A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations, M2AN Math. Model. Numer. Anal. 35 (2001), no. 2, 355–387. MR 1825703, DOI 10.1051/m2an:2001119
Tamir Tassa, Regularity of weak solutions of the nonlinear Fokker-Planck equation, Math. Res. Lett. 3 (1996), no. 4, 475–490. MR 1406013, DOI 10.4310/MRL.1996.v3.n4.a6
A. I. Vol′pert, Spaces $\textrm {BV}$ and quasilinear equations, Mat. Sb. (N.S.) 73(115) (1967), 255–302 (Russian). MR 216338
A. I. Vol′pert and S. I. Hudjaev, The Cauchy problem for second order quasilinear degenerate parabolic equations, Mat. Sb. (N.S.) 78(120) (1969), 374–396 (Russian). MR 264232
Zhuo Qun Wu and Jing Xue Yin, Some properties of functions in $BV_x$ and their applications to the uniqueness of solutions for degenerate quasilinear parabolic equations, Northeast. Math. J. 5 (1989), no. 4, 395–422. MR 1053519