On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions
Karlsen, Kenneth Hvistendahl1 ; Risebro, Nils Henrik1 ; Storrøsten, Erlend Briseid1
1 Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 499-539.

We analyze upwind difference methods for strongly degenerate convection-diffusion equations in several spatial dimensions. We prove that the local L 1 -error between the exact and numerical solutions is 𝒪 ( Δ x 2 / ( 19 + d ) ) , where d is the spatial dimension and Δx is the grid size. The error estimate is robust with respect to vanishing diffusion effects. The proof makes effective use of specific kinetic formulations of the difference method and the convection-diffusion equation. This paper is a continuation of [K.H. Karlsen, N.H. Risebro E.B. Storrøsten, Math. Comput. 83 (2014) 2717–2762], in which the one-dimensional case was examined using the Kružkov−Carrillo entropy framework.

DOI : 10.1051/m2an/2015057
Classification : 65M06, 65M15, 35K65, 35L65
Mots-clés : Degenerate convection-diffusion equations, entropy conditions, finite difference methods, error estimates
Karlsen, Kenneth Hvistendahl 1 ; Risebro, Nils Henrik 1 ; Storrøsten, Erlend Briseid 1

1 Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway 
@article{M2AN_2016__50_2_499_0,
     author = {Karlsen, Kenneth Hvistendahl and Risebro, Nils Henrik and Storr{\o}sten, Erlend Briseid},
     title = {On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {499--539},
     publisher = {EDP-Sciences},
     volume = {50},
     number = {2},
     year = {2016},
     doi = {10.1051/m2an/2015057},
     mrnumber = {3482553},
     zbl = {1342.65182},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2015057/}
}
TY  - JOUR
AU  - Karlsen, Kenneth Hvistendahl
AU  - Risebro, Nils Henrik
AU  - Storrøsten, Erlend Briseid
TI  - On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2016
SP  - 499
EP  - 539
VL  - 50
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2015057/
DO  - 10.1051/m2an/2015057
LA  - en
ID  - M2AN_2016__50_2_499_0
ER  - 
%0 Journal Article
%A Karlsen, Kenneth Hvistendahl
%A Risebro, Nils Henrik
%A Storrøsten, Erlend Briseid
%T On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2016
%P 499-539
%V 50
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2015057/
%R 10.1051/m2an/2015057
%G en
%F M2AN_2016__50_2_499_0
Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik; Storrøsten, Erlend Briseid. On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 499-539. doi : 10.1051/m2an/2015057. http://archive.numdam.org/articles/10.1051/m2an/2015057/

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) 1–67. | DOI | MR | Zbl

B. Andreianov and N. Igbida, On uniqueness techniques for degenerate convection-diffusion problems. Int. J. Dyn. Syst. Differ. Eq. 4 (2012) 3–34. | MR | Zbl

D. Aregba-Driollet, R. Natalini and S. Tang, Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems. Math. Comput. 73 (2004) 63–94. | DOI | MR | Zbl

G. Barles and E.R. Jakobsen, Error bounds for monotone approximation schemes for Hamilton−Jacobi−Bellman equations. SIAM J. Numer. Anal. 43 (2005) 540–558. | DOI | MR | Zbl

F. Bouchut and B. Perthame, Kružkov’s estimates for scalar conservation laws revisited. Trans. Amer. Math. Soc. 350 (1998) 2847–2870. | DOI | MR | Zbl

F. Bouchut, F. R. Guarguaglini and R. Natalini, Diffusive BGK approximations for nonlinear multidimensional parabolic equations. Indiana Univ. Math. J. 49 (2000) 723–749. | DOI | MR | Zbl

L.A. Caffarelli and P. E. Souganidis. A rate of convergence for monotone finite difference approximations to fully nonlinear, uniformly elliptic PDEs. Commun. Pure Appl. Math. 61 (2008) 1–17. | DOI | MR | Zbl

J. Carrillo, Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147 (1999) 269–361. | DOI | MR | Zbl

A. Chambolle and B.J. Lucier, Un principe du maximum pour des opérateurs monotones. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) 823–827. | DOI | MR | Zbl

G.-Q. Chen and B. Perthame, Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20 (2003) 645–668. | DOI | Numdam | MR | Zbl

G.-Q. Chen and K.H. Karlsen, Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients. Commun. Pure Appl. Anal. 4 (2005) 241–266. | DOI | MR | Zbl

G.-Q. Chen and K.H. Karlsen, L1-framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations. Trans. Amer. Math. Soc. 358 (2006) 937–963. | DOI | MR | Zbl

B. Cockburn, Continuous dependence and error estimation for viscosity methods. Acta Numer. 12 (2003) 127–180. | DOI | MR | Zbl

M.G. Crandall and T.M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces. Amer. J. Math. 93 (1971) 265–298. | DOI | MR | Zbl

M.G. Crandall and P.-L. Lions, Two approximations of solutions of Hamilton-Jacobi equations. Math. Comput. 43 (1984) 1–19. | DOI | MR | Zbl

C.M. Dafermos, Hyperbolic conservation laws in continuum physics. Vol. 325 of Grundl. Math. Wiss. [Fundamental Principles of Mathematical Sciences]. 3rd edition Springer-Verlag, Berlin (2010). | MR | Zbl

S. Evje and K.H. Karlsen, Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations. Numer. Math. 86 (2000) 377–417. | DOI | MR | Zbl

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. | DOI | MR | Zbl

S. Evje and K.H. Karlsen, An error estimate for viscous approximate solutions of degenerate parabolic equations. J. Nonlin. Math. Phys. 9 (2002) 262–281. | DOI | MR | Zbl

R. Eymard, T. Gallouët and R. Herbin, Error estimate for approximate solutions of a nonlinear convection-diffusion problem. Adv. Differ. Equ. 7 (2002) 419–440. | MR | Zbl

R. Eymard, T. Gallouët, R. Herbin and A. Michel, Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math. 92 (2002) 41–82. | DOI | MR | Zbl

H. Holden, K.H. Karlsen, K.-A. Lie and N.H. Risebro, Splitting methods for partial differential equations with rough solutions. EMS Series of Lect. Math. Analysis and MATLAB programs. European Mathematical Society (EMS), Zürich (2010). | MR | Zbl

K.H. Karlsen and N.H. Risebro, Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients. ESAIM: M2AN 35 (2001) 239–269. | DOI | Numdam | MR | Zbl

K.H. Karlsen, U. Koley N.H. Risebro, An error estimate for the finite difference approximation to degenerate convection-diffusion equations. Numer. Math. 121 (2012) 367–395. | DOI | MR | Zbl

K.H. Karlsen, N.H. Risebro E.B. Storrøsten, L1 error estimates for difference approximations of degenerate convection-diffusion equations. Math. Comput. 83 (2014) 2717–2762. | DOI | MR | Zbl

S.N. Kružkov, First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 (1970) 228–255. | MR | Zbl

N.V. Krylov, The rate of convergence of finite-difference approximations for Bellman equations with Lipschitz coefficients. Appl. Math. Optim. 52 (2005) 365–399. | DOI | MR | Zbl

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) 1489–1502, 1627. | MR | Zbl

P.-L. Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Amer. Math. Soc. 7 (1994) 169–191. | DOI | MR | Zbl

C. Makridakis and B. Perthame, Optimal rate of convergence for anisotropic vanishing viscosity limit of a scalar balance law. SIAM J. Math. Anal. 34 (2003) 1300–1307. | DOI | MR | Zbl

M. Ohlberger, A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations. ESAIM: M2AN 35 (2001) 355–387. | DOI | Numdam | MR | Zbl

N.H. Pavel, Differential equations, flow invariance and applications. Vol. 113 of Res. Notes Math. Pitman (Advanced Publishing Program), Boston, MA (1984). | MR | Zbl

B. Perthame, Uniqueness and error estimates in first order quasilinear conservation laws via the kinetic entropy defect measure. J. Math. Pures Appl. 77 (1998) 1055–1064. | DOI | MR | Zbl

K. Sato, On the generators of non-negative contraction semigroups in Banach lattices. J. Math. Soc. Japan 20 (1968) 423–436. | DOI | MR | Zbl

A.I. Vol’Pert and S.I. Hudjaev, The Cauchy problem for second order quasilinear degenerate parabolic equations. Mat. Sb. (N.S.) 78 (1969) 374–396. | MR | Zbl

Z.Q. Wu and J.X. Yin, Some properties of functions in BVx and their applications to the uniqueness of solutions for degenerate quasilinear parabolic equations. Northeast. Math. J. 5 (1989) 395–422. | MR | Zbl

Cité par Sources :