On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions

Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik; Storrøsten, Erlend Briseid

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 $𝒪\left(\Delta x^{2/(19+d)}\right)$, where $d$ is the spatial dimension and $\Delta x$ 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.

Classification : 65M06, 65M15, 35K65, 35L65
Mots clés : Degenerate convection-diffusion equations, entropy conditions, finite difference methods, error estimates

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 BV_x and their applications to the uniqueness of solutions for degenerate quasilinear parabolic equations. Northeast. Math. J. 5 (1989) 395–422. | MR | Zbl