On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions. (English) Zbl 1342.65182
This work aims to derive the error estimates for a class of finite difference methods of nonlinear, possibly strongly degenerate, convection-diffusion problems; particularly this is aimed at entropy solutions to convection-diffusion equations. The monotone methods make use of an upwind discretisation of the convection term and a centred discretisation for the parabolic term. It is shown that the local \(L^1\)-error is \(O(\Delta x ^{2/(19+d)})\), where \(d\) is the spatial dimension. The proof provided makes use of specific kinetic formulations of the difference method and the convection-diffusion equation.
Reviewer: Charis Harley (Johannesburg)
MSC:
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 35K65 | Degenerate parabolic equations |
Keywords:
degenerate convection-diffusion equations; entropy conditions; finite difference methods; error estimatesReferences:
| [1] | 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. · Zbl 1207.35020 · doi:10.1142/S0219891610002062 |
| [2] | B. Andreianov and N. Igbida, On uniqueness techniques for degenerate convection-diffusion problems. Int. J. Dyn. Syst. Differ. Eq.4 (2012) 3-34. · Zbl 1263.35007 |
| [3] | D. Aregba-Driollet, R. Natalini and S. Tang, Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems. Math. Comput.73 (2004) 63-94. · Zbl 1031.65093 · doi:10.1090/S0025-5718-03-01549-7 |
| [4] | G. Barles and E.R. Jakobsen, Error bounds for monotone approximation schemes for Hamiltonacobiellman equations. SIAM J. Numer. Anal.43 (2005) 540-558. · Zbl 1092.65077 · doi:10.1137/S003614290343815X |
| [5] | F. Bouchut and B. Perthame, Kružkov’s estimates for scalar conservation laws revisited. Trans. Amer. Math. Soc.350 (1998) 2847-2870. · Zbl 0955.65069 · doi:10.1090/S0002-9947-98-02204-1 |
| [6] | F. Bouchut, F. R. Guarguaglini and R. Natalini, Diffusive BGK approximations for nonlinear multidimensional parabolic equations. Indiana Univ. Math. J.49 (2000) 723-749. · Zbl 0964.35011 · doi:10.1512/iumj.2000.49.1811 |
| [7] | 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. · Zbl 1140.65075 · doi:10.1002/cpa.20208 |
| [8] | J. Carrillo, Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal.147 (1999) 269-361. · Zbl 0935.35056 · doi:10.1007/s002050050152 |
| [9] | 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. · Zbl 0935.47034 · doi:10.1016/S0764-4442(98)80020-7 |
| [10] | G.-Q. Chen and B. Perthame, Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire20 (2003) 645-668. · Zbl 1031.35077 · doi:10.1016/S0294-1449(02)00014-8 |
| [11] | 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. · Zbl 1084.35034 · doi:10.3934/cpaa.2005.4.241 |
| [12] | G.-Q. Chen and K.H. Karlsen, L^{1}-framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations. Trans. Amer. Math. Soc.358 (2006) 937-963. · Zbl 1096.35072 · doi:10.1090/S0002-9947-04-03689-X |
| [13] | B. Cockburn, Continuous dependence and error estimation for viscosity methods. Acta Numer.12 (2003) 127-180. · Zbl 1048.65090 · doi:10.1017/S0962492902000107 |
| [14] | 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. · Zbl 0226.47038 · doi:10.2307/2373376 |
| [15] | M.G. Crandall and P.-L. Lions, Two approximations of solutions of Hamilton-Jacobi equations. Math. Comput.43 (1984) 1-19. · Zbl 0557.65066 · doi:10.1090/S0025-5718-1984-0744921-8 |
| [16] | 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). · Zbl 1196.35001 |
| [17] | S. Evje and K.H. Karlsen, Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations. Numer. Math.86 (2000) 377-417. · Zbl 0963.65094 · doi:10.1007/s002110000156 |
| [18] | 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. · Zbl 0985.65100 · doi:10.1137/S0036142998336138 |
| [19] | 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. · Zbl 1183.35180 · doi:10.2991/jnmp.2002.9.3.3 |
| [20] | 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. · Zbl 1173.35563 |
| [21] | 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. · Zbl 1005.65099 · doi:10.1007/s002110100342 |
| [22] | 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). · Zbl 1191.35005 |
| [23] | K.H. Karlsen and N.H. Risebro, Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients. ESAIM: M2AN35 (2001) 239-269. · Zbl 1032.76048 · doi:10.1051/m2an:2001114 |
| [24] | 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. · Zbl 1248.65098 · doi:10.1007/s00211-011-0433-9 |
| [25] | K.H. Karlsen, N.H. Risebro E.B. Storrøsten, L^{1} error estimates for difference approximations of degenerate convection-diffusion equations. Math. Comput.83 (2014) 2717-2762. · Zbl 1300.65067 · doi:10.1090/S0025-5718-2014-02818-4 |
| [26] | S.N. Kružkov, First order quasilinear equations with several independent variables. Mat. Sb. (N.S.)81 (1970) 228-255. |
| [27] | N.V. Krylov, The rate of convergence of finite-difference approximations for Bellman equations with Lipschitz coefficients. Appl. Math. Optim.52 (2005) 365-399. · Zbl 1087.65100 · doi:10.1007/s00245-005-0832-3 |
| [28] | 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. |
| [29] | 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. · Zbl 0820.35094 · doi:10.1090/S0894-0347-1994-1201239-3 |
| [30] | 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. · Zbl 1035.35068 · doi:10.1137/S0036141002407995 |
| [31] | M. Ohlberger, A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations. ESAIM: M2AN35 (2001) 355-387. · Zbl 0992.65100 · doi:10.1051/m2an:2001119 |
| [32] | N.H. Pavel, Differential equations, flow invariance and applications. Vol. 113 of Res. Notes Math. Pitman (Advanced Publishing Program), Boston, MA (1984). |
| [33] | 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. · Zbl 0919.35088 · doi:10.1016/S0021-7824(99)80003-8 |
| [34] | K. Sato, On the generators of non-negative contraction semigroups in Banach lattices. J. Math. Soc. Japan20 (1968) 423-436. · Zbl 0167.13601 · doi:10.2969/jmsj/02030423 |
| [35] | 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. |
| [36] | 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. |
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