Uniform discrete Sobolev estimates of solutions to finite difference schemes for singular limits of nonlinear PDEs
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 727-757.

Uniform discrete Sobolev space estimates are proven for a class of finite-difference schemes for singularly-perturbed hyperbolic-parabolic systems. The estimates obtained improve previous results even when the PDEs do not involve singular perturbations. These estimates are used in a companion paper to prove the convergence of solutions as the discretization parameter and/or the singular perturbation parameter tends to zero.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016038
Classification : 65M10
Mots clés : Uniform estimates, finite-difference methods, discrete Sobolev spaces, fully-discrete sharp Gårding inequality, singular limits
Even-Dar Mandel, Liat 1, 2 ; Schochet, Steven 2

1 Department of Mathematics and Computer Science, Open University, 43107 Raanana, Israel.
2 School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel.
@article{M2AN_2017__51_2_727_0,
     author = {Even-Dar Mandel, Liat and Schochet, Steven},
     title = {Uniform discrete {Sobolev} estimates of solutions to finite difference schemes for singular limits of nonlinear {PDEs}},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {727--757},
     publisher = {EDP-Sciences},
     volume = {51},
     number = {2},
     year = {2017},
     doi = {10.1051/m2an/2016038},
     mrnumber = {3626417},
     zbl = {1368.65163},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2016038/}
}
TY  - JOUR
AU  - Even-Dar Mandel, Liat
AU  - Schochet, Steven
TI  - Uniform discrete Sobolev estimates of solutions to finite difference schemes for singular limits of nonlinear PDEs
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2017
SP  - 727
EP  - 757
VL  - 51
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2016038/
DO  - 10.1051/m2an/2016038
LA  - en
ID  - M2AN_2017__51_2_727_0
ER  - 
%0 Journal Article
%A Even-Dar Mandel, Liat
%A Schochet, Steven
%T Uniform discrete Sobolev estimates of solutions to finite difference schemes for singular limits of nonlinear PDEs
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2017
%P 727-757
%V 51
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2016038/
%R 10.1051/m2an/2016038
%G en
%F M2AN_2017__51_2_727_0
Even-Dar Mandel, Liat; Schochet, Steven. Uniform discrete Sobolev estimates of solutions to finite difference schemes for singular limits of nonlinear PDEs. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 727-757. doi : 10.1051/m2an/2016038. http://www.numdam.org/articles/10.1051/m2an/2016038/

G. Alì and L. Chen, The zero-electron-mass limit in the Euler-Poisson system for both well- and ill-prepared initial data. Nonlinearity 24 (2011) 2745–2761. | DOI | MR | Zbl

F. Ancona and A. Marson, A locally quadratic Glimm functional and sharp convergence rate of the Glimm scheme for nonlinear hyperbolic systems. Arch. Ration. Mech. Anal. 196 (2010) 455–487. | DOI | MR | Zbl

M. Bessemoulin-Chatard, C. Chainais-Hillairet and F. Filbet, On discrete functional inequalities for some finite volume schemes. IMA J. Numer. Anal. 35 (2015) 1125–1149. | DOI | MR | Zbl

A. Bressan, Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem. Vol. 20 of Oxford Lect. Ser. Math. Appl. Oxford University Press, Oxford (2000). | MR | Zbl

L. Chen, D. Donatelli and P. Marcati, Incompressible type limit analysis of a hydrodynamic model for charge-carrier transport. SIAM J. Math. Anal. 45 (2013) 915–933. | DOI | MR | Zbl

F. Coquel and P. Lefloch, Convergence de schémas aux différences finies pour des lois de conservation à plusieurs dimensions d’espace. C. R. Acad. Sci. Paris Sér. I Math. 310 (1990) 455–460. | MR | Zbl

S. Cordier and E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics. Commun. Partial Differ. Eq. 25 (2000) 1099–1113. | DOI | MR | Zbl

S. Dellacherie, Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number. J. Comput. Phys. 229 (2010) 978–1016. | DOI | MR | Zbl

L. Even-Dar Mandel and S. Schochet, Convergence of solutions to finite difference schemes for singular limits of nonlinear evolutionary PDEs. To appear in ESAIM: M2AN (2016) DOI: 10.1051/m2an/2016029. | Numdam | MR

U.S. Fjordholm, S. Mishra and E. Tadmor, Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws. SIAM J. Numer. Anal. 50 (2012) 544–573. | DOI | MR | Zbl

G.B. Folland, Introduction to partial differential equations. Princeton University Press, Princeton, N.J. (1976). | MR | Zbl

G.B. Folland, Fourier analysis and its applications. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA (1992). | MR | Zbl

A. Friedman, Partial differential equations, original edn. Robert E. Krieger Publishing Co., Huntington, New York (1976). | MR

I. Gallagher, Applications of Schochet’s methods to parabolic equations. J. Math. Pures Appl. 77 (1998) 989–1054. | DOI | MR | Zbl

A. Glitzky and J.A. Griepentrog, Discrete Sobolev-Poincaré inequalities for Voronoi finite volume approximations. SIAM J. Numer. Anal. 48 (2010) 372–391. | DOI | MR | Zbl

M. Goldberg, Stable difference schemes for parabolic systems – a numerical radius approach. SIAM J. Numer. Anal. 35 (1998) 478–493. | DOI | MR | Zbl

E. Grenier, Pseudo-differential energy estimates of singular perturbations. Commun. Pure Appl. Math. 50 (1997) 821–865. | DOI | MR | Zbl

B. Gustafsson, H.O. Kreiss and J. Oliger, Time dependent problems and difference methods. Pure and Applied Mathematics. John Wiley & Sons Inc., New York (1995). | MR | Zbl

F. John, On integration of parabolic equations by difference methods. I. Linear and quasi-linear equations for the infinite interval. Commun. Pure Appl. Math. 5 (1952) 155–211. | DOI | MR | Zbl

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34 (1981) 481–524. | DOI | MR | Zbl

R. Klein, Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics. I. One-dimensional flow. J. Comput. Phys. 121 (1995) 213–237. | DOI | MR | Zbl

R. Klein, N. Botta, T. Schneider, C.D. Munz, S. Roller, A. Meister, L. Hoffmann and T. Sonar, Asymptotic adaptive methods for multi-scale problems in fluid mechanics. J. Engrg. Math. 39 (2001) 261–343. | DOI | MR | Zbl

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. | DOI | MR | Zbl

P.D. Lax and L. Nirenberg, On stability for difference schemes: A sharp form of Gȧrding’s inequality. Commun. Pure Appl. Math. 19 (1966) 473–492. | DOI | MR | Zbl

R.J. LeVeque, Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002). | MR | Zbl

A. Majda, Compressible fluid flow and systems of conservation laws in several space variables. Vol. 53 of Appl. Math. Sci. Springer-Verlag, New York (1984). | MR | Zbl

D. Michelson, Stability theory of difference approximations for multidimensional initial-boundary value problems. Math. Comput. 40 (1983) 1–45. | DOI | MR | Zbl

C.D. Munz, S. Roller, R. Klein and K.J. Geratz, The extension of incompressible flow solvers to the weakly compressible regime. Comput. Fluids 32 (2003) 173–196. | DOI | MR | Zbl

S. Noelle, G. Bispen, K.R. Arun, M. Lukáčová-Medviďová and C.D. Munz, A weakly asymptotic preserving low Mach number scheme for the Euler equations of gas dynamics. SIAM J. Sci. Comput. 36 (2014) B989–B1024. | DOI | MR | Zbl

S. Osher and E. Tadmor, On the convergence of difference approximations to scalar conservation laws. Math. Comput. 50 (1988) 19–51. | DOI | MR | Zbl

A.A. Samarskii, The theory of difference schemes. Vol. 240 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., New York (2001). | MR | Zbl

S. Schochet, The incompressible limit in nonlinear elasticity. Commun. Math. Phys. 102 (1985) 207–215. | DOI | MR | Zbl

S. Schochet, Fast singular limits of hyperbolic PDEs. J. Differ. Eq. 114 (1994) 476–512. | DOI | MR | Zbl

S. Schochet, The mathematical theory of low Mach number flows. ESAIM: M2AN 39 (2005) 441–458. | DOI | Numdam | MR | Zbl

S. Schochet, Convergence of finite-volume schemes to smooth solutions of multidimensional hyperbolic systems. In preparation (2017).

S. Schochet and M.I. Weinstein, The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence. Commun. Math. Phys. 106 (1986) 569–580. | DOI | MR | Zbl

J. Schütz and S. Noelle, Flux splitting for stiff equations: a notion on stability. J. Sci. Comput. 64 (2015) 522–540. | DOI | MR | Zbl

E.L. Smirnova, The stability of parabolic difference schemes with variable coefficients. Russian original published (1975). Vestnik Leningrad. Univ. Math. (1980) 373–381. | MR | Zbl

E.L. Smirnova, The invertibility and stability of difference operators with variable coefficients. Russian original published (1977). Vestnik Leningrad. Univ. Math. (1982) 209–215. | MR | Zbl

G. Strang, Accurate partial difference methods. II. Non-linear problems. Numer. Math. 6 (1964) 37–46. | DOI | MR | Zbl

E. Tadmor, Local error estimates for discontinuous solutions of nonlinear hyperbolic equations. SIAM J. Numer. Anal. 28 (1991) 891–906. | DOI | MR | Zbl

M.E. Taylor, Pseudodifferential operators. Vol. 34 of Princeton Mathematical Series. Princeton University Press, Princeton, N.J. (1981). | MR | Zbl

M.E. Taylor, Partial differential equations, II. Qualitative studies of linear equations. Vol. 116 of Appl. Math. Sci. Springer-Verlag, New York (1996). | MR | Zbl

K. Tomoeda, Convergence of difference approximations for quasilinear hyperbolic systems. Hiroshima Math. J. 11 (1981) 465–491. | DOI | MR | Zbl

R. Vaillancourt, A simple proof of Lax–Nirenberg theorems. Commun. Pure Appl. Math. 23 (1970) 151–163. | DOI | MR | Zbl

D. Wang and C. Yu, Incompressible limit for the compressible flow of liquid crystals. J. Math. Fluid Mech. 16 (2014) 771–786. | DOI | MR | Zbl

O.B. Widlund, On the stability of parabolic difference schemes. Math. Comput. 19 (1965) 1–13. | DOI | MR | Zbl

Y.L. Zhou, Applications of discrete functional analysis to the finite difference method. International Academic Publishers, Beijing (1991). | MR | Zbl

Y.L. Zhou, On the general interpolation formulas for discrete functional spaces. I. J. Comput. Math. 11 (1993) 188–192. | MR | Zbl

Y.L. Zhou, General interpolation formulas for spaces of discrete functions with nonuniform meshes. J. Comput. Math. 13 (1995) 70–93. | MR | Zbl

Cité par Sources :