Convergence of solutions to finite difference schemes for singular limits of nonlinear evolutionary PDEs
Even-Dar Mandel, L.1 ; Schochet, S.1
1 School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 587-614.

Solutions of certain finite-difference schemes for singularly-perturbed evolutionary PDEs converge as the perturbation parameter and/or the discretization parameters tend to zero. Under suitable hypotheses a sharp convergence rate of order one-half in the time step, uniform in the perturbation parameter, is obtained.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016029
Classification : 65M12
Mots-clés : Rate of convergence, discrete Sobolev spaces, singular limits
Even-Dar Mandel, L. 1 ; Schochet, S. 1

1 School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel 
@article{M2AN_2017__51_2_587_0,
     author = {Even-Dar Mandel, L. and Schochet, S.},
     title = {Convergence of solutions to finite difference schemes for singular limits of nonlinear evolutionary {PDEs}},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {587--614},
     publisher = {EDP-Sciences},
     volume = {51},
     number = {2},
     year = {2017},
     doi = {10.1051/m2an/2016029},
     mrnumber = {3626412},
     zbl = {1368.65158},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2016029/}
}
TY  - JOUR
AU  - Even-Dar Mandel, L.
AU  - Schochet, S.
TI  - Convergence of solutions to finite difference schemes for singular limits of nonlinear evolutionary PDEs
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2017
SP  - 587
EP  - 614
VL  - 51
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2016029/
DO  - 10.1051/m2an/2016029
LA  - en
ID  - M2AN_2017__51_2_587_0
ER  - 
%0 Journal Article
%A Even-Dar Mandel, L.
%A Schochet, S.
%T Convergence of solutions to finite difference schemes for singular limits of nonlinear evolutionary PDEs
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2017
%P 587-614
%V 51
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2016029/
%R 10.1051/m2an/2016029
%G en
%F M2AN_2017__51_2_587_0
Even-Dar Mandel, L.; Schochet, S. Convergence of solutions to finite difference schemes for singular limits of nonlinear evolutionary PDEs. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 587-614. doi : 10.1051/m2an/2016029. http://archive.numdam.org/articles/10.1051/m2an/2016029/

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

G. Browning and H.O. Kreiss, Problems with different time scales for nonlinear partial differential equations. SIAM J. Appl. Math. 42 (1982) 704–718. | DOI | MR | Zbl

C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput. 38 (1982) 67–86. | DOI | 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

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

P.J. Davis, Interpolation and approximation. Dover Publications Inc., New York (1975). | MR | Zbl

S. De Marchi and M. Vianello, Peano’s kernel theorem for vector-valued functions and some applications. Numer. Funct. Anal. Optim. 17 (1996) 57–64. | DOI | MR | Zbl

L. Even-Dar Mandel and S. Schochet, Uniform discrete Sobolev estimates of solutions to finite difference schemes for singular limits of nonlinear PDEs. To appear in ESAIM: M2AN (2017). | DOI | Numdam | MR

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

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

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

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

T. Kato, A short introduction to perturbation theory for linear operators. Springer-Verlag, New York (1982). | MR | Zbl

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

R. Klein, Multiple spatial scales in engineering and atmospheric low Mach number flows. ESAIM: M2AN 39 (2005) 537–559. | DOI | Numdam | 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

V.S. Ryaben'kii, S.V. Tsynkov, A theoretical introduction to numerical analysis. Chapman & Hall/CRC, Boca Raton, FL (2007). | MR | Zbl

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

S. Schochet, Symmetric hyperbolic systems with a large parameter. Comm. Partial Differ. Eq. 11 (1986) 1627–1651. | 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 (2016).

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

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

J.C. Strikwerda, Finite difference schemes and partial differential equations. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA (1989). | MR | Zbl

M.E. Taylor, Partial differential equations, III. Vol. 117 of Appl. Math. Sci. Springer-Verlag, New York (1997). | MR | Zbl

J.W. Thomas, Numerical partial differential equations: finite difference methods. Vol. 22 of Texts Appl. Math. Springer-Verlag, New York (1995). | MR | Zbl

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

Cité par Sources :