G. B. Alalykin, S. K. Godunov, L. L. Kireyeva, and L. A. Pliner, Solution of One-Dimensional Problems in Gas Dynamics on Moving Grids, 1970.

N. S. Bakhvalov, The optimization of methods of solving boundary value problems with a boundary layer, USSR Computational Mathematics and Mathematical Physics, vol.9, issue.4, pp.139-166, 1969.
DOI : 10.1016/0041-5553(69)90038-X

S. Barbeiro, J. A. Ferreira, and R. D. Grigorieff, Supraconvergence of a finite difference scheme for solutions in Hs(0, L), IMA Journal of Numerical Analysis, vol.25, issue.4, pp.797-811, 2005.
DOI : 10.1093/imanum/dri018

C. Budd and V. Dorodnitsyn, Symmetry-adapted moving mesh schemes for the nonlinear Schr??dinger equation, Journal of Physics A: Mathematical and General, vol.34, issue.48, pp.10387-10400, 2001.
DOI : 10.1088/0305-4470/34/48/305

C. J. Budd, B. Leimkuhler, and M. D. Piggott, Scaling invariance and adaptivity, Applied Numerical Mathematics, vol.39, issue.3-4, pp.261-288, 2001.
DOI : 10.1016/S0168-9274(00)00036-2
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.40.933

C. J. Budd and M. D. Piggott, The geometric integration of scale-invariant ordinary and partial differential equations, Journal of Computational and Applied Mathematics, vol.128, issue.1-2, pp.399-422, 2001.
DOI : 10.1016/S0377-0427(00)00521-5

M. Chhay, E. Hoarau, A. Hamdouni, and P. Sagaut, Comparison of some Lie-symmetry-based integrators, Journal of Computational Physics, vol.230, issue.5, pp.2174-2188, 2011.
DOI : 10.1016/j.jcp.2010.12.015
URL : https://hal.archives-ouvertes.fr/hal-01298900

L. M. Degtyarev, V. V. Drozdov, and T. S. Ivanova, The method of adaptive grids for the solution of singularly perturbed one dimensional boundary value problems, Differ. Uravn, vol.23, issue.15, pp.1160-1169, 1987.

V. A. Dorodnitsyn, FINITE DIFFERENCE MODELS ENTIRELY INHERITING CONTINUOUS SYMMETRY OF ORIGINAL DIFFERENTIAL EQUATIONS, International Journal of Modern Physics C, vol.05, issue.04, pp.5723-734, 1994.
DOI : 10.1142/S0129183194000830

E. Emmrich and R. D. Grigorieff, Supraconvergence of a Finite Difference Scheme for Elliptic Boundary Value Problems of the Third Kind in Fractional Order Sobolev Spaces, Computational Methods in Applied Mathematics, vol.6, issue.2, 2006.
DOI : 10.2478/cmam-2006-0008

R. Eymard, J. Fuhrmann, and K. Gärtner, A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local dirichlet problems, Numerische Mathematik, vol.102, issue.3, pp.463-495, 2004.
DOI : 10.1007/s00211-005-0659-5

P. A. Farrell, J. J. Miller, E. O-'riordan, and G. I. Shishkin, A Uniformly Convergent Finite Difference Scheme for a Singularly Perturbed Semilinear Equation, SIAM Journal on Numerical Analysis, vol.33, issue.3, pp.1135-1149, 1996.
DOI : 10.1137/0733056

R. P. Fedorenko, Introduction to Computational Physics, 1994.

J. A. Ferreira and P. Oliveira, Convergence properties of numerical discretizations and regridding methods, Journal of Computational and Applied Mathematics, vol.45, issue.3, pp.321-330, 1993.
DOI : 10.1016/0377-0427(93)90049-H

J. A. Ferreira and R. D. Grigorieff, On the supraconvergence of elliptic finite difference schemes, Applied Numerical Mathematics, vol.28, issue.2-4, pp.275-292, 1998.
DOI : 10.1016/S0168-9274(98)00048-8

J. A. Ferreira and R. D. Grigorieff, Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform Grids. Numerical Functional Analysis and Optimization, pp.5-6539, 2006.

B. García-archilla and J. M. Sanz-serna, A finite difference formula for the discretization of d 3

W. Huang and R. D. Russell, Adaptive Moving Mesh Methods, Applied Mathematical Sciences, vol.174, issue.5, 2011.
DOI : 10.1007/978-1-4419-7916-2

H. B. Keller, Numerical Methods in Boundary-Layer Theory, Annual Review of Fluid Mechanics, vol.10, issue.1, pp.417-433, 1978.
DOI : 10.1146/annurev.fl.10.010178.002221

G. S. Khakimzyanov, D. Dutykh, D. E. Mitsotakis, and N. Y. Shokina, Numerical solution of conservation laws on moving grids, pp.1-28

P. D. Lax and R. D. Richtmyer, Survey of the stability of linear finite difference equations, Communications on Pure and Applied Mathematics, vol.5, issue.2, pp.267-293, 1956.
DOI : 10.1002/cpa.3160090206

A. G. Nikitin, On the principal eigenfunction of a singularly perturbed Sturm-Liouville problem, Zh. Vychisl. Mat. Mat. Fiz, vol.39, issue.4 6, pp.588-591, 1999.

B. V. Numerov, A Method of Extrapolation of Perturbations, Monthly Notices of the Royal Astronomical Society, vol.84, issue.8, pp.592-602, 1924.
DOI : 10.1093/mnras/84.8.592

B. V. Numerov, Note on the numerical integration of d 2 x dt 2 = f (x, t) Astronomische Nachrichten, pp.359-364, 1927.

K. Phaneendra, S. Rakmaiah, and M. C. Reddy, Computational Method for Singularly Perturbed Boundary Value Problems with Dual Boundary Layer, Procedia Engineering, vol.127, issue.4, pp.370-376, 2015.
DOI : 10.1016/j.proeng.2015.11.383
URL : http://doi.org/10.1016/j.proeng.2015.11.383

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, and P. B. Kramer, Numerical recipes: the art of scientific computing, p.22, 2007.

P. L. Roe, Computational fluid dynamics???retrospective and prospective, International Journal of Computational Fluid Dynamics, vol.19, issue.8, pp.581-594, 2005.
DOI : 10.1080/10618560600585315

H. Roos, Ten ways to generate the Il'in and related schemes, Journal of Computational and Applied Mathematics, vol.53, issue.1, pp.43-59, 1994.
DOI : 10.1016/0377-0427(92)00124-R

A. A. Samarskii, The Theory of Difference Schemes, p.18, 2001.

H. Schlichting and K. Gersten, Boundary-Layer Theory, 2000.
DOI : 10.1007/978-3-662-52919-5

G. Strang and A. Iserles, Barriers to Stability, SIAM Journal on Numerical Analysis, vol.20, issue.6, pp.1251-1257, 1983.
DOI : 10.1137/0720096

V. G. Sudobicher and S. M. Shugrin, Flow along a dry channel, Izv. Akad. Nauk SSSR, vol.13, issue.3 4, pp.116-122, 1968.

A. N. Tikhonov and A. A. Samarskii, Homogeneous difference schemes, USSR Computational Mathematics and Mathematical Physics, vol.1, issue.1, pp.5-63, 1961.
DOI : 10.1016/0041-5553(62)90005-8

A. N. Tikhonov and A. A. Samarskii, Homogeneous difference schemes on non-uniform nets, USSR Computational Mathematics and Mathematical Physics, vol.2, issue.5, pp.812-832, 1962.
DOI : 10.1016/0041-5553(63)90505-6

Y. B. Zel-'dovich and Y. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, 2002.

G. Khakimzyanov, Novosibirsk 630090, Russia E-mail address: Khak@ict.nsc.ru URL: http://www.ict.nsc.ru/ru/structure/Persons