RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Zh. Vychisl. Mat. Mat. Fiz.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Zh. Vychisl. Mat. Mat. Fiz., 1969, Volume 9, Number 4, Pages 841–859 (Mi zvmmf7128)  

This article is cited in 169 scientific papers (total in 170 papers)

On the optimization of the methods for solving boundary value problems in the presence of a boundary layer

N. S. Bakhvalov

Moscow

Full text: PDF file (1782 kB)

English version:
USSR Computational Mathematics and Mathematical Physics, 1969, 9:4, 139–166

Bibliographic databases:

UDC: 518:517.944/.947
MSC: Primary 65N22; Secondary 65L10
Received: 31.01.1969

Citation: N. S. Bakhvalov, “On the optimization of the methods for solving boundary value problems in the presence of a boundary layer”, Zh. Vychisl. Mat. Mat. Fiz., 9:4 (1969), 841–859; U.S.S.R. Comput. Math. Math. Phys., 9:4 (1969), 139–166

Citation in format AMSBIB
\Bibitem{Bak69}
\by N.~S.~Bakhvalov
\paper On the optimization of the methods for solving boundary value problems in the presence of a boundary layer
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 1969
\vol 9
\issue 4
\pages 841--859
\mathnet{http://mi.mathnet.ru/zvmmf7128}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=0255066}
\zmath{https://zbmath.org/?q=an:0208.19103}
\transl
\jour U.S.S.R. Comput. Math. Math. Phys.
\yr 1969
\vol 9
\issue 4
\pages 139--166
\crossref{https://doi.org/10.1016/0041-5553(69)90038-X}


Linking options:
  • http://mi.mathnet.ru/eng/zvmmf7128
  • http://mi.mathnet.ru/eng/zvmmf/v9/i4/p841

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. I. A. Blatov, “The projection method for singularly perturbed boundary value problems”, U.S.S.R. Comput. Math. Math. Phys., 30:4 (1990), 47–56  mathnet  crossref  mathscinet  zmath
    2. I. P. Boglaev, “Numerical solution of a quasilinear parabolic equation with a boundary layer”, U.S.S.R. Comput. Math. Math. Phys., 30:3 (1990), 55–63  mathnet  crossref  mathscinet  zmath
    3. V. D. Liseikin, “The use of special transformations in the numerical solution of boundary layer problems”, U.S.S.R. Comput. Math. Math. Phys., 30:1 (1990), 43–53  mathnet  crossref  mathscinet  zmath
    4. G. I. Shishkin, “Grid approximation of a singularly perturbed boundary-value problem for a quasi-linear elliptic equation in the completely degenerate case”, U.S.S.R. Comput. Math. Math. Phys., 31:12 (1991), 33–46  mathnet  mathscinet  zmath  isi
    5. G. I. Shishkin, “A grid approximation of singularly perturbed parabolic equations degenerate on the boundary”, U.S.S.R. Comput. Math. Math. Phys., 31:10 (1991), 53–63  mathnet  mathscinet  zmath  isi
    6. U.S.S.R. Comput. Math. Math. Phys., 31:4 (1991), 28–36  mathnet  mathscinet  zmath  isi
    7. G. I. Shishkin, “A difference scheme for a singularly perturbed parabolic equation degenerating on the boundary”, Comput. Math. Math. Phys., 32:5 (1992), 621–636  mathnet  mathscinet  zmath  isi
    8. G. I. Shishkin, “A difference approximation of a singularly perturbed boundary-value problem for quasilinear elliptic equations degenerating into first-order equations”, Comput. Math. Math. Phys., 32:4 (1992), 467–480  mathnet  mathscinet  zmath  isi
    9. G. I. Shishkin, “The method of additive separation of singularities for quasilinear singularly perturbed elliptic and parabolic equations”, Comput. Math. Math. Phys., 34:12 (1994), 1541–1558  mathnet  mathscinet  zmath  isi
    10. G. I. Shishkin, “A grid approximation of singularly perturbed quasilinear elliptic and parabolic equations which degenerate into equations without spatial derivatives”, Comput. Math. Math. Phys., 34:11 (1994), 1403–1419  mathnet  mathscinet  zmath  isi
    11. K. V. Emel'yanov, “Applying optimal difference grids to problems with singular perturbations”, Comput. Math. Math. Phys., 34:6 (1994), 809–814  mathnet  mathscinet  zmath  isi
    12. G. I. Shishkin, “A grid approximation of the method of additive separation of singularities for a singularly perturbed equation of parabolic type”, Comput. Math. Math. Phys., 34:5 (1994), 621–637  mathnet  mathscinet  zmath  isi
    13. I. A. Savin, “On the rate of convergence, uniform with respect to a small parameter, of a difference scheme for an ordinary differential equation”, Comput. Math. Math. Phys., 35:11 (1995), 1417–1422  mathnet  mathscinet  zmath  isi
    14. G. I. Shishkin, “Mesh approximation of singularly perturbed boundary-value problems for systems of elliptic and parabolic equations”, Comput. Math. Math. Phys., 35:4 (1995), 429–446  mathnet  mathscinet  zmath  isi
    15. V. B. Andreev, I. A. Savin, “On the convergence, uniform with respect to the small parameter, of A. A. Samarskii's monotone scheme and its modifications”, Comput. Math. Math. Phys., 35:5 (1995), 581–591  mathnet  mathscinet  zmath  isi
    16. Kopteva N.V., “Uniform convergence with respect to a small parameter of a four-point scheme for the one-dimensional stationary convection-diffusion equation”, Differ Equ, 32:7 (1996), 958–964  mathnet  mathscinet  zmath  isi
    17. G. I. Shishkin, “Approximation of the solutions and diffusion flows of singularly perturbed boundary-value problems with discontinuous initial conditions”, Comput. Math. Math. Phys., 36:9 (1996), 1233–1250  mathnet  mathscinet  zmath  isi
    18. V. B. Andreev, N. V. Kopteva, “A study of difference schemes with the first derivative approximated by a central difference ratio”, Comput. Math. Math. Phys., 36:8 (1996), 1065–1078  mathnet  mathscinet  zmath  isi
    19. G. I. Shishkin, “Grid approximation of parabolic equations with singular initial conditions”, Comput. Math. Math. Phys., 36:3 (1996), 341–356  mathnet  mathscinet  zmath  isi
    20. G. I. Shishkin, “Locally one-dimensional difference schemes for singularly perturbed parabolic equations”, Comput. Math. Math. Phys., 36:2 (1996), 165–180  mathnet  mathscinet  zmath  isi
    21. V. D. Liseǐkin, “A survey of methods for constructing structured adaptive grids”, Comput. Math. Math. Phys., 36:1 (1996), 1–32  mathnet  mathscinet  zmath  isi
    22. Stynes M., ORiordan E., “A uniformly convergent galerkin method on a Shishkin mesh for a convection-diffusion problem”, J Math Anal Appl, 214:1 (1997), 36–54  crossref  mathscinet  zmath  isi
    23. S. A. Ivanenko, G. P. Prokopov, “Methods of adaptive harmonic grid generation”, Comput. Math. Math. Phys., 37:6 (1997), 627–645  mathnet  mathscinet  zmath
    24. G. I. Shishkin, “Singularly perturbed boundary value problems with concentrated sources and discontinuous initial conditions”, Comput. Math. Math. Phys., 37:4 (1997), 417–434  mathnet  mathscinet  zmath
    25. G. I. Shishkin, “Grid approximation of a singularly perturbed Neumann problem for parabolic equations in the case of a discontinuous boundary function”, Comput. Math. Math. Phys., 37:3 (1997), 370–373  mathnet  mathscinet  zmath
    26. N. V. Kopteva, “On the uniform in small parameter convergence of a weighted scheme for the one-dimensional time-dependent convection–diffusion equation”, Comput. Math. Math. Phys., 37:10 (1997), 1173–1180  mathnet  mathscinet  zmath
    27. T. A. Yakubenko, “Error estimation for a numerical solution to a boundary value problem with a boundary layer”, Comput. Math. Math. Phys., 37:8 (1997), 913–918  mathnet  mathscinet  zmath
    28. G. I. Shishkin, “Grid approximations of singularly perturbed systems for parabolic convection-diffusion equations with counterflow”, Sib. zhurn. vychisl. matem., 1:3 (1998), 281–297  mathnet  mathscinet  zmath
    29. Andreev V.B., Kopteva N.V., “On the convergence, uniform with respect to a small parameter, of monotone three-point finite-difference approximations”, Differ Equ, 34:7 (1998), 921–929  mathscinet  zmath  isi
    30. G. I. Shishkin, “Finite-difference approximations for singularly perturbed elliptic equations”, Comput. Math. Math. Phys., 38:12 (1998), 1909–1921  mathnet  mathscinet  zmath
    31. G. I. Shishkin, “Approximation of singularly perturbed elliptic equations with convective terms in the case of a flow impinging on an impermeable wall”, Comput. Math. Math. Phys., 38:11 (1998), 1768–1782  mathnet  mathscinet  zmath
    32. V. D. Liseikin, “A method of algebraic adaptation”, Comput. Math. Math. Phys., 38:10 (1998), 1624–1640  mathnet  mathscinet  zmath
    33. A. I. Zadorin, “Numerical solution of a boundary value problem for a set of equations with a small parameter”, Comput. Math. Math. Phys., 38:8 (1998), 1201–1211  mathnet  mathscinet  zmath
    34. V. B. Andreev, “Convergence of a modified Samarskij's monotonic scheme on a smoothly condensing grid”, Comput. Math. Math. Phys., 38:8 (1998), 1212–1224  mathnet  mathscinet  zmath
    35. G. I. Shishkin, “A grid approximation for the Riemann problem in the case of the Burgers equation”, Comput. Math. Math. Phys., 38:8 (1998), 1361–1363  mathnet  mathscinet  zmath
    36. K. V. Emel'yanov, “Application of one-dimensional optimal grids to two-dimensional singularly perturbed problems”, Comput. Math. Math. Phys., 38:3 (1998), 411–418  mathnet  mathscinet  zmath
    37. A. I. Zadorin, “Perenos kraevogo usloviya iz beskonechnosti pri chislennom reshenii uravnenii vtorogo poryadka s malym parametrom”, Sib. zhurn. vychisl. matem., 2:1 (1999), 21–35  mathnet  zmath
    38. B. P. Kolobov, Yu. I. Molorodov, “Raschet optimalnykh uzlov kollokatsii dlya resheniya parabolicheskikh uravnenii skhemoi vysokogo poryadka tochnosti”, Sib. zhurn. vychisl. matem., 2:4 (1999), 351–360  mathnet  zmath
    39. G. I. Shishkin, “Increasing the accuracy of approximate solutions by residual correction for singularly perturbed equations with convective terms”, Russian Math. (Iz. VUZ), 43:5 (1999), 77–89  mathnet  mathscinet  zmath
    40. N. V. Kopteva, “Uniform convergence with respect to a small parameter of a scheme with central difference on refining grids”, Comput. Math. Math. Phys., 39:10 (1999), 1594–1610  mathnet  mathscinet  zmath
    41. N. S. Bakhvalov, “Automatic construction of integration mesh for boundary value problems with boundary layers”, Comput. Math. Math. Phys., 39:8 (1999), 1238–1243  mathnet  mathscinet  zmath
    42. G. I. Shishkin, “Singularly perturbed boundary value problems with locally perturbed initial conditions: Equations with convective terms”, Comput. Math. Math. Phys., 39:2 (1999), 249–265  mathnet  mathscinet  zmath
    43. Clavero C., Gracia J.L., Lisbona F., “High order methods on Shishkin meshes for singular perturbation problems of convection-diffusion type”, Numer Algorithms, 22:1 (1999), 73–97  crossref  mathscinet  zmath  isi
    44. Linss T., “An upwind difference scheme on a novel Shishkin-type mesh for a linear convection-diffusion problem”, J Comput Appl Math, 110:1 (1999), 93–104  crossref  mathscinet  zmath  isi
    45. G. I. Shishkin, “Approximation of systems of convection-diffusion elliptic equations with parabolic boundary layers”, Comput. Math. Math. Phys., 40:11 (2000), 1582–1595  mathnet  mathscinet  zmath
    46. G. I. Shishkin, “Grid approximation of singularly perturbed boundary value problems on locally condensing grids: Convection-diffusion equations”, Comput. Math. Math. Phys., 40:5 (2000), 680–691  mathnet  mathscinet  zmath
    47. I. A. Brayanov, L. G. Volkov, “Uniform in a small parameter convergence of Samarskii's monotone scheme and its modification for the convection-diffusion equation with a concentrated source”, Comput. Math. Math. Phys., 40:4 (2000), 534–550  mathnet  mathscinet  zmath
    48. Andreev V.B., Kopteva N.V., “Uniform with respect to a small parameter convergence of difference schemes for a convection-diffusion problem”, Analytical and Numerical Methods for Convection-Dominated and Singularly Perturbed Problems, 2000, 133–139  isi
    49. Linss T., “A novel Shishkin-type mesh for convection-diffusion problems”, Analytical and Numerical Methods for Convection-Dominated and Singularly Perturbed Problems, 2000, 199–204  isi
    50. Liseikin V., “On the Method of Coordinate Transformations for the Numerical Solution of Singularly Perturbed Systems of Ordinary Differential Equations”, Dokl. Math., 62:2 (2000), 283–287  mathscinet  zmath  isi
    51. G. I. Shishkin, “Grid approximation of a wave equation singularly perturbed with respect to the space variable”, Russian Math. (Iz. VUZ), 45:1 (2001), 63–77  mathnet  mathscinet  zmath  elib
    52. G. I. Shishkin, “Metod dekompozitsii dlya singulyarno vozmuschennykh parabolicheskikh uravnenii konvektsii-diffuzii s razryvnymi nachalnymi usloviyami”, Sib. zhurn. vychisl. matem., 4:1 (2001), 85–106  mathnet  zmath
    53. Kopteva N., Linss T., “Uniform second-order pointwise convergence of a central difference approximation for a quasilinear convection-diffusion problem”, J Comput Appl Math, 137:2 (2001), 257–267  crossref  mathscinet  zmath  isi
    54. Kopteva N., “Maximum norm a posteriori error estimates for a one-dimensional convection-diffusion problem”, SIAM J Numer Anal, 39:2 (2001), 423–441  crossref  mathscinet  zmath  isi
    55. Linss T., “Sufficient conditions for uniform convergence on layer-adapted grids”, Appl Numer Math, 37:1–2 (2001), 241–255  crossref  mathscinet  zmath  isi
    56. Linss T., “Uniform pointwise convergence of finite difference schemes using grid equidistribution”, Computing, 66:1 (2001), 27–39  crossref  mathscinet  zmath  isi
    57. Kopteva N., “Uniform pointwise convergence of difference schemes for convection-diffusion problems on layer-adapted meshes”, Computing, 66:2 (2001), 179–197  crossref  mathscinet  zmath  isi
    58. Vulanovic R., “A higher-order scheme for quasilinear boundary value problems with two small parameters”, Computing, 67:4 (2001), 287–303  crossref  mathscinet  zmath  isi
    59. Kopteva N., Stynes M., “Approximation of derivatives in a convection-diffusion two-point boundary value problem”, Appl Numer Math, 39:1 (2001), 47–60  crossref  mathscinet  zmath  isi
    60. A. N. Minailos, “Computation of equations up to a prescribed accuracy with respect to singular terms and defect of differential equations”, Comput. Math. Math. Phys., 41:10 (2001), 1489–1505  mathnet  mathscinet  zmath
    61. G. I. Shishkin, “Mesh approximation of singularly perturbed equations with convective terms for the perturbation of data”, Comput. Math. Math. Phys., 41:5 (2001), 649–664  mathnet  mathscinet  zmath  elib
    62. Comput. Math. Math. Phys., 41:6 (2001), 898–909  mathnet  mathscinet  zmath
    63. V. D. Liseikin, “On the numerical solution of singularly perturbed problems with turning points”, Comput. Math. Math. Phys., 41:1 (2001), 55–83  mathnet  mathscinet  zmath  elib
    64. G. I. Shishkin, “Piecewise-uniform grids, optimal with respect to the order of convergence, for singularly perturbed convection-diffusion equations”, Russian Math. (Iz. VUZ), 46:3 (2002), 56–68  mathnet  mathscinet  zmath
    65. G. I. Shishkin, “Setochnye approksimatsii s uluchshennoi skorostyu skhodimosti dlya singulyarno vozmuschennykh ellipticheskikh uravnenii v oblastyakh s kharakteristicheskimi granitsami”, Sib. zhurn. vychisl. matem., 5:1 (2002), 71–92  mathnet  zmath
    66. Lenferink W., “A second order scheme for a time-dependent, singularly perturbed convection-diffusion equation”, J Comput Appl Math, 143:1 (2002), 49–68  crossref  mathscinet  zmath  isi
    67. I. R. Rafatov, S. N. Sklyar, “Finite-difference scheme for singularly perturbed boundary value problems associated with solutions to spherically symmetric elliptic equations”, Comput. Math. Math. Phys., 42:9 (2002), 1331–1340  mathnet  mathscinet  zmath
    68. G. I. Shishkin, “Grid approximation of improved convergence order for a singularly perturbed elliptic convection-diffusion equation”, Proc. Steklov Inst. Math. (Suppl.), 2003no. , suppl. 1, S184–S202  mathnet  mathscinet  zmath  elib
    69. K. V. Emel'yanov, “On an approximate solution of a one-dimensional linear singularly perturbed problem”, Proc. Steklov Inst. Math. (Suppl.), 2003no. , suppl. 2, S45–S54  mathnet  mathscinet  zmath  elib
    70. G. I. Shishkin, “An improved piecewise uniform mesh for a singularly perturbed elliptic reaction-diffusion equation”, Proc. Steklov Inst. Math. (Suppl.), 2003no. , suppl. 2, S138–S147  mathnet  mathscinet  zmath  elib
    71. G. I. Shishkin, “Approximation of solutions and derivative of singularly perturbed elliptic equation of convection-diffusion”, Comput. Math. Math. Phys., 43:5 (2003), 641–657  mathnet  mathscinet  zmath
    72. G. I. Shishkin, “The Schwarz grid method for singularly perturbed convection-diffusion parabolic equations in the case of coherent and incoherent grids on subdomains”, Comput. Math. Math. Phys., 43:2 (2003), 242–254  mathnet  mathscinet  zmath  elib
    73. V. G. Zverev, “On a special difference scheme for the solution of boundary value problems of heat and mass transfer”, Comput. Math. Math. Phys., 43:2 (2003), 255–267  mathnet  mathscinet  zmath
    74. Linss T., “Layer-adapted meshes for convection-diffusion problems”, Comput Methods Appl Mech Engrg, 192:9–10 (2003), 1061–1105  crossref  mathscinet  zmath  isi
    75. Ansari A.R., Hegarty A.F., “Numerical solution of a convection diffusion problem with Robin boundary conditions”, J Comput Appl Math, 156:1 (2003), 221–238  crossref  mathscinet  zmath  isi
    76. V. B. Andreev, “On the uniform convergence of a classical difference scheme on an irregular grid for the one-dimensional singularly perturbed reaction-diffusion equation”, Comput. Math. Math. Phys., 44:3 (2004), 449–464  mathnet  mathscinet  zmath
    77. P. W. Hemker, G. I. Shishkin, L. P. Shishkina, “High-order accurate decomposition of the Richardson method for a singularly perturbed elliptic reaction-diffusion equation”, Comput. Math. Math. Phys., 44:2 (2004), 309–316  mathnet  mathscinet  zmath
    78. Shishkin G.I., “Discrete approximations of solutions and derivatives for a singularly perturbed parabolic convection-diffusion equation”, J Comput Appl Math, 166:1 (2004), 247–266  crossref  mathscinet  zmath  isi
    79. Andreev V.B., “On the theory of difference schemes for singularly perturbed equations”, Differ Equ, 40:7 (2004), 959–970  mathnet  crossref  mathscinet  zmath  isi
    80. G. I. Shishkin, “Grid approximation of the domain and solution decomposition method with improved convergence rate for singularly perturbed elliptic equations in domains with characteristic boundaries”, Comput. Math. Math. Phys., 45:7 (2005), 1155–1171  mathnet  mathscinet  zmath
    81. M. K. Kerimov, “Academician Nikolai Sergeevich Bakhvalov (on the occasion of his seventieth birthday)”, Comput. Math. Math. Phys., 45:4 (2005), 539–549  mathnet  mathscinet  zmath  elib
    82. K. V. Emel'yanov, “Exponentially fitted scheme for a singularly perturbed problem”, Comput. Math. Math. Phys., 45:4 (2005), 645–652  mathnet  mathscinet  zmath
    83. E. V. Glushkov, N. V. Glushkova, D. V. Timofeev, “Solution of singularly perturbed convection–diffusion problems by the local Green's function method”, Comput. Math. Math. Phys., 45:3 (2005), 444–453  mathnet  mathscinet  zmath  elib  elib
    84. G. I. Shishkin, “Grid approximation in a half plane for singularly perturbed elliptic equations with convective terms that grow at infinity”, Comput. Math. Math. Phys., 45:2 (2005), 285–301  mathnet  mathscinet  zmath  elib  elib
    85. G. I. Shishkin, “Grid approximation of a singularly perturbed elliptic equation with convective terms in the presence of various boundary layers”, Comput. Math. Math. Phys., 45:1 (2005), 104–119  mathnet  mathscinet  zmath  elib  elib
    86. Kopteva N., Madden N., Stynes M., “Grid equidistribution for reaction-diffusion problems in one dimension”, Numer Algorithms, 40:3 (2005), 305–322  crossref  mathscinet  zmath  isi  elib
    87. Shishkin G.I., Shishkina L.P., “A higher-order Richardson method for a quasilinear singularly perturbed elliptic reaction-diffusion equation”, Differ Equ, 41:7 (2005), 1030–1039  mathnet  crossref  mathscinet  zmath  isi  elib
    88. Linss T., “Sufficient conditions for uniform convergence on layer-adapted meshes for one-dimensional reaction-diffusion problems”, Numer Algorithms, 40:1 (2005), 23–32  crossref  mathscinet  zmath  isi
    89. Shishkin G.I., “Grid approximation of parabolic convection-diffusion equations with piecewise smooth initial conditions”, Doklady Mathematics, 72:3 (2005), 850–853  mathscinet  zmath  isi  elib
    90. G. I. Shishkin, “Grid approximation of singularly perturbed parabolic reaction-diffusion equations on large domains with respect to the space and time variables”, Comput. Math. Math. Phys., 46:11 (2006), 1953–1971  mathnet  crossref  mathscinet
    91. G. I. Shishkin, “The use of solutions on embedded grids for the approximation of singularly perturbed parabolic convection-diffusion equations on adapted grids”, Comput. Math. Math. Phys., 46:9 (2006), 1539–1559  mathnet  crossref  mathscinet
    92. G. I. Shishkin, “A method of asymptotic constructions of improved accuracy for a quasilinear singularly perturbed parabolic convection-diffusion equation”, Comput. Math. Math. Phys., 46:2 (2006), 231–250  mathnet  crossref  mathscinet  zmath
    93. Zh. Zh. Bai, L. A. Krukier, T. S. Martynova, “Two-step iterative methods for solving the stationary convection-diffusion equation with a small parameter at the highest derivative on a uniform grid”, Comput. Math. Math. Phys., 46:2 (2006), 282–293  mathnet  crossref  mathscinet  zmath
    94. G. I. Shishkin, “Grid approximation of singularly perturbed parabolic convection-diffusion equations with a piecewise-smooth initial condition”, Comput. Math. Math. Phys., 46:1 (2006), 49–72  mathnet  crossref  mathscinet  zmath
    95. G. I. Shishkin, “Metod povyshennoi tochnosti dlya kvazilineinogo singulyarno vozmuschennogo ellipticheskogo uravneniya konvektsii-diffuzii”, Sib. zhurn. vychisl. matem., 9:1 (2006), 81–108  mathnet  zmath
    96. G. I. Shishkin, “Richardson's method for increasing the accuracy of difference solutions of singularly perturbed elliptic convection-diffusion equations”, Russian Math. (Iz. VUZ), 50:2 (2006), 57–71  mathnet  mathscinet  zmath
    97. Shishkin G.I., Shishkina L.P., “The Richardson extrapolation technique for quasilinear parabolic singularly perturbed convection-diffusion equations”, International Workshop on Multi-Rate Processes and Hysteresis, Journal of Physics Conference Series, 55, 2006, 203–213  crossref  isi
    98. G. I. Shishkin, “Grid approximation of singularly perturbed parabolic equations with piecewise continuous initial-boundary conditions”, Proc. Steklov Inst. Math. (Suppl.), 259, suppl. 2 (2007), S213–S230  mathnet  crossref  elib
    99. D. A. Serkov, “Ctrategii minimaksnogo riska (sozhaleniya) v sisteme s prostymi dvizheniyami”, Tr. IMM UrO RAN, 13, no. 3, 2007, 121–135  mathnet  elib
    100. V. G. Zverev, “Raznostnye skhemy povyshennogo poryadka tochnosti dlya chislennogo resheniya zhestkogo obyknovennogo differentsialnogo uravneniya s lineinymi koeffitsientami”, Matem. modelirovanie, 19:9 (2007), 94–104  mathnet  mathscinet  zmath
    101. A. I. Zadorin, “Metod interpolyatsii dlya zadachi s pogranichnym sloem”, Sib. zhurn. vychisl. matem., 10:3 (2007), 267–275  mathnet
    102. G. I. Shishkin, “Necessary conditions for $\varepsilon$-uniform convergence of finite difference schemes for parabolic equations with moving boundary layers”, Comput. Math. Math. Phys., 47:10 (2007), 1636–1655  mathnet  crossref  mathscinet
    103. G. I. Shishkin, “Approximation of systems of singularly perturbed elliptic reaction-diffusion equations with two parameters”, Comput. Math. Math. Phys., 47:5 (2007), 797–828  mathnet  crossref  mathscinet
    104. Linss T., Madden N., “Parameter uniform approximations for time-dependent reaction-diffusion problems”, Numer Methods Partial Differential Equations, 23:6 (2007), 1290–1300  crossref  mathscinet  zmath  isi
    105. Shishkin G.I., “Using the technique of majorant functions in approximation of a singular perturbed parabolic convection-diffusion equation on adaptive grids”, Russian J Numer Anal Math Modelling, 22:3 (2007), 263–289  crossref  mathscinet  zmath  isi  elib
    106. Kopteva N., “Maximum norm a posteriori error estimates for a ID singularly perturbed semilinear reaction-diffusion problem”, IMA J Numer Anal, 27:3 (2007), 576–592  crossref  mathscinet  zmath  isi  elib
    107. Linss T., “Maximum-norm error analysis of a non-monotone FEM for a singularly perturbed reaction-diffusion problem”, BIT Numerical Mathematics, 47:2 (2007), 379–391  crossref  mathscinet  zmath  isi  elib
    108. Vulanovic R., “The layer-resolving transformation and mesh generation for quasilinear singular perturbation problems”, J Comput Appl Math, 203:1 (2007), 177–189  crossref  mathscinet  zmath  isi  elib
    109. Song Q.S., Yin G., Zhang Z. ., “An epsilon-uniform finite element method for singularly perturbed two-point boundary value problems”, Int J Numer Anal Model, 4:1 (2007), 127–140  mathscinet  zmath  isi  elib
    110. Shishkin G.I., “Grid approximation of singularly perturbed parabolic reaction-diffusion equations with piecewise smooth initial-boundary conditions”, Math Model Anal, 12:2 (2007), 235–254  crossref  mathscinet  zmath  isi  elib
    111. G. I. Shishkin, “Setochnaya approksimatsiya singulyarno vozmuschennogo kvazilineinogo parabolicheskogo uravneniya konvektsii-diffuzii na apriorno adaptiruyuschikhsya setkakh”, Uchen. zap. Kazan. gos. un-ta. Ser. Fiz.-matem. nauki, 149, no. 4, Izd-vo Kazanskogo un-ta, Kazan, 2007, 146–172  mathnet
    112. A. I. Zadorin, “Refined-mesh interpolation method for functions with a boundary-layer component”, Comput. Math. Math. Phys., 48:9 (2008), 1634–1645  mathnet  crossref  mathscinet  isi
    113. G. I. Shishkin, “Conditioning of finite difference schemes for a singularly perturbed convection-diffusion parabolic equation”, Comput. Math. Math. Phys., 48:5 (2008), 769–785  mathnet  crossref  mathscinet  zmath  isi
    114. G. I. Shishkin, L. P. Shishkina, “Approximation of a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle”, Comput. Math. Math. Phys., 48:4 (2008), 627–640  mathnet  crossref  mathscinet  zmath  isi
    115. V. B. Andreev, “Uniform grid approximation of nonsmooth solutions to the mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a rectangle”, Comput. Math. Math. Phys., 48:1 (2008), 85–108  mathnet  crossref  mathscinet  zmath  isi
    116. G. I. Shishkin, “Grid approximation of a parabolic convection-diffusion equation on a priori adapted grids: $\varepsilon$-uniformly convergent schemes”, Comput. Math. Math. Phys., 48:6 (2008), 956–974  mathnet  crossref  zmath  isi
    117. Shishkin G.I., “Grid Approximation of Singularly Perturbed Parabolic Equations with Moving Boundary Layers”, Math Model Anal, 13:3 (2008), 421–442  crossref  mathscinet  zmath  isi  elib
    118. Shishkin G., “Optimal difference schemes on piecewise-uniform meshes for a singularly perturbed parabolic convection-diffusion equation”, Math Model Anal, 13:1 (2008), 99–112  crossref  mathscinet  zmath  isi  elib
    119. Chen L., Xu J., “Stability and accuracy of adapted finite element methods for singularly perturbed problems”, Numer Math, 109:2 (2008), 167–191  crossref  mathscinet  zmath  isi  elib
    120. Shishkin G.I., “A Finite Difference Scheme on a Priori Adapted Meshes for a Singularly Perturbed Parabolic Convection-Diffusion Equation”, Numer. Math.-Theory Methods Appl., 1:2 (2008), 214–234  mathscinet  zmath  isi
    121. Shishkina L., Shishkin G., “Robust Numerical Method for a System of Singularly Perturbed Parabolic Reaction-Diffusion Equations on a Rectangle”, Math. Model. Anal., 13:2 (2008), 251–261  crossref  mathscinet  zmath  isi  elib
    122. G. I. Shishkin, “The Richardson scheme for the singularly perturbed parabolic reaction-diffusion equation in the case of a discontinuous initial condition”, Comput. Math. Math. Phys., 49:8 (2009), 1348–1368  mathnet  crossref  zmath  isi
    123. G. I. Shishkin, “Approximation of singularly perturbed parabolic equations in unbounded domains subject to piecewise smooth boundary conditions in the case of solutions that grow at infinity”, Comput. Math. Math. Phys., 49:10 (2009), 1748–1764  mathnet  crossref  isi
    124. O'Riordan E., Stynes J., Stynes M., “An Iterative Numerical Algorithm for a Strongly Coupled System of Singularly Perturbed Convection-Diffusion Problems”, Numerical Analysis and its Applications - 4th International Conference, NAA 2008, Lecture Notes in Computer Science, 5434, 2009, 104–115  crossref  mathscinet  zmath  isi
    125. G. I. Shishkin, L. P. Shishkina, “Improved difference scheme of the solution decomposition method for a singularly perturbed reaction-diffusion equation”, Proc. Steklov Inst. Math. (Suppl.), 272, suppl. 1 (2011), S197–S214  mathnet  crossref  isi  elib
    126. G. I. Shishkin, L. P. Shishkina, “A Richardson scheme of an increased order of accuracy for a semilinear singularly perturbed elliptic convection-diffusion equation”, Comput. Math. Math. Phys., 50:3 (2010), 437–456  mathnet  crossref  mathscinet  adsnasa  isi
    127. I. A. Blatov, N. V. Dobrobog, “Conditional $\varepsilon$-uniform convergence of adaptation algorithms in the finite element method for singularly perturbed problems”, Comput. Math. Math. Phys., 50:9 (2010), 1476–1493  mathnet  crossref  mathscinet  adsnasa  isi
    128. G. I. Shishkin, L. P. Shishkina, “A Richardson scheme of the decomposition method for solving singularly perturbed parabolic reaction-diffusion equation”, Comput. Math. Math. Phys., 50:12 (2010), 2003–2022  mathnet  crossref  adsnasa
    129. Kadalbajoo M.K., Gupta V., “A brief survey on numerical methods for solving singularly perturbed problems”, Applied Mathematics and Computation, 217:8 (2010), 3641–3716  crossref  mathscinet  zmath  isi
    130. Kopteva N., O'Riordan E., “Shishkin Meshes in the Numerical Solution of Singularly Perturbed Differential Equations”, Int J Numer Anal Model, 7:3 (2010), 393–415  mathscinet  zmath  isi  elib
    131. Vulkov L.G., Zadorin A.I., “Two-Grid Algorithms for an Ordinary Second Order Equation with an Exponential Boundary Layer in the Solution”, Int J Numer Anal Model, 7:3 (2010), 580–592  mathscinet  zmath  isi  elib
    132. Linss T., “Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems Introduction”, Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems, Lecture Notes in Mathematics, 1985, 2010, 1  crossref  mathscinet  isi
    133. A. I. Zadorin, S. V. Tikhovskaya, “Analysis of a difference scheme for a singular perturbation Cauchy problem on refined grids”, Num. Anal. Appl., 4:1 (2011), 36–45  mathnet  crossref
    134. G. I. Shishkin, L. P. Shishkina, “Improved approximations of the solution and derivatives to a singularly perturbed reaction-diffusion equation based on the solution decomposition method”, Comput. Math. Math. Phys., 51:6 (2011), 1020–1049  mathnet  crossref  mathscinet  isi
    135. A. I. Zadorin, N. A. Zadorin, “Interpolyatsiya funktsii s pogransloinymi sostavlyayuschimi i ee primenenie v dvukhsetochnom metode”, Sib. elektron. matem. izv., 8 (2011), 247–267  mathnet
    136. Shishkin G.I., Shishkina L.P., “Iterative Newton solution method for the Richardson scheme for a semilinear singular perturbed elliptic convection-diffusion equation”, Russian J Numer Anal Math Modelling, 26:4 (2011), 427–445  crossref  mathscinet  zmath  isi  elib
    137. A. I. Zadorin, S. V. Tihovskaya, “Difference scheme on an uniform mesh for a singularly perturbed Cauchy problem”, J. Math. Sci., 195:6 (2013), 865–872  mathnet  crossref
    138. N. G. Bandurin, N. A. Gureeva, “Software package for the numerical solution of systems of essentially nonlinear ordinary integro-differential-algebraic equations”, Math. Models Comput. Simul., 4:5 (2012), 455–463  mathnet  crossref  mathscinet  elib
    139. Vulanovic R., “Stability of a finite-difference discretization of a singular perturbation problem”, Linear Algebra Appl, 436:2 (2012), 326–334  crossref  mathscinet  zmath  isi  elib
    140. K. V. Emelyanov, “Raznostnaya skhema podgonki dlya singulyarno vozmuschennoi zadachi s tochkoi povorota”, Tr. IMM UrO RAN, 18, no. 2, 2012, 80–91  mathnet  elib
    141. G. I. Shishkin, “Obuslovlennost raznostnoi skhemy metoda dekompozitsii resheniya dlya singulyarno vozmuschennogo uravneniya konvektsii-diffuzii”, Tr. IMM UrO RAN, 18, no. 2, 2012, 291–304  mathnet  elib
    142. U. H. Zhemuhov, “Uniform grid approximation of nonsmooth solutions with improved convergence for a singularly perturbed convection-diffusion equation with characteristic layers”, Comput. Math. Math. Phys., 52:9 (2012), 1239–1259  mathnet  crossref  mathscinet  isi  elib  elib
    143. A. I. Zadorin, S. V. Tikhovskaya, “Solution of second order nonlinear singular perturbation ordinary differential equation based on the Samarskii scheme”, Num. Anal. Appl., 6:1 (2013), 9–23  mathnet  crossref  mathscinet  elib
    144. A. I. Zadorin, S. V. Tikhovskaya, “Dvukhsetochnyi metod dlya nelineinoi singulyarno vozmuschennoi kraevoi zadachi na setke Shishkina”, Sib. zhurn. industr. matem., 16:1 (2013), 42–55  mathnet  mathscinet
    145. G. I. Shishkin, “Conditioning and stability of finite difference schemes on uniform meshes for a singularly perturbed parabolic convection-diffusion equation”, Comput. Math. Math. Phys., 53:4 (2013), 431–454  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    146. K. V. Emelyanov, “O raznostnoi skheme pervogo poryadka tochnosti dlya singulyarno vozmuschennoi zadachi s tochkoi povorota”, Tr. IMM UrO RAN, 19, no. 3, 2013, 120–135  mathnet  mathscinet  elib
    147. Shishkin G.I., “Data Perturbation Stability of Difference Schemes on Uniform Grids for a Singularly Perturbed Convection-Diffusion Equation”, Russ. J. Numer. Anal. Math. Model, 28:4 (2013), 381–417  crossref  mathscinet  isi  elib
    148. Zheng Q., Li X.-zh., Liu Yu.-f., “Uniform Second-Order Hybrid Schemes on Bakhvalov-Shishkin Mesh For Quasi-Linear Convection-Diffusion Problems”, Applied Mechanics, Fluid and Solid Mechanics, Advanced Materials Research, 871, ed. Tan J., Trans Tech Publications Ltd, 2014, 135–140  crossref  isi  elib
    149. V. B. Andreev, “Estimating the smoothness of the regular component of the solution to a one-dimensional singularly perturbed convection-diffusion equation”, Comput. Math. Math. Phys., 55:1 (2015), 19–30  mathnet  crossref  crossref  isi  elib  elib
    150. G. I. Shishkin, L. P. Shishkina, “Difference scheme of highest accuracy order for a singularly perturbed reaction-diffusion equation based on the solution decomposition method”, Proc. Steklov Inst. Math. (Suppl.), 292, suppl. 1 (2016), 262–275  mathnet  crossref  mathscinet  isi  elib
    151. Zheng Q., Li X., Gao Yu., “Uniformly Convergent Hybrid Schemes For Solutions and Derivatives in Quasilinear Singularly Perturbed Bvps”, Appl. Numer. Math., 91 (2015), 46–59  crossref  mathscinet  zmath  isi  elib
    152. S. V. Tikhovskaya, “Issledovanie dvukhsetochnogo metoda povyshennoi tochnosti dlya ellipticheskogo uravneniya reaktsii–diffuzii s pogranichnymi sloyami”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 157, no. 1, Izd-vo Kazanskogo un-ta, Kazan, 2015, 60–74  mathnet  elib
    153. G. I. Shishkin, L. P. Shishkina, “A higher order accurate solution decomposition scheme for a singularly perturbed parabolic reaction-diffusion equation”, Comput. Math. Math. Phys., 55:3 (2015), 386–409  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    154. A. A. Belov, N. N. Kalitkin, “Numerical simulations of boundary layer problems”, Math. Models Comput. Simul., 8:4 (2016), 341–347  mathnet  crossref  mathscinet  elib
    155. Zadorin A.I. Tikhovskaya S.V. Zadorin N.A., “a Two-Grid Method For Elliptic Problem With Boundary Layers”, Appl. Numer. Math., 93:SI (2015), 270–278  crossref  mathscinet  zmath  isi  elib
    156. Tikhovskaya S.V., Zadorin A.I., “a Two-Grid Method With Richardson Extrapolation For a Semilinear Convection-Diffusion Problem”, Application of Mathematics in Technical and Natural Sciences (Amitans'15), AIP Conference Proceedings, 1684, ed. Todorov M., Amer Inst Physics, 2015, 090007  crossref  isi
    157. I. A. Blatov, E. V. Kitaeva, “Convergence of the adapting grid method of Bakhvalov's type for singularly perturbed boundary value problems”, Num. Anal. Appl., 9:1 (2016), 34–44  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    158. Zh. O. Dombrovskaya, “Metod konechnykh raznostei vo vremennoi oblasti dlya kusochno-odnorodnykh dielektricheskikh sred”, Model. i analiz inform. sistem, 23:5 (2016), 539–547  mathnet  crossref  mathscinet  elib
    159. Tikhovskaya S.V., Zadorin A.I., “Analysis of polynomial interpolation of the function of two variables with large gradients in the parabolic boundary layers”, APPLICATION OF MATHEMATICS IN TECHNICAL AND NATURAL SCIENCES: 8th International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences - AMiTaNS?16 (Albena, Bulgaria, 22?27 June 2016), AIP Conference Proceedings, 1773, ed. Todorov M., Amer Inst Physics, 2016, 100008  crossref  isi
    160. Gracia J.L., O'Riordan E., “Numerical approximation of solution derivatives of singularly perturbed parabolic problems of convection-diffusion type”, Math. Comput., 85:298 (2016), 581–599  crossref  isi
    161. A. I. Zadorin, N. A. Zadorin, “Polinomialnaya interpolyatsiya funktsii dvukh peremennykh s bolshimi gradientami v pogranichnykh sloyakh”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 158, no. 1, Izd-vo Kazanskogo un-ta, Kazan, 2016, 40–50  mathnet  elib
    162. I. A. Blatov, A. I. Zadorin, E. V. Kitaeva, “Cubic spline interpolation of functions with high gradients in boundary layers”, Comput. Math. Math. Phys., 57:1 (2017), 7–25  mathnet  crossref  crossref  isi  elib
    163. I. A. Blatov, A. I. Zadorin, E. V. Kitaeva, “About the uniform convergence of parabolic spline interpolation on the class of functions with large gradients in the boundary layer”, Num. Anal. Appl., 10:2 (2017), 108–119  mathnet  crossref  crossref  isi  elib
    164. Tikhovskaya S., “Solving a Singularly Perturbed Elliptic Problem By a Multigrid Algorithm With Richardson Extrapolation”, Numerical Analysis and Its Applications (NAA 2016), Lecture Notes in Computer Science, 10187, eds. Dimov I., Farago I., Vulkov L., Springer International Publishing Ag, 2017, 674–681  crossref  isi
    165. Khakimzyanov G., Dutykh D., “On Supraconvergence Phenomenon For Second Order Centered Finite Differences on Non-Uniform Grids”, J. Comput. Appl. Math., 326 (2017), 1–14  crossref  isi
    166. V. B. Andreev, “Hölder estimates for the regular component of the solution to a singularly perturbed convection-diffusion equation”, Comput. Math. Math. Phys., 57:12 (2017), 1935–1972  mathnet  crossref  crossref  isi  elib
    167. I. A. Blatov, A. I. Zadorin, E. V. Kitaeva, “Parabolic spline interpolation for functions with large gradient in the boundary layer”, Siberian Math. J., 58:4 (2017), 578–590  mathnet  crossref  crossref  isi  elib  elib
    168. I. A. Blatov, A. I. Zadorin, E. V. Kitaeva, “On the parameter-uniform convergence of exponential spline interpolation in the presence of a boundary layer”, Comput. Math. Math. Phys., 58:3 (2018), 348–363  mathnet  crossref  crossref  isi  elib
    169. V. D. Liseikin, V. I. Paasonen, “Compact difference schemes and layer-resolving grids for the numerical modeling of problems with boundary and interior layers”, Num. Anal. Appl., 12:1 (2019), 37–50  mathnet  crossref  crossref  isi  elib
    170. Zadorin A., Tikhovskaya S., “Formulas of Numerical Differentiation on a Uniform Mesh For Functions With the Exponential Boundary Layer”, Int. J. Numer. Anal. Model., 16:4 (2019), 590–608  mathscinet  isi
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
    Number of views:
    This page:1223
    Full text:503
    First page:1

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020