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

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

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

N. S. Bakhvalov

Moscow

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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

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} 

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This publication is cited in the following articles:
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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
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
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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
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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
32. V. D. Liseikin, “A method of algebraic adaptation”, Comput. Math. Math. Phys., 38:10 (1998), 1624–1640
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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
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
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
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
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
55. Linss T., “Sufficient conditions for uniform convergence on layer-adapted grids”, Appl Numer Math, 37:1–2 (2001), 241–255
56. Linss T., “Uniform pointwise convergence of finite difference schemes using grid equidistribution”, Computing, 66:1 (2001), 27–39
57. Kopteva N., “Uniform pointwise convergence of difference schemes for convection-diffusion problems on layer-adapted meshes”, Computing, 66:2 (2001), 179–197
58. Vulanovic R., “A higher-order scheme for quasilinear boundary value problems with two small parameters”, Computing, 67:4 (2001), 287–303
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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
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
62. Comput. Math. Math. Phys., 41:6 (2001), 898–909
63. V. D. Liseikin, “On the numerical solution of singularly perturbed problems with turning points”, Comput. Math. Math. Phys., 41:1 (2001), 55–83
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
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
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
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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
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
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
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
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
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
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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
79. Andreev V.B., “On the theory of difference schemes for singularly perturbed equations”, Differ Equ, 40:7 (2004), 959–970
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81. M. K. Kerimov, “Academician Nikolai Sergeevich Bakhvalov (on the occasion of his seventieth birthday)”, Comput. Math. Math. Phys., 45:4 (2005), 539–549
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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
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
86. Kopteva N., Madden N., Stynes M., “Grid equidistribution for reaction-diffusion problems in one dimension”, Numer Algorithms, 40:3 (2005), 305–322
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
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
89. Shishkin G.I., “Grid approximation of parabolic convection-diffusion equations with piecewise smooth initial conditions”, Doklady Mathematics, 72:3 (2005), 850–853
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
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
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
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
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
95. G. I. Shishkin, “Metod povyshennoi tochnosti dlya kvazilineinogo singulyarno vozmuschennogo ellipticheskogo uravneniya konvektsii-diffuzii”, Sib. zhurn. vychisl. matem., 9:1 (2006), 81–108
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
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
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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
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
104. Linss T., Madden N., “Parameter uniform approximations for time-dependent reaction-diffusion problems”, Numer Methods Partial Differential Equations, 23:6 (2007), 1290–1300
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
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
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
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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
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
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
112. A. I. Zadorin, “Refined-mesh interpolation method for functions with a boundary-layer component”, Comput. Math. Math. Phys., 48:9 (2008), 1634–1645
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
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
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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
117. Shishkin G.I., “Grid Approximation of Singularly Perturbed Parabolic Equations with Moving Boundary Layers”, Math Model Anal, 13:3 (2008), 421–442
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
119. Chen L., Xu J., “Stability and accuracy of adapted finite element methods for singularly perturbed problems”, Numer Math, 109:2 (2008), 167–191
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
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
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
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
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
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
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
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
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
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
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
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
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