Uniform high-order difference schemes for a singularly perturbed two-point boundary value problem
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- Math. Comp. 48 (1987), 551-564 Request permission
Abstract:
A family of uniformly accurate finite-difference schemes for the model problem $- \varepsilon u”+ a(x)u’+ b(x)u = f(x)$ is constructed using a general finite-difference framework of Lynch and Rice [Math. Comp., v. 34, 1980, pp. 333-372] and Doedel [SIAM J. Numer. Anal., v. 15, 1978, pp. 450-465], A scheme of order ${h^p}$ (uniform in $\varepsilon$) is constructed to be exact on a collection of functions of the type $\{ 1,x, \ldots ,{x^p},\exp (\frac {1}{\varepsilon }\smallint a),x\exp (\frac {1}{\varepsilon }\smallint a), \ldots ,{x^{p - 1}}\exp (\frac {1}{\varepsilon }\smallint a)\}$. The high order is achieved by using extra evaluations of the source term f. The details of the construction of such a scheme (for general p) and a complete discretization error analysis, which uses the stability results of Niederdrenk and Yserentant [Numer. Math., v. 41, 1983, pp. 223-253], are given. Numerical experiments exhibiting uniform orders ${h^p}$, $p = 1,2,3, \text {and}\;4$, are presented.References
- L. R. Abrahamsson, H. B. Keller, and H. O. Kreiss, Difference approximations for singular perturbations of systems of ordinary differential equations, Numer. Math. 22 (1974), 367–391. MR 388784, DOI 10.1007/BF01436920
- Leif Abrahamsson and Stanley Osher, Monotone difference schemes for singular perturbation problems, SIAM J. Numer. Anal. 19 (1982), no. 5, 979–992. MR 672572, DOI 10.1137/0719071
- D. N. de G. Allen and R. V. Southwell, Relaxation methods applied to determine the motion, in two dimensions, of a viscous fluid past a fixed cylinder, Quart. J. Mech. Appl. Math. 8 (1955), 129–145. MR 70367, DOI 10.1093/qjmam/8.2.129
- U. Ascher and R. Weiss, Collocation for singular perturbation problems. I. First order systems with constant coefficients, SIAM J. Numer. Anal. 20 (1983), no. 3, 537–557. MR 701095, DOI 10.1137/0720035
- U. Ascher and R. Weiss, Collocation for singular perturbation problems. II. Linear first order systems without turning points, Math. Comp. 43 (1984), no. 167, 157–187. MR 744929, DOI 10.1090/S0025-5718-1984-0744929-2
- O. Axelsson, Stability and error estimates of Galerkin finite element approximations for convection-diffusion equations, IMA J. Numer. Anal. 1 (1981), no. 3, 329–345. MR 641313, DOI 10.1093/imanum/1.3.329
- Alan E. Berger, Jay M. Solomon, Melvyn Ciment, Stephen H. Leventhal, and Bernard C. Weinberg, Generalized OCI schemes for boundary layer problems, Math. Comp. 35 (1980), no. 151, 695–731. MR 572850, DOI 10.1090/S0025-5718-1980-0572850-8
- Alan E. Berger, Jay M. Solomon, and Melvyn Ciment, An analysis of a uniformly accurate difference method for a singular perturbation problem, Math. Comp. 37 (1981), no. 155, 79–94. MR 616361, DOI 10.1090/S0025-5718-1981-0616361-0
- R. C. Y. Chin and R. Krasny, A hybrid asymptotic-finite element method for stiff two-point boundary value problems, SIAM J. Sci. Statist. Comput. 4 (1983), no. 2, 229–243. MR 697177, DOI 10.1137/0904018
- James Alan Cochran, On the uniqueness of solutions of linear differential equations, J. Math. Anal. Appl. 22 (1968), 418–426. MR 224895, DOI 10.1016/0022-247X(68)90183-2
- Eusebius J. Doedel, The construction of finite difference approximations to ordinary differential equations, SIAM J. Numer. Anal. 15 (1978), no. 3, 450–465. MR 483481, DOI 10.1137/0715029
- E. P. Doolan, J. J. H. Miller, and W. H. A. Schilders, Uniform numerical methods for problems with initial and boundary layers, Boole Press, Dún Laoghaire, 1980. MR 610605 K. V. Emeĺjanov, "A truncated difference scheme for a linear singularly perturbed boundary value problem," Soviet Math. Dokl., v. 25, 1982, pp. 168-172.
- Joseph E. Flaherty and William Mathon, Collocation with polynomial and tension splines for singularly-perturbed boundary value problems, SIAM J. Sci. Statist. Comput. 1 (1980), no. 2, 260–289. MR 594760, DOI 10.1137/0901018
- Joseph E. Flaherty and R. E. O’Malley Jr., The numerical solution of boundary value problems for stiff differential equations, Math. Comput. 31 (1977), no. 137, 66–93. MR 0657396, DOI 10.1090/S0025-5718-1977-0657396-0
- Joseph E. Flaherty and Robert E. O’Malley Jr., Numerical methods for stiff systems of two-point boundary value problems, SIAM J. Sci. Statist. Comput. 5 (1984), no. 4, 865–886. MR 765211, DOI 10.1137/0905061
- R. Bruce Kellogg and Alice Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comp. 32 (1978), no. 144, 1025–1039. MR 483484, DOI 10.1090/S0025-5718-1978-0483484-9
- Barbro Kreiss and Heinz-Otto Kreiss, Numerical methods for singular perturbation problems, SIAM J. Numer. Anal. 18 (1981), no. 2, 262–276. MR 612142, DOI 10.1137/0718019
- Heinz-Otto Kreiss, Difference methods for stiff ordinary differential equations, SIAM J. Numer. Anal. 15 (1978), no. 1, 21–58. MR 486570, DOI 10.1137/0715003
- Stephen H. Leventhal, An operator compact implicit method of exponential type, J. Comput. Phys. 46 (1982), no. 1, 138–165. MR 665807, DOI 10.1016/0021-9991(82)90008-0
- Robert E. Lynch and John R. Rice, A high-order difference method for differential equations, Math. Comp. 34 (1980), no. 150, 333–372. MR 559190, DOI 10.1090/S0025-5718-1980-0559190-8
- J. J. H. Miller (ed.), Boundary and interior layers—computational and asymptotic methods, Boole Press, Dún Laoghaire, 1980. MR 589347
- J. J. H. Miller (ed.), Computational and asymptotic methods for boundary and interior layers, Boole Press Conference Series, vol. 4, Boole Press, Dún Laoghaire, 1982. MR 737565
- J. J. H. Miller (ed.), BAIL III, Boole Press Conference Series, vol. 6, Boole Press, Dún Laoghaire, 1984. Proceddings of the third international confernce on boundary and interior layers—computational and asymptotic methods; Held at Trinity College, Dublin, June 20–22, 1984. MR 774603
- Klaus Niederdrenk and Harry Yserentant, Die gleichmäßige Stabilität singulär gestörter diskreter und kontinuierlicher Randwertprobleme, Numer. Math. 41 (1983), no. 2, 223–253 (German, with English summary). MR 703123, DOI 10.1007/BF01390214
- Koichi Niijima, A uniformly convergent difference scheme for a semilinear singular perturbation problem, Numer. Math. 43 (1984), no. 2, 175–198. MR 736079, DOI 10.1007/BF01390122
- Robert E. O’Malley Jr., Introduction to singular perturbations, Applied Mathematics and Mechanics, Vol. 14, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0402217
- Eugene O’Riordan, Singularly perturbed finite element methods, Numer. Math. 44 (1984), no. 3, 425–434. MR 757497, DOI 10.1007/BF01405573
- Eugene O’Riordan and Martin Stynes, A finite element method for a singularly perturbed boundary value problem in conservative form, BAIL III (Dublin, 1984) Boole Press Conf. Ser., vol. 6, Boole, Dún Laoghaire, 1984, pp. 271–275. MR 774621
- Stanley Osher, Nonlinear singular perturbation problems and one-sided difference schemes, SIAM J. Numer. Anal. 18 (1981), no. 1, 129–144. MR 603435, DOI 10.1137/0718010
- John R. Rice, The approximation of functions. Vol. 2: Nonlinear and multivariate theory, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0244675
- Donald R. Smith, The multivariable method in singular perturbation analysis, SIAM Rev. 17 (1975), 221–273. MR 361331, DOI 10.1137/1017032
- M. van Veldhuizen, Higher order methods for a singularly perturbed problem, Numer. Math. 30 (1978), no. 3, 267–279. MR 501937, DOI 10.1007/BF01411843
- Richard Weiss, An analysis of the box and trapezoidal schemes for linear singularly perturbed boundary value problems, Math. Comp. 42 (1984), no. 165, 41–67. MR 725984, DOI 10.1090/S0025-5718-1984-0725984-2
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Math. Comp. 48 (1987), 551-564
- MSC: Primary 65L10; Secondary 34B05, 34E15
- DOI: https://doi.org/10.1090/S0025-5718-1987-0878690-0
- MathSciNet review: 878690