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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Generalized OCI schemes for boundary layer problems
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by Alan E. Berger, Jay M. Solomon, Melvyn Ciment, Stephen H. Leventhal and Bernard C. Weinberg PDF
Math. Comp. 35 (1980), 695-731 Request permission

Abstract:

A family of tridiagonal formally fourth-order difference schemes is developed for a class of singular perturbation problems. These schemes have no cell Reynolds number limitation and satisfy a discrete maximum principle. Error estimates and numerical results for this family of methods are given, and are compared with those for several other schemes.
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
  • 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
  • A. E. BERGER, J. M. SOLOMON & M. CIMENT, "On a uniformly accurate difference method for a singular perturbation problem." (In preparation.)
  • T. H. Chong, A variable mesh finite difference method for solving a class of parabolic differential equations in one space variable, SIAM J. Numer. Anal. 15 (1978), no. 4, 835–857. MR 501973, DOI 10.1137/0715055
  • I. CHRISTIE & A. R. MITCHELL, "Upwinding of high order Galerkin methods in conduction-convection problems," Internat. J. Numer. Methods Engrg., v. 12, 1978, pp. 1764-1771.
  • Melvyn Ciment, Stephen H. Leventhal, and Bernard C. Weinberg, The operator compact implicit method for parabolic equations, J. Comput. Phys. 28 (1978), no. 2, 135–166. MR 505588, DOI 10.1016/0021-9991(78)90031-1
  • Fred Dorr, The numerical solution of singular perturbations of boundary value problems, SIAM J. Numer. Anal. 7 (1970), 281–313. MR 267781, DOI 10.1137/0707021
  • Byron L. Ehle, $A$-stable methods and Padé approximations to the exponential, SIAM J. Math. Anal. 4 (1973), 671–680. MR 331787, DOI 10.1137/0504057
  • T. M. EL-MISTIKAWY & M. J. WERLE, "Numerical method for boundary layers with blowing—the exponential box scheme," AIAA J., v. 16, 1978, pp. 749-751. J. C. HEINRICH, P. S. HUYAKORN, O. C. ZIENKIEWICZ & A. R. MITCHELL, "An upwind finite element scheme for two-dimensional convective transport equation," Internat. J. Numer. Methods Engrg., v. 11, 1977. pp. 131-143. J. C. HEINRICH & O. C. ZIENKIEWICZ, "Quadratic finite element schemes for two-dimensional convective-transport problems," Internat. J. Numer. Methods Engrg., v. 11, 1977, pp. 1831-1844.
  • Richard S. Hirsh and David H. Rudy, The role of diagonal dominance and cell Reynolds number in implicit difference methods for fluid mechanics problems, J. Comput. Phys. 16 (1974), 304–310. MR 381512, DOI 10.1016/0021-9991(74)90098-9
  • Thomas J. R. Hughes, Wing Kam Liu, and Alec Brooks, Finite element analysis of incompressible viscous flows by the penalty function formulation, J. Comput. Phys. 30 (1979), no. 1, 1–60. MR 524162, DOI 10.1016/0021-9991(79)90086-X
  • A. M. Il′in, A difference scheme for a differential equation with a small parameter multiplying the highest derivative, Mat. Zametki 6 (1969), 237–248 (Russian). MR 260195
  • Eugene Isaacson and Herbert Bishop Keller, Analysis of numerical methods, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0201039
  • 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
  • Heinz-Otto Kreiss, Difference approximations for singular perturbation problems, Numerical solutions of boundary value problems for ordinary differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1974) Academic Press, New York, 1975, pp. 199–211. MR 0405869
  • Heinz-Otto Kreiss and Nancy Nichols, Numerical methods for singular perturbation problems, Computing methods in applied sciences (Second Internat. Sympos.,Versailles, 1975) Lecture Notes in Phys., Vol. 58, Springer, Berlin, 1976, pp. 544–558. MR 0445849
  • D. C. L. Lam and R. B. Simpson, Centered differencing and the box scheme for diffusion convection problems, J. Comput. Phys. 22 (1976), no. 4, 486–500. MR 475566, DOI 10.1016/0021-9991(76)90045-0
  • J. J. H. MILLER, "Some finite difference schemes for a singular perturbation problem," in Constructive Function Theory, Proc. Internat. Conf. on Constr. Fcn. Theory, Blagoevgrad, 30 May-4 June 1977. (To appear.)
  • John J. H. Miller, Sufficient conditions for the convergence, uniformly in $\varepsilon$, of a three-point difference scheme for a singular perturbation problem, Numerical treatment of differential equations in applications (Proc. Meeting, Math. Res. Center, Oberwolfach, 1977) Lecture Notes in Math., vol. 679, Springer, Berlin, 1978, pp. 85–91. MR 515572
  • Carl E. Pearson, On a differential equation of boundary layer type, J. Math. and Phys. 47 (1968), 134–154. MR 228189
  • Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
  • Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, 2nd ed., Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0220455
  • Patrick J. Roache, Computational fluid dynamics, Hermosa Publishers, Albuquerque, N.M., 1976. With an appendix (“On artificial viscosity”) reprinted from J. Computational Phys. 10 (1972), no. 2, 169–184; Revised printing. MR 0411358
  • Donald R. Smith, The multivariable method in singular perturbation analysis, SIAM Rev. 17 (1975), 221–273. MR 361331, DOI 10.1137/1017032
  • B. K. SWARTZ, "The construction of finite difference analogs of some finite element schemes," in Mathematical Aspects of Finite Elements in Partial Differential Equations (C. de Boor, Ed.), Academic Press, New York, 1974, pp. 279-312.
  • 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
  • M. van Veldhuizen, Higher order schemes of positive type for singular perturbation problems, Numerical analysis of singular perturbation problems (Proc. Conf., Math. Inst., Catholic Univ., Nijmegen, 1978) Academic Press, London-New York, 1979, pp. 361–383. MR 556526
  • Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502
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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Math. Comp. 35 (1980), 695-731
  • MSC: Primary 65L10; Secondary 65M10
  • DOI: https://doi.org/10.1090/S0025-5718-1980-0572850-8
  • MathSciNet review: 572850