Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Generalized OCI schemes for boundary layer problems

Authors: Alan E. Berger, Jay M. Solomon, Melvyn Ciment, Stephen H. Leventhal and Bernard C. Weinberg
Journal: Math. Comp. 35 (1980), 695-731
MSC: Primary 65L10; Secondary 65M10
MathSciNet review: 572850
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] 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 0388784
  • [2] 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 0070367
  • [3] A. E. BERGER, J. M. SOLOMON & M. CIMENT, "On a uniformly accurate difference method for a singular perturbation problem." (In preparation.)
  • [4] 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 0501973
  • [5] 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.
  • [6] 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, 10.1016/0021-9991(78)90031-1
  • [7] Fred Dorr, The numerical solution of singular perturbations of boundary value problems, SIAM J. Numer. Anal. 7 (1970), 281–313. MR 0267781
  • [8] Byron L. Ehle, 𝐴-stable methods and Padé approximations to the exponential, SIAM J. Math. Anal. 4 (1973), 671–680. MR 0331787
  • [9] 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.
  • [10] 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.
  • [11] 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.
  • [12] 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. Computational Phys. 16 (1974), 304–310. MR 0381512
  • [13] 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, 10.1016/0021-9991(79)90086-X
  • [14] 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 0260195
  • [15] Eugene Isaacson and Herbert Bishop Keller, Analysis of numerical methods, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0201039
  • [16] 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 0483484, 10.1090/S0025-5718-1978-0483484-9
  • [17] 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
  • [18] Heinz-Otto Kreiss and Nancy Nichols, Numerical methods for singular perturbation problems, Computing methods in applied sciences (Second Internat. Sympos.,Versailles, 1975) Springer, Berlin, 1976, pp. 544–558. Lecture Notes in Phys., Vol. 58. MR 0445849
  • [19] D. C. L. Lam and R. B. Simpson, Centered differencing and the box scheme for diffusion convection problems, J. Computational Phys. 22 (1976), no. 4, 486–500. MR 0475566
  • [20] 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.)
  • [21] John J. H. Miller, Sufficient conditions for the convergence, uniformly in 𝜖, 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
  • [22] Carl E. Pearson, On a differential equation of boundary layer type, J. Math. and Phys. 47 (1968), 134–154. MR 0228189
  • [23] Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
  • [24] Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0220455
  • [25] 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
  • [26] Donald R. Smith, The multivariable method in singular perturbation analysis, SIAM Rev. 17 (1975), 221–273. MR 0361331
  • [27] 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.
  • [28] M. van Veldhuizen, Higher order methods for a singularly perturbed problem, Numer. Math. 30 (1978), no. 3, 267–279. MR 0501937
  • [29] 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
  • [30] Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65L10, 65M10

Retrieve articles in all journals with MSC: 65L10, 65M10

Additional Information

Article copyright: © Copyright 1980 American Mathematical Society