Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

The numerical solution of second-order boundary value problems on nonuniform meshes


Authors: Thomas A. Manteuffel and Andrew B. White
Journal: Math. Comp. 47 (1986), 511-535, S53
MSC: Primary 65L10
DOI: https://doi.org/10.1090/S0025-5718-1986-0856700-3
MathSciNet review: 856700
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we examine the solution of second-order, scalar boundary value problems on nonuniform meshes. We show that certain commonly used difference schemes yield second-order accurate solutions despite the fact that their truncation error is of lower order. This result illuminates a limitation of the standard stability, consistency proof of convergence for difference schemes defined on nonuniform meshes. A technique of reducing centered-difference approximations of first-order systems to discretizations of the underlying scalar equation is developed. We treat both vertex-centered and cell-centered difference schemes and indicate how these results apply to partial differential equations on Cartesian product grids.


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

  • [1] C. M. Ablow & S. Schechter, "Campylotropic coordinates," J. Comput. Phys., v. 27, 1978, pp. 351-362. MR 0483454 (58:3455)
  • [2] 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., v. 15, 1978, pp. 835-857. MR 0501973 (58:19182)
  • [3] Melvyn Ciment, Stephen W. Leventhal & Bernard C. Weinberg, "The operator compact implicit method for parabolic equations," J. Comput. Phys., v. 28, 1978, pp. 135-166. MR 505588 (80a:65212a)
  • [4] Stephen F. Davis & Joseph E. Flaherty, "An adaptive finite element method for initial-boundary value problems for partial differential equations," SIAM J. Sci. Statist. Comput., v. 3, 1982, pp. 6-27. MR 651864 (83d:65259)
  • [5] V. E. Denny & R. B. Landis, "A new method for solving two-point boundary value problems using optimal node distribution," J. Comput. Phys., v. 9, 1972, pp. 120-137. MR 0295583 (45:4649)
  • [6] Eugenia Kaunax de Rivas, "On the use of nonuniform grids in finite difference equations," J. Comput. Phys., v. 10, 1972, pp. 202-210.
  • [7] Eusebius J. Doedel, "The construction of finite difference approximations to ordinary differential equations," SIAM J. Numer. Anal., v. 15, 1978, pp. 450-465. MR 0483481 (58:3482)
  • [8] Eusebius J. Doedel, "Finite difference methods for nonlinear two-point boundary value problems," SIAM J. Numer. Anal., v. 16, 1979, pp. 173-185. MR 526482 (82d:65058)
  • [9] Rolf D. Grigorieff, "Some stability inequalities for compact finite difference schemes," typescript.
  • [10] J. D. Hoffman, "Relationship between the truncation errors of centered finite-difference approximations on uniform and nonuniform meshes," J. Comput. Phys., v. 46, 1982, pp. 469-474. MR 673711 (83j:65031)
  • [11] Eugene Isaacson & H. B. Keller, Analysis of Numerical Methods, Wiley, New York, 1966. MR 0201039 (34:924)
  • [12] Herbert B. Keller, "Accurate difference methods for linear ordinary differential systems subject to linear constraints," SIAM J. Numer. Anal., v. 6, 1969, pp. 8-30. MR 0253562 (40:6776)
  • [13] H. B. Keller & A. B. White, Jr., "Difference methods for boundary value problems in ordinary differential equations," SIAM J. Numer. Anal., v. 12, 1975, pp. 791-802. MR 0413513 (54:1627)
  • [14] Herbert B. Keller, Numerical Solution of Two Point Boundary Value Problems. SIAM, Regional Conference Series in Applied Mathematics, 24. MR 0433897 (55:6868)
  • [15] H.-O. Kreiss, "Difference approximations for boundary and eigenvalue problems for ordinary differential equations," Math. Comp., v. 26, 1972, pp. 605-624. MR 0373296 (51:9496)
  • [16] H.-O. Kreiss, T. A. Manteuffel, B. Swartz, B. Wendroff & A. B. White, Jr., "Supra-convergent schemes on irregular grids," Math. Comp., v. 47, 1986, pp. 537-554. MR 856701 (88b:65082)
  • [17] David Levermore, private communication.
  • [18] R. E. Lynch & J. R. Rice, The HODIE method, Report C5D-TR 188, Dept. of Computer Science, Purdue University, W. Lafayette, Indiana.
  • [19] T. A. Manteuffel & A. B. White, Jr., "On the efficient solution of systems of second-order boundary value problems," SIAM J. Numer. Anal. (To appear.)
  • [20] M. R. Osborne, "Minimizing truncation error in finite difference approximations to ordinary differential equations," Math. Comp., v. 21, 1967, pp. 133-145. MR 0223107 (36:6156)
  • [21] Carl E. Pearson, "On a differential equation of boundary layer type," J. Math. Phys., v. 47, 1968, pp. 134-154. MR 0228189 (37:3773)
  • [22] Blair K. Swartz, "The construction and comparison of finite difference analogs of some finite element schemes," in Mathematical Aspects of Finite Elements Partial Differential Equation (1974).
  • [23] A. N. Tikhonov & A. A. Samarskiĭ, "Homogeneous difference schemes on nonuniform nets," Zh. Vychisl. Mat. i Mat. Fiz., v. 1, 1962, English translation (Russian).
  • [24] A. B. White, Jr., "On selection of equidistributing meshes for two-point boundary value problems," SIAM J. Numer. Anal., v. 16, 1979, pp. 472-502. MR 530482 (83e:65138)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65L10

Retrieve articles in all journals with MSC: 65L10


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1986-0856700-3
Keywords: Boundary value problems, compact difference schemes, irregular grids, nonuniform mesh, difference quotients, truncation error
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society