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

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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.

**[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)**

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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