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

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]  E. Doedel.  The Construction of Finite Difference Approximations to Ordinary Differential Equations , 1978 .

[2]  J. Flaherty,et al.  An Adaptive Finite Element Method for Initial-Boundary Value Problems for Partial Differential Equations , 1982 .

[3]  M. Ciment,et al.  Review. The Operator Compact Implicit Method for Parabolic Equations , 1978 .

[4]  Joe D. Hoffman,et al.  Relationship between the truncation errors of centered finite-difference approximations on uniform and nonuniform meshes , 1982 .

[5]  V. E. Denny,et al.  A new method for solving two-point boundary value problems using optimal node distribution , 1972 .

[6]  H. Keller.  Numerical Solution of Two Point Boundary Value Problems , 1976 .

[7]  C. Pearson.  On a Differential Equation of Boundary Layer Type , 1968 .

[8]  A. A. Samarskii,et al.  Homogeneous difference schemes on non-uniform nets☆ , 1963 .

[9]  H. Keller,et al.  Analysis of Numerical Methods , 1969 .

[10]  B. Swartz.  The Construction and Comparison of Finite Difference Analogs of Some Finite Element Schemes , 1974 .

[11]  H. Keller,et al.  Difference Methods for Boundary Value Problems in Ordinary Differential Equations , 1975 .

[12]  Andrew B. White,et al.  Supra-convergent schemes on irregular grids , 1986 .

[13]  M. R. Osborne.  Minimising truncation error in finite difference approximations to ordinary differential equations , 1967 .

[14]  Heinz-Otto Kreiss,et al.  Difference approximations for boundary and eigenvalue problems for ordinary differential equations , 1972 .

[15]  H. Keller.  Accurate Difference Methods for Nonlinear Two-Point Boundary Value Problems , 1974 .

[16]  Eugenia Kálnay de Rivas.  On the use of nonuniform grids in finite-difference equations , 1972 .

[17]  H. Keller,et al.  Accurate Difference Methods for Linear Ordinary Differential Systems Subject to Linear Constraints , 1969 .

[18]  A Variable Mesh Finite Difference Method for Solving a Class of Parabolic Differential Equations in One Space Variable , 1978 .

[19]  Eusebius J. Doedel,et al.  FINITE DIFFERENCE COLLOCATION METHODS FOR NONLINEAR TWO POINT BOUNDARY VALUE PROBLEMS , 1979 .

[20]  J. A. White,et al.  On Selection of Equidistributing Meshes for Two-Point Boundary-Value Problems , 1979 .

[21]  Thomas A. Manteuffel,et al.  On the efficient numerical solution of systems of second order boundary value problems , 1986 .

[22]  Rolf Dieter Grigorieff.  Some Stability Inequalities for Compact Finite Difference Schemes , 1988 .