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 limitatio...

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Bibliographic Details
Published in:Mathematics of computation 1986, Vol.47 (176), p.511-535
Main Authors: Manteuffel, Thomas A., White, Andrew B.
Format: Article
Language:English
Subjects:
Approximation
Boundary conditions
Boundary value problems
Differential equations
Error rates
Exact sciences and technology
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Ordinary differential equations
Partial differential equations
Property lines
Sciences and techniques of general use
Taylor series
Truncation errors
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Summary: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.
ISSN:0025-5718
1088-6842
DOI:10.1090/s0025-5718-1986-0856700-3