Quadrature rules for singular integrals with application to Schwarz-Christoffel mappings

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Original languageEnglish
Pages (from-to)15-30
Journal / PublicationJournal of Computational Physics
Issue number1
Publication statusPublished - Mar 1988
Externally publishedYes


Numerical quadrature rules for singular integrals are presented and error bounds are derived. The rules are simple modifications of composite Newton-Cotes formulas. For singularities of type xα, α > -1, the lowest order rule (modified midpoint rule) has error terms of order Δ2, Δ2+α, and Δ2 log ( 1 Δ), where Δ is the subinterval length. The rule proposed by Davis for integration of the Schwarz-Christoffel equation for conformal mapping of polygons is shown to have error terms of the same order. For polygons with sharp corners, i.e., α close to -1, the number of integration subintervals required for the Schwarz-Christoffel equation can be reduced by several orders of magnitude by use of higher order rules given here. Explicit formulas are given for four rules of most likely utility; they are extensions of the midpoint, trapezoidal, Simpson's, and 4-point rules. © 1988.

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