Abstract
The classical composite midpoint rectangle rule for computing Cauchy principal value integrals on an interval is studied. By using a piecewise constant interpolant to approximate the density function, an extended error expansion and its corresponding superconvergence results are obtained. The superconvergence phenomenon shows that the convergence rate of the midpoint rectangle rule is higher than that of the general Riemann integral when the singular point coincides with some priori known points. Finally, several numerical examples are presented to demonstrate the accuracy and effectiveness of the theoretical analysis. This research is meaningful to improve the accuracy of the collocation method for singular integrals.
Highlights
Singular integrals, especially Cauchy principal value integrals, are usually encountered in the fields of Boundary Element Method (BEM) [1,2,3], for example, the fluid mechanics, the elasticity and fracture mechanics, the acoustics, and the electromagnetics
Especially Cauchy principal value integrals, are usually encountered in the fields of Boundary Element Method (BEM) [1,2,3], for example, the fluid mechanics, the elasticity and fracture mechanics, the acoustics, and the electromagnetics. In these fields, much attention has been paid to the Cauchy principal value integrals [4,5,6,7,8]
We further study the accuracy of the midpoint rectangle rule
Summary
Especially Cauchy principal value integrals, are usually encountered in the fields of Boundary Element Method (BEM) [1,2,3], for example, the fluid mechanics, the elasticity and fracture mechanics, the acoustics, and the electromagnetics. To improve the accuracy of boundary element analysis, an efficient method called general (composite) Newton–Cotes rule has been studied and used to compute Cauchy principal value integrals and Hadamard finite-part integrals [26, 27]. We will focus on the superconvergence phenomenon of midpoint rectangle rules for Cauchy principal integrals with the density function (f(x)/(x − s)) being replaced by the approximation function (f(xi)/(xi − s)), i 0, 1, .
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