Asymptotic error expansions and splitting extrapolation algorithm for two classes of two-dimensional Cauchy principal-value integrals

Abstract

This paper proposes numerical quadrature rules for two-dimensional Cauchy principal-value integrals of the forms ∫∫Ωf(x,y)(x−s)2+(y−t)2dydx and ∫∫Ωf(x,y)(x−s)(y−t)dydx. The derivation of these quadrature rules is based on the Euler–Maclaurin error expansion of a modified trapezoidal rule for one-dimensional Cauchy singular integrals. The corresponding error estimations are investigated, and the convergence rates O(hm2μ+hn2μ) are obtained for the proposed quadrature rules, where hm and hn are partition sizes in x and y directions, μ is a positive integer determined by integrand. To further improve accuracy, a splitting extrapolation algorithm is developed based on the asymptotic error expansions. Several numerical tests are performed to verify the effectiveness of the proposed methods.