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

This paper proposes numerical quadrature rules for two-dimensional Cauchy principal-value integrals of the forms ∫ ∫ Ω f (x , y) (x − s) 2 + (y − t) 2 d y d x and ∫ ∫ Ω f (x , y) (x − s) (y − t) d y d x. 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 (h m 2 μ + h n 2 μ) are obtained for the proposed quadrature rules, where h m and h n 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. [ABSTRACT FROM AUTHOR]