High-order corrected trapezoidal quadrature rules for the coulomb potential in three dimensions

Abstract

We construct correction coefficients for high-order trapezoidal quadrature rules to evaluate three-dimensional singular integrals of the form, J ( v ) = ∫ D v ( x , y , z ) x 2 + y 2 + z 2 dx dy dz , where the domain D is a cube containing the point of singularity (0,0,0) and v is a C ∞ function defined on ℝ 3. The procedure employed here is a generalization to 3-D of the method of central corrections for logarithmic singularities [1] in one dimension, and [2] in two dimensions. As in one and two dimensions, the correction coefficients for high-order trapezoidal rules for J( v) are independent of the number of sampling points used to discretize the cube D. When v is compactly supported in D, the approximation is the trapezoidal rule plus a local weighted sum of the values of v around the point of singularity. These quadrature rules provide an efficient, stable and accurate way of approximating J( v). We demonstrate the performance of these quadratures of orders up to 17 for highly oscillatory functions v. These type of integrals appear in scattering calculations in 3-D.