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

J. C. Aguilar, Yu Chen

Research output: Contribution to journalArticle

Abstract

We construct correction coefficients for high-order trapezoidal quadrature rules to evaluate three-dimensional singular integrals of the form, J(v)=∫Dv(x,y,z)x2+y2+z2dxdydz, 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.

Original languageEnglish (US)
Pages (from-to)625-631
Number of pages7
JournalComputers and Mathematics with Applications
Volume49
Issue number4
DOIs
StatePublished - Feb 2005

Fingerprint

Trapezoidal Rule
Coulomb Potential
Quadrature Rules
Three-dimension
Singularity
Higher Order
One Dimension
Regular hexahedron
Two Dimensions
Singular Integrals
Scattering
Coefficient
Sampling
Weighted Sums
Quadrature
3D
Logarithmic
Three-dimensional
Evaluate
Approximation

Keywords

  • Boundary corrections
  • Correction coefficients
  • Coulomb potential
  • Quadratures
  • Trapezoidal rule

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics
  • Modeling and Simulation

Cite this

High-order corrected trapezoidal quadrature rules for the coulomb potential in three dimensions. / Aguilar, J. C.; Chen, Yu.

In: Computers and Mathematics with Applications, Vol. 49, No. 4, 02.2005, p. 625-631.

Research output: Contribution to journalArticle

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