### 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 language | English (US) |
---|---|

Pages (from-to) | 625-631 |

Number of pages | 7 |

Journal | Computers and Mathematics with Applications |

Volume | 49 |

Issue number | 4 |

DOIs | |

State | Published - Feb 2005 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

*Computers and Mathematics with Applications*, vol. 49, no. 4, pp. 625-631. https://doi.org/10.1016/j.camwa.2004.01.018

}

TY - JOUR

T1 - High-order corrected trapezoidal quadrature rules for the coulomb potential in three dimensions

AU - Aguilar, J. C.

AU - Chen, Yu

PY - 2005/2

Y1 - 2005/2

N2 - 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.

AB - 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.

KW - Boundary corrections

KW - Correction coefficients

KW - Coulomb potential

KW - Quadratures

KW - Trapezoidal rule

UR - http://www.scopus.com/inward/record.url?scp=17944368481&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=17944368481&partnerID=8YFLogxK

U2 - 10.1016/j.camwa.2004.01.018

DO - 10.1016/j.camwa.2004.01.018

M3 - Article

AN - SCOPUS:17944368481

VL - 49

SP - 625

EP - 631

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 4

ER -