Methods and Algorithms for Scientific Computing

High-Order Quadrature Methods for Implicitly Defined Surfaces and Volumes in Hyperrectangles

Abstract

A high-order accurate numerical quadrature algorithm is presented for the evaluation of integrals over curved surfaces and volumes which are defined implicitly via a fixed isosurface of a given function restricted to a given hyperrectangle. By converting the implicitly defined geometry into the graph of an implicitly defined height function, the approach leads to a recursive algorithm on the number of spatial dimensions which requires only one-dimensional root finding and one-dimensional Gaussian quadrature. The computed quadrature scheme yields strictly positive quadrature weights and inherits the high-order accuracy of Gaussian quadrature: a range of different convergence tests demonstrate orders of accuracy up to 20th order. Also presented is an application of the quadrature algorithm to a high-order embedded boundary discontinuous Galerkin method for solving partial differential equations on curved domains.

Keywords

  1. quadrature
  2. integration
  3. implicit surfaces
  4. level set function
  5. level set methods
  6. high order

MSC codes

  1. 65D30
  2. 65N30

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Supplementary Material


PLEASE NOTE: These supplementary files have not been peer-reviewed.


Index of Supplementary Materials

Title of paper: High-order Quadrature Methods for Implicitly Defined Surfaces and Volumes in Hyperrectangles

Author: R. I. Saye

File: supp.pdf

Type: PDF file

Contents: Additional convergence tests and analysis of the high-order quadrature scheme.

Justfication: The supplementary material of this paper contains additional convergence tests which examine the quadrature scheme in scenarios which are more application-specific and may be omitted on a first reading.

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Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A993 - A1019
ISSN (online): 1095-7197

History

Submitted: 23 April 2014
Accepted: 14 January 2015
Published online: 23 April 2015

Keywords

  1. quadrature
  2. integration
  3. implicit surfaces
  4. level set function
  5. level set methods
  6. high order

MSC codes

  1. 65D30
  2. 65N30

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