A High-Order Quadrature Method for Implicitly Defined Hypersurfaces and Regions

TL;DR

This paper introduces a high-order quadrature algorithm for accurately computing integrals over implicitly defined hypersurfaces and regions, utilizing a domain subdivision and Gaussian quadrature with positive weights.

Contribution

The paper develops a novel high-order quadrature method that simplifies integration over implicit hypersurfaces using a single root-finding step and guarantees positive weights.

Findings

Numerical tests confirm high-order convergence.

Method achieves high accuracy with positive weights.

Applicable to curved surfaces and regions defined by level sets.

Abstract

This paper presents a high-order accurate numerical quadrature algorithm for evaluating integrals over curved surfaces and regions defined implicitly via a level set of a given function restricted to a hyperrectangle. The domain is divided into small tetrahedrons, and by employing the change of variables formula, the approach yields an algorithm requiring only one-dimensional root finding and standard Gaussian quadrature. The resulting quadrature scheme guarantees strictly positive weights and inherits the high-order accuracy of Gaussian quadrature. Numerical convergence tests confirm the method's high-order accuracy.