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

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.