High-Order, Efficient, Numerical Algorithms for Integration in Manifolds Implicitly Defined by Level Sets

New numerical algorithms are devised for high-order, efficient quadrature in domains arising from the intersection of a hyperrectangle and a manifold implicitly defined by level sets. By casting the manifold locally as the graph of a function (implicitly evaluated through a recurrence relation for the zero level set), a recursion stack is set up in which the interface and integrand information of a single dimension after another will be treated. Efficient means for the resulting dimension reduction process are developed, including maps for identifying lower-dimensional hyperrectangle facets, algorithms for minimal coordinate-flip vertex traversal, which, together with our multilinear-form-based derivative approximation algorithms, are used for checking a proposed integration direction on a facet, as well as algorithms for detecting interface-free sub-hyperrectangles. The multidimensional quadrature nodes generated by this method are inside their respective domains (hence, the method does not require any extension of the integrand) and the quadrature weights inherit any positivity of the underlying single-dimensional quadrature method, if present. The accuracy and efficiency of the method are demonstrated through convergence and timing studies for test cases in spaces of up to seven dimensions. The strengths and weaknesses of the method in high dimensional spaces are discussed.