EXACT INTEGRATIONS OF POLYNOMIALS AND SYMMETRICAL QUADRATURE-FORMULASOVER ARBITRARY POLYHEDRAL GRIDS

This paper is concerned with two important elements in the high-order accurate spatial discretization of finite-volume equations over arbitr ary grids. One element is the integration of basis functions over arbi trary domains, which is used in expressing various spatial integrals i n terms of discrete unknowns. The other consists of quadrature approxi mations to those integrals. Only polynomial basis functions applied to polyhedral and polygonal grids are treated here. Nontriangular polygo nal faces are subdivided into a union of planar triangular facets, and the resulting triangulated polyhedron is subdivided into a union of t etrahedra. The straight line segment, triangle, and tetrahedron are th us the fundamental shapes that are the building blocks for all integra tions and quadrature approximations, Integrals of products up to the f ifth order are derived in a unified manner for the three fundamental s hapes in terms of the position vectors of vertices, Results are given both in terms of tensor products and products of Cartesian coordinates , The exact polynomial integrals are used to obtain symmetric quadratu re approximations of any degree of precision up to five for arbitrary integrals over the three Fundamental domains, Using a coordinate-free formulation, simple and rational procedures are developed to derive vi rtually. all quadrature formulas, including some previously unpublishe d Four symmetry groups of quadrature points are introduced to derive G auss formulas, while their limiting forms are used to derive Lobatto f ormulas, Representative Gauss and Lobatto formulas are tabulated, The relative efficiency of their application to polyhedral and polygonal g rids is detailed, The extension to higher degrees of precision is disc ussed, (C) 1998 Academic Press.