High-order ENO schemes for unstructured meshes based on least-squares reconstruction

High-order accurate schemes for conservation laws for unstructured meshes are not nearly so well advanced as such schemes for structured meshes. Consequently, little or nothing is known about the possible practical advantages of high-order discretization on unstructured meshes. This article is part of an ongoing effort to develop high-order schemes for unstructured meshes to the point where meaningful information can be obtained about the trade-offs involved in using spatial discretizations of higher than second-order accuracy on unstructured meshes. This article describes a high-order accurate ENO reconstruction scheme, called DD-L{sub 2}-ENO, for use with vertex-centered upwind flow solution algorithms on unstructured meshes. The solution of conservation equations in this context can be broken naturally into three phases: (1) solution reconstruction, in which a polynomial approximation of the solution is obtained in each control volume. (2) Flux integration around each control volume, using an appropriate flux function and a quadrature rule with accuracy commensurate with that of the reconstruction. (3) Time evolution, which may be implicit, explicit, multigrid, or some hybrid