High-order Finite Volume Methods and Multiresolution Reproducing Kernels

Abstract

This paper presents a review of some of the most successful higher-order numerical schemes for the compressible Navier-Stokes equations on unstructured grids. A suitable candidate scheme would need to be able to handle potentially discontinuous flows, arising from the predominantly hyperbolic character of the equations, and at the same time be well suited for elliptic problems, in order to deal with the viscous terms. Within this context, we explore the performance of Moving Least-Squares (MLS) approximations in the construction of higher order finite volume schemes on unstructured grids. The scope of the application of MLS is threefold: 1) computation of high order derivatives of the field variables for a Godunov-type approach to hyperbolic problems or terms of hyperbolic character, 2) direct reconstruction of the fluxes at cell edges, for elliptic problems or terms of elliptic character, and 3) multiresolution shock detection and selective limiting. The proposed finite volume method is formulated within a continuous spatial representation framework, provided by the MLS approximants, which is "broken" locally (inside each cell) into piecewise polynomial expansions, in order to make use of the specialized finite volume technology for hyperbolic problems. This approach is in contrast with the usual practice in the finite volume literature, which proceeds bottomup, starting from a piecewise constant spatial representa-L. Cueto-Felgueroso ( ) tion. Accuracy tests show that the proposed method achieves the expected convergence rates. Representative simulations show that the methodology is applicable to problems of engineering interest, and very competitive when compared to other existing procedures.

On the accuracy of finite volume and discontinuous Galerkin discretizations for compressible flow on unstructured grids

“…The first discretization analyzed in this study is a high-order unstructured-grid FV method for convection-dominated flows developed by the authors [7][8][9]. The key ingredient of this scheme is a point-wise local high-order approximation framework, which provides a continuous representation of the solution, and thus allows a simple and efficient discretization of equations with high-order terms.…”

“…In convection-dominated problems, where the character of the equations is predominantly hyperbolic, this centered approach can lead to unstable computations. For this latter type of problems, we introduce a 'broken' reconstruction, u hb I , which approximates u h (x) (and, therefore, u(x)) locally inside each cell I , and is discontinuous across cell interfaces [7,9]. In general, we require the order of accuracy of the broken reconstruction to be the same as that of the original continuous reconstruction.…”

“…For acoustic applications, Eldredge, Colonius and Leonard have used a vortex particle method for the calculation of a corotating vortex pair [6]. In [7][8][9] a method that uses a meshfree technique (moving least squares (MLS)) has been proposed to calculate high-order derivatives of the field variables, achieving a truly multidimensional high-order approach with a FV scheme. In this method, the spatial FV discretization uses the MLS approximation as a kind of 'shape functions' for unstructured grids.…”

“…Notice that the FV-MLS method works well in problems with non-smooth solutions [7][8][9], by using shock limiter techniques designed for FV solvers.…”

“…The approximate derivatives of u(x) can be expressed in terms of the derivatives of the MLS shape functions, which are functions of the derivatives of the polynomial basis p( x−x I h ) and the derivatives of C(x) [7,9]. For example, the first-, second-and third-order derivatives of u(x), evaluated at x I , are given by…”