On Obtaining High-Order Finite-Volume Solutions to the Euler Equations on Unstructured Meshes

Abstract

High-order finite volume schemes for unstructured meshes first appeared in the literature at least as early as Barth and Frederickson's 1990 paper. 1 However, the approach has not gained a wide following, perhaps because of the difficulties in achieving a genuinely high-order accurate solution, especially when dealing with curved boundaries. In this paper, we document in detail our approach to constructing a high-order solver. In addition to reconstruction, we discuss flux integration and curved boundary treatment. We describe how we tackle each of these issues, provide useful testing procedures to verify code correctness, and illustrate the results of mis-handling various small but important details.

High-Resolution Finite Volume Methods on Unstructured Grids for Turbulence and Aeroacoustics

“…However, by using these techniques it is no easy to find convergence orders higher than two. The k-exact reconstruction [18,163,164] is based on the computation of a polynomial expansion inside each cells that preserves the mean of the variable in that cell. This polynomial expansion reconstructs exactly polynomials up to order k. The coefficients defining this polynomial are chosen by minimization, in the Least-Squares sense, of the difference between the averages of the reconstructing polynomial and the actual averages.…”

A naturally anti-diffusive compressible two phases Kapila model with boundedness preservation coupled to a high order finite volume solver

“…This reconstruction can be found in some mean-conservation restrictions used in other cell averaged finite volume method schemes, see [31,32] for instance. In our case, the mean-conservation correction appears a posteriori in the reconstruction contrary to the k-exact method for example, in which a priori mean-conservation restriction is imposed.…”

Effect of discretization order on preconditioning and convergence of a high-order unstructured Newton-GMRES solver for the Euler equations

“…While ''everyone knows" that curved boundaries must be used to obtain genuinely high-order solutions, there is essentially no discussion in the literature on what level of accuracy is required, or how to get it. We have shown [37,33,35] that the common knowledge is correct: the order of accuracy of the boundary shape must equal the order of accuracy of the solver. In addition, we have shown that Gauss point locations along the boundary must be spaced by arc length rather than being projections of Gauss points from a polygonal boundary representation onto the curved boundary [35].…”

“…We have shown [37,33,35] that the common knowledge is correct: the order of accuracy of the boundary shape must equal the order of accuracy of the solver. In addition, we have shown that Gauss point locations along the boundary must be spaced by arc length rather than being projections of Gauss points from a polygonal boundary representation onto the curved boundary [35]. Finally, treatment of shocks and other solution discontinuities requires particular attention for high-order schemes, both to eliminate overshoots at discontinuities and to avoid ruining accuracy in smooth regions of the flow while retaining good convergence properties.…”

“…While accurate reconstruction -by whatever means -is of course the foundation of a high-order finite-volume method, accurate flux integration, accurate boundary treatment and robust treatment of discontinuities are equally important, and receive little or no treatment in the literature. High-order accurate flux integration, for instance, requires Gauss quadrature along all control volume interfaces [35]. While ''everyone knows" that curved boundaries must be used to obtain genuinely high-order solutions, there is essentially no discussion in the literature on what level of accuracy is required, or how to get it.…”