Nodal Points and the Nonlinear Stability of High-Order Methods for Unsteady Flow Problems on Tetrahedral Meshes

Abstract

High-order methods have the potential to efficiently generate accurate solutions to fluid dynamics problems of practical interest. However, high-order methods are less robust than lower-order methods, as they are less dissipative, making them more susceptible to spurious oscillations and aliasing driven instabilities that arise during simulations of nonlinear phenomena. An effective approach for addressing this issue comes from noting that, for nonlinear problems, the stability of high-order nodal methods is significantly effected by the locations of the nodal points. In fact, in 1D and 2D, it has been shown that placing the nodal points at the locations of quadrature points reduces aliasing errors and improves the robustness of high-order schemes. In this paper, the authors perform an investigation of a particular set of nodal points whose locations coincide with quadrature points. Analysis is performed in order to determine the conditioning of these points and their suitability for interpolation. Thereafter, numerical experiments are performed on several canonical 3D problems in order to show that this set of nodal points is effective in reducing aliasing errors and promoting nonlinear stability.

Simulation of the Taylor–Green Vortex Using High-Order Flux Reconstruction Schemes

“…Furthermore, Jameson, Vincent and Castonguay [18] showed that the aliasing error associated with the ESFR schemes could be minimized in 1D by choosing the solution points to be the Gaussian quadrature points. Williams, Castonguay, Vincent and Jameson [19] devised new quadrature schemes in order to enhance nonlinear stability in simplex elements (triangles and tetrahedra).…”

“…It was shown for the ESFR schemes in 1D that the error is minimized by choosing the solution points to be the Gaussian quadrature points [18]. Recently, enhanced nonlinear stability has also been achieved in simplex elements by devising new quadrature schemes [19]. Even so, the aliasing error is still present and can become significant at higher orders of approximation, so some additional control over aliasing errors is sought.…”

An Analysis of Solution Point Coordinates for Flux Reconstruction Schemes on Triangular Elements

“…Following [6,7,17] we will evaluate the numerical performance of our solution points by modelling an isentropic Euler vortex in a free-stream. The initial conditions for this numerical experiment are given by…”

“…Following Williams we will refer to these herein as the Williams-Shunn (WS) points. Further, in [7] Williams and Jameson proceeded to analyse the points by Shunn and Ham [12] for solving unsteady flow problems on tetrahedral meshes. (In these experiments the corresponding set of WS points were used on the faces of the tetrahedra.)…”

“…2 we give a short overview of previous work on the subject. Existing point sets proposed by Hesthaven and Warburton [2] and Williams and Jameson [7] are discussed. We also take the opportunity to elaborate on the constraints that candidate points must fulfil.…”