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.