A Comparative Study from Spectral Analyses of High-Order Methods with Non-Constant Advection Velocities

Abstract

The response of numerical methods to flow properties dynamic changes is a key ingredient of numerical modeling calibration. In this context, a range of spectral analyses and canonical fluid mechanics problems are explored to compare the responses of the high-order spectral difference scheme and of the flux reconstruction methods. Spatial eigen-analysis (Hu et al. in J Comput Phys 151(2):921–946, 1999; Hu and Atkins in J Comput Phys 182(2):516–545, 2002), based on spatially evolving oscillations, is used to get useful insights for problems with inflow/outflow boundary conditions (i.e., non-periodic), while non-modal analysis (Fernandez et al. in Comput Methods Appl Mech Eng 346:43–62, 2019. https://doi.org/10.1016/j.cma.2018.11.027 ) focusses on the short-term dynamics of the discretised system. These two approaches are also extended to the general scalar conservation law with non-constant advection velocity. All these tools are used to obtain more insight about the expected behavior of the mentioned numerical methods for typical engineering applications. On this regard, despite the lack of a mathematical proof of stability in the non-linear case, the selected numerical schemes, for which an extensive literature is available for applications to Euler and Navier–Stokes equations, will be assumed to be stable. Findings are then verified considering one-dimensional linear advection, and two- and three-dimensional flows modeled with the compressible Euler equations. Implications in the accurate control of numerical dissipation for turbulent flows is also addressed. Within the context of high-order methods, this work complement the former temporal spectral analyses of the discontinuous Galerkin approach discretising the linear advection equation.