On the properties of high-order least-squares finite-volume schemes

Abstract

High-order finite-volume schemes based on polynomial least-squares methods are an active research topic for the discretization of hyperbolic equations as they allow to obtain high-order spatial discretization schemes in arbitrary meshes. However, few studies have analyzed their performance in good-quality/near-to-uniform meshes, which are commonly used as a meshing strategy in zones where turbulent effects are important. In this paper, the theoretical numerical properties of commonly used least-squares (LSQ) k-exact high-order finite volume schemes are studied in one-dimensional and in several two-dimensional meshes (with some remarks regarding their properties in three-dimensional meshes). These results are compared to those obtained using fully-constrained polynomial reconstructions only compatible with structured meshes. The numerical properties of the schemes are investigated through the von Neumann analysis methodology applied to the one-dimensional and two-dimensional finite volume formulation, including temporal discretization errors. This analysis is also extended to non-uniform and unstructured two-dimensional meshes. At last, the schemes are tested with several numerical experiments using the linear advection, the Euler equations and the Navier-Stokes equations. The analysis of both studies yields similar conclusions regarding the numerical errors and stability of the different studied schemes showing that the high-order least-squares finite volume schemes yield stable and robust results across different uniform and non-uniform unstructured meshes. However, their performance is heavily degraded in simulations with low mesh resolution compared to schemes specially catered to structured meshes. On the other hand, the latter schemes lack stability and robustness in general structured meshes and its formulation cannot be straightforwardly extended to unstructured meshes. Moreover, this work shows that the use of weighted-LSQ can drastically improve the results of LSQ schemes in under-resolved simulations.