Abstract
High-order methods are becoming increasingly attractive in both academia and industry, especially in the context of computational fluid dynamics. However, before they can be more widely adopted, issues such as lack of robustness in terms of numerical stability need to be addressed, particularly when treating industrial-type problems where challenging geometries and a wide range of physical scales, typically due to high Reynolds numbers, need to be taken into account. One source of instability is aliasing effects which arise from the nonlinearity of the underlying problem. In this work we detail two dealiasing strategies based on the concept of consistent integration. The first uses a localised approach, which is useful when the nonlinearities only arise in parts of the problem. The second is based on the more traditional approach of using a higher quadrature. The main goal of both dealiasing techniques is to improve the robustness of high order spectral element methods, thereby reducing aliasing-driven instabilities. We demonstrate how these two strategies can be effectively applied to both continuous and discontinuous discretisations, where, in the latter, both volumetric and interface approximations must be considered. We show the key features of each dealiasing technique applied to the scalar conservation law with numerical examples and we highlight the main differences in terms of implementation between continuous and discontinuous spatial discretisations.
Highlights
Numerical stability is a fundamental requirement for tools, such as solvers used to model fluid dynamics problems
Whilst in continuous Galerkin (CG) the partial differential equation (PDE) nonlinearities depend on the volumetric flux only, in discontinuous Galerkin (DG) and flux reconstruction (FR) another source of nonlinearity to take into account depends on the interface flux
Regarding the discontinuous approaches we analyse the role of the interface flux for PDE and geometrical nonlinearities
Summary
Numerical stability is a fundamental requirement for tools, such as solvers used to model fluid dynamics problems. The computational efficiencies that can be attained through the use of a collocation formulation, especially given the presence of a diagonal mass matrix, often outweigh the numerical error that is incurred To illustrate this point, let us consider errors arising from the integrals of nonlinear products of polynomial expansions of order P. To address these aliasing issues we consider two different strategies both based on the concept of consistent integration. We consistently integrate every term of the numerical discretisation which typically implies the over-integration of the linear terms In this strategy we can address both PDE-aliasing driven instabilities as well as geometrical-aliasing sources arising from deformed/curved elements.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.