[Paper Review] Generalised Lebesgue Stable Flux Reconstruction

Will Trojak|arXiv (Cornell University)|May 31, 2018

Computational Fluid Dynamics and Aerodynamics24 references3 citations

TL;DR

This paper introduces Generalised Lebesgue Stable Flux Reconstruction (GLSFR), a novel family of correction functions for Flux Reconstruction methods that ensures energy stability in the Lebesgue norm, enabling highly arbitrary yet stable schemes. It demonstrates through von Neumann analysis and turbulent Taylor-Green vortex simulations that GLSFR can achieve better accuracy than Nodal DG in kinetic energy dissipation error, with optimal correction functions reducing error by up to 30% on refined grids.

ABSTRACT

A unique set of correction functions for Flux Reconstruction is presented, with there derivation stemming from proving the existence of energy stability in the Lebesgue norm. The set is shown to be incredibly arbitrary with the only union to existing correction function sets being show to be for DG. Von Neumann analysis of both advection and coupled advection-diffusion is used to show that once coupled to a temporal integration method, good CFL performance can be achieved and the correction function may have better dispersion and dissipation for application to implicit LES. Lastly, the turbulent Taylor-Green vortex test case is then used to show that correction functions can be found that improve the accuracy of the scheme when compared to the error levels of Discontinuous Galerkin.

Motivation & Objective

• To address the incomplete understanding of stable correction functions in Flux Reconstruction (FR) methods, particularly for non-standard or non-DG configurations.
• To derive a new family of correction functions that guarantee energy stability in the Lebesgue norm, extending beyond existing frameworks.
• To evaluate whether these arbitrary correction functions can maintain mass conservation and temporal stability when coupled with Runge-Kutta time integration.
• To assess the practical performance of GLSFR in turbulent flow simulations, particularly in capturing kinetic energy dissipation accurately.
• To determine if GLSFR can outperform standard Nodal DG via FR in real-world fluid dynamics applications.

Proposed method

• Derive energy stability for FR schemes using the Lebesgue norm instead of Sobolev norms, simplifying the stability proof and enabling broader correction function families.
• Define correction functions $ h_L $ and $ h_R $ satisfying boundary conditions $ h_L(-1)=1, h_L(1)=0 $, and $ h_R(-1)=0, h_R(1)=1 $, while ensuring Lebesgue stability.
• Apply von Neumann analysis to linear advection and advection-diffusion equations to assess temporal stability and dispersion/dissipation characteristics under Runge-Kutta time integration.
• Use the turbulent incompressible Taylor-Green vortex at $ Re = 1600 $ to evaluate error in time-averaged kinetic energy dissipation rate against DNS data.
• Systematically vary correction function parameters (e.g., $ h_{l1} $, $ ilde{h}_{l0} $) to identify optimal configurations minimizing dissipation error.
• Compare GLSFR results with those from Nodal DG via FR using identical grid resolutions and time steps.

Experimental results

Research questions

• RQ1 Can energy stability in the Lebesgue norm be established for a broader class of correction functions in FR, beyond existing known sets?
• RQ2 Does the derived family of correction functions maintain mass conservation and temporal stability when coupled with explicit Runge-Kutta time integration?
• RQ3 Can GLSFR correction functions reduce error in turbulent kinetic energy dissipation compared to Nodal DG via FR in high-order simulations?
• RQ4 Are there specific correction function parameters that yield improved dispersion and dissipation characteristics suitable for implicit LES?
• RQ5 Is the performance improvement of GLSFR consistent across different grid resolutions in turbulent flow problems?

Key findings

• The GLSFR framework establishes a new family of correction functions that are Lebesgue-stable and extend beyond existing stable sets, with only Nodal DG via FR as a common intersection point.
• Von Neumann analysis confirms a region of temporal stability for both upwinded and central flux interfaces when using Runge-Kutta time integration, enabling practical application.
• Optimal GLSFR correction functions reduce the error in turbulent kinetic energy dissipation by up to 30% compared to Nodal DG via FR on $ 24^3 $ grids.
• For both $ 16^3 $ and $ 24^3 $ grids, GLSFR correction functions were found in a consistent region of parameter space that outperformed Nodal DG in error minimization.
• The optimal correction functions lie in the left half-plane of $ ilde{h}_{l0} $, indicating a preference for less dissipative schemes to minimize error, contrary to CFL stability trends.
• The kinetic energy dissipation rate is better matched by GLSFR than DG at the critical transition phase ($ t \approx 7.5 $) where turbulence structures form, especially at higher resolution.