Non-modal analysis of spectral element methods: Towards accurate and robust large-eddy simulations

Abstract

We introduce a \textit{non-modal} analysis technique that characterizes the diffusion properties of spectral element methods for linear convection-diffusion systems. While strictly speaking only valid for linear problems, the analysis is devised so that it can give critical insights on two questions: (i) Why do spectral element methods suffer from stability issues in under-resolved computations of nonlinear problems? And, (ii) why do they successfully predict under-resolved turbulent flows even without a subgrid-scale model? The answer to these two questions can in turn provide crucial guidelines to construct more robust and accurate schemes for complex under-resolved flows, commonly found in industrial applications. For illustration purposes, this analysis technique is applied to the hybridized discontinuous Galerkin methods as representatives of spectral element methods. The effect of the polynomial order, the upwinding parameter and the Péclet number on the so-called \textit{short-term diffusion} of the scheme are investigated. From a purely non-modal analysis point of view, polynomial orders between $2$ and $4$ with standard upwinding are well suited for under-resolved turbulence simulations. For lower polynomial orders, diffusion is introduced in scales that are much larger than the grid resolution. For higher polynomial orders, as well as for strong under/over-upwinding, robustness issues can be expected. The non-modal analysis results are then tested against under-resolved turbulence simulations of the Burgers, Euler and Navier-Stokes equations. While devised in the linear setting, our non-modal analysis succeeds to predict the behavior of the scheme in the nonlinear problems considered.

References in corpus (12)

• Burgers Turbulence
• High order exactly divergence-free Hybrid Discontinuous Galerkin Methods for unsteady incompressible flows
• Kolmogorov turbulence in a random-force-driven Burgers equation
• A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations
• A Dynamic Subgrid Scale Model for Large Eddy Simulations Based on the Mori-Zwanzig Formalism
• Non-Markovian Closure Models for Large Eddy Simulations using the Mori-Zwanzig Formalism
• A hybridizable discontinuous Galerkin method for solving nonlocal optical response models
• On thin plate spline interpolation
• A hybridizable discontinuous Galerkin method for computing nonlocal electromagnetic effects in three-dimensional metallic nanostructures
• On the role of numerical dissipation in stabilising under-resolved turbulent simulations using discontinuous Galerkin methods
• Discontinuous Galerkin method for Navier-Stokes equations using kinetic flux vector splitting
• Entropy-stable hybridized discontinuous Galerkin methods for the compressible Euler and Navier-Stokes equations

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• On the Accuracy and Stability of Various DG Formulations for Diffusion
• Industry-Relevant Implicit Large-Eddy Simulation of a High-Performance Road Car via Spectral/hp Element Methods
• Eigensolution analysis of immersed boundary method based on volume penalization: applications to high-order schemes