1,530 research outputs found
Non-modal analysis of spectral element methods: Towards accurate and robust large-eddy simulations
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 P\'eclet
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 and 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
An explicit predictor-corrector solver with applications to Burgers' equation
Forward Euler's explicit, finite-difference formula of extrapolation, is used as a predictor and a convex formula as a corrector to integrate differential equations numerically. An application has been made to Burger's equation
On the fourth-order accurate compact ADI scheme for solving the unsteady Nonlinear Coupled Burgers' Equations
The two-dimensional unsteady coupled Burgers' equations with moderate to
severe gradients, are solved numerically using higher-order accurate finite
difference schemes; namely the fourth-order accurate compact ADI scheme, and
the fourth-order accurate Du Fort Frankel scheme. The question of numerical
stability and convergence are presented. Comparisons are made between the
present schemes in terms of accuracy and computational efficiency for solving
problems with severe internal and boundary gradients. The present study shows
that the fourth-order compact ADI scheme is stable and efficient
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