Our paper titled Spectral/hp element methods' linear
mechanism of (apparent) energy transfer in Fourier space:
insights into dispersion analysis for implicit LES
was just accepted for publication in Journal of Computational
Broadly speaking, physical systems can be described by
partial differential equations (PDEs). These are equations
that depend on more than one independent variable (e.g,
they may depend on time and space). In addition, they
contain changes (or derivatives) of the physical (or
prognostic) variables with respect to the independent
variables). PDEs frequently exhibit highly nonlinear
and multiscale behaviour, and the usually do not have
analytical solutions. Hence, to solve them it is common
to rely on numerical approximations.
These approximations may suffer from numerical errors,
that might produce an unphysical solution. In this paper,
we build on previous work on the topic, and furhter characterize
some of these errors for spectral element numerical methods.
In recent years, different dispersion-diffusion (eigen)analyses
have been developed and used to assess various spectral element
methods (SEMs) with regards to accuracy and stability, both of
which are very important aspects for under-resolved computations
of transitional and turbulent flows. Not surprisingly, eigenanalysis
has been used recurrently to probe the inner-workings of SEM-based
implicit LES approaches, where numerical dissipation acts alone
in lieu of a subgrid model. In this study we present and discuss
an intriguing linear mechanism that causes energy transfer across
Fourier modes as seen in the energy spectrum of SEM computations.
Despite its linear nature, this mechanism has not been considered
in eigenanalyses so far, possibly due to its connection to the
often overlooked multiple eigencurves feature of periodic eigenanalysis.
As we unveil the mechanism in the simplified context of linear advection,
we point out how its effects might take place in actual turbulence
simulations. In particular, we highlight how taking it into account
in eigenanalysis can improve dissipation estimates in wavenumber
space, potentially allowing for a superior correlation between
dissipation estimates and energy spectra measured in SEM-based
eddy-resolving turbulence computations.