1,130 research outputs found
Entropy Solution Theory for Fractional Degenerate Convection-Diffusion Equations
We study a class of degenerate convection diffusion equations with a
fractional nonlinear diffusion term. These equations are natural
generalizations of anomalous diffusion equations, fractional conservations
laws, local convection diffusion equations, and some fractional Porous medium
equations. In this paper we define weak entropy solutions for this class of
equations and prove well-posedness under weak regularity assumptions on the
solutions, e.g. uniqueness is obtained in the class of bounded integrable
functions. Then we introduce a monotone conservative numerical scheme and prove
convergence toward an Entropy solution in the class of bounded integrable
functions of bounded variation. We then extend the well-posedness results to
non-local terms based on general L\'evy type operators, and establish some
connections to fully non-linear HJB equations. Finally, we present some
numerical experiments to give the reader an idea about the qualitative behavior
of solutions of these equations
An adaptive pseudospectral method for discontinuous problems
The accuracy of adaptively chosen, mapped polynomial approximations is studied for functions with steep gradients or discontinuities. It is shown that, for steep gradient functions, one can obtain spectral accuracy in the original coordinate system by using polynomial approximations in a transformed coordinate system with substantially fewer collocation points than are necessary using polynomial expansion directly in the original, physical, coordinate system. It is also shown that one can avoid the usual Gibbs oscillation associated with steep gradient solutions of hyperbolic pde's by approximation in suitably chosen coordinate systems. Continuous, high gradient solutions are computed with spectral accuracy (as measured in the physical coordinate system). Discontinuous solutions associated with nonlinear hyperbolic equations can be accurately computed by using an artificial viscosity chosen to smooth out the solution in the mapped, computational domain. Thus, shocks can be effectively resolved on a scale that is subgrid to the resolution available with collocation only in the physical domain. Examples with Fourier and Chebyshev collocation are given
Chebyshev pseudospectral methods for conservation laws with source terms and applications to multiphase flow
Pseudospectral methods are well known to produce superior results for the solution of partial differential equations whose solutions have a certain amount of regularity. Recent advances have made possible the use of spectral methods for the solution of conservation laws whose solutions may contain shocks. We use a recently described Super Spectral Viscosity method to obtain stable approximations of Systems of Nonlinear Hyperbolic Conservation Laws. A recently developed postprocessing method, which is theoretically capable of completely removing the Gibbs phenomenon from the Super Spectral Viscosity approximation, is examined. The postprocessing method has shown great promise when applied in some simple cases. We discuss its application to more complicated problems and examine the possibility of the method being used as a black box postprocessing method. Applications to multiphase fluid flow are made
Mathematical Aspects of Hydrodynamics
The workshop dealt with the partial differential equations that describe fluid motion and related topics.
These topics included both inviscid and viscous fluids in two and three dimensions. Some talks addressed
aspects of fluid dynamics such as the construction of wild weak solutions, compressible shock formation,
inviscid limit and behavior of boundary layers, as well as both polymer/fluid and structure/fluid interaction
On the decay of Burgers turbulence
This work is devoted to the decay ofrandom solutions of the unforced Burgers
equation in one dimension in the limit of vanishing viscosity. The initial
velocity is homogeneous and Gaussian with a spectrum proportional to at
small wavenumbers and falling off quickly at large wavenumbers. In physical
space, at sufficiently large distances, there is an ``outer region'', where the
velocity correlation function preserves exactly its initial form (a power law)
when is not an even integer. When the spectrum, at long times, has
three scaling regions : first, a region at very small \ms1 with a
time-independent constant, stemming from this outer region, in which the
initial conditions are essentially frozen; second, a region at
intermediate wavenumbers, related to a self-similarly evolving ``inner region''
in physical space and, finally, the usual region, associated to the
shocks. The switching from the to the region occurs around a wave
number , while the switching from to
occurs around (ignoring logarithmic
corrections in both instances). The key element in the derivation of the
results is an extension of the Kida (1979) log-corrected law for the
energy decay when to the case of arbitrary integer or non-integer .
A systematic derivation is given in which both the leading term and estimates
of higher order corrections can be obtained. High-resolution numerical
simulations are presented which support our findings.Comment: In LaTeX with 11 PostScript figures. 56 pages. One figure contributed
by Alain Noullez (Observatoire de Nice, France
Localness of energy cascade in hydrodynamic turbulence, II. Sharp spectral filter
We investigate the scale-locality of subgrid-scale (SGS) energy flux and
inter-band energy transfers defined by the sharp spectral filter. We show by
rigorous bounds, physical arguments and numerical simulations that the spectral
SGS flux is dominated by local triadic interactions in an extended turbulent
inertial-range. Inter-band energy transfers are also shown to be dominated by
local triads if the spectral bands have constant width on a logarithmic scale.
We disprove in particular an alternative picture of ``local transfer by
nonlocal triads,'' with the advecting wavenumber mode at the energy peak.
Although such triads have the largest transfer rates of all {\it individual}
wavenumber triads, we show rigorously that, due to their restricted number,
they make an asymptotically negligible contribution to energy flux and
log-banded energy transfers at high wavenumbers in the inertial-range. We show
that it is only the aggregate effect of a geometrically increasing number of
local wavenumber triads which can sustain an energy cascade to small scales.
Furthermore, non-local triads are argued to contribute even less to the
space-average energy flux than is implied by our rigorous bounds, because of
additional cancellations from scale-decorrelation effects. We can thus recover
the -4/3 scaling of nonlocal contributions to spectral energy flux predicted by
Kraichnan's ALHDIA and TFM closures. We support our results with numerical data
from a pseudospectral simulation of isotropic turbulence with
phase-shift dealiasing. We conclude that the sharp spectral filter has a firm
theoretical basis for use in large-eddy simulation (LES) modeling of turbulent
flows.Comment: 42 pages, 9 figure
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