70,679 research outputs found
High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws
The article first studies the propagation of well prepared high frequency
waves with small amplitude \eps near constant solutions for entropy solutions
of multidimensional nonlinear scalar conservation laws. Second, such
oscillating solutions are used to highlight a conjecture of Lions, Perthame,
Tadmor, (1994), about the maximal regularizing effect for nonlinear
conservation laws. For this purpose, a new definition of nonlinear flux is
stated and compared to classical definitions. Then it is proved that the
smoothness expected in Sobolev spaces cannot be exceeded.Comment: 28 p
Computing Nearly Singular Solutions Using Pseudo-Spectral Methods
In this paper, we investigate the performance of pseudo-spectral methods in
computing nearly singular solutions of fluid dynamics equations. We consider
two different ways of removing the aliasing errors in a pseudo-spectral method.
The first one is the traditional 2/3 dealiasing rule. The second one is a high
(36th) order Fourier smoothing which keeps a significant portion of the Fourier
modes beyond the 2/3 cut-off point in the Fourier spectrum for the 2/3
dealiasing method. Both the 1D Burgers equation and the 3D incompressible Euler
equations are considered. We demonstrate that the pseudo-spectral method with
the high order Fourier smoothing gives a much better performance than the
pseudo-spectral method with the 2/3 dealiasing rule. Moreover, we show that the
high order Fourier smoothing method captures about more effective
Fourier modes in each dimension than the 2/3 dealiasing method. For the 3D
Euler equations, the gain in the effective Fourier codes for the high order
Fourier smoothing method can be as large as 20% over the 2/3 dealiasing method.
Another interesting observation is that the error produced by the high order
Fourier smoothing method is highly localized near the region where the solution
is most singular, while the 2/3 dealiasing method tends to produce oscillations
in the entire domain. The high order Fourier smoothing method is also found be
very stable dynamically. No high frequency instability has been observed.Comment: 26 pages, 23 figure
Computation of Spiral Spectra
A computational linear stability analysis of spiral waves in a
reaction-diffusion equation is performed on large disks. As the disk radius R
increases, eigenvalue spectra converge to the absolute spectrum predicted by
Sandstede and Scheel. The convergence rate is consistent with 1/R, except
possibly near the edge of the spectrum. Eigenfunctions computed on large disks
are compared with predicted exponential forms. Away from the edge of the
absolute spectrum the agreement is excellent, while near the edge computed
eigenfunctions deviate from predictions, probably due to finite-size effects.
In addition to eigenvalues associated with the absolute spectrum, computations
reveal point eigenvalues. The point eigenvalues and associated eigenfunctions
responsible for both core and far-field breakup of spiral waves are shown.Comment: 20 pages, 13 figures, submitted to SIAD
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