301 research outputs found
Convenient total variation diminishing conditions for nonlinear difference schemes
Convenient conditions for nonlinear difference schemes to be total-variation diminishing (TVD) are reviewed. It is shown that such schemes share the TVD property, provided their numerical fluxes meet a certain positivity condition at extrema values but can be arbitrary otherwise. The conditions are invariant under different incremental representations of the nonlinear schemes, and thus provide a simplified generalization of the TVD conditions due to Harten and others
The convergence of spectral methods for nonlinear conservation laws
The convergence of the Fourier method for scalar nonlinear conservation laws which exhibit spontaneous shock discontinuities is discussed. Numerical tests indicate that the convergence may (and in fact in some cases must) fail, with or without post-processing of the numerical solution. Instead, a new kind of spectrally accurate vanishing viscosity is introduced to augment the Fourier approximation of such nonlinear conservation laws. Using compensated compactness arguments, it is shown that this spectral viscosity prevents oscillations, and convergence to the unique entropy solution follows
Local error estimates for discontinuous solutions of nonlinear hyperbolic equations
Let u(x,t) be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data. Suppose u sub epsilon(x,t) is the solution of an approximate viscosity regularization, where epsilon greater than 0 is the small viscosity amplitude. It is shown that by post-processing the small viscosity approximation u sub epsilon, pointwise values of u and its derivatives can be recovered with an error as close to epsilon as desired. The analysis relies on the adjoint problem of the forward error equation, which in this case amounts to a backward linear transport with discontinuous coefficients. The novelty of this approach is to use a (generalized) E-condition of the forward problem in order to deduce a W(exp 1,infinity) energy estimate for the discontinuous backward transport equation; this, in turn, leads one to an epsilon-uniform estimate on moments of the error u(sub epsilon) - u. This approach does not follow the characteristics and, therefore, applies mutatis mutandis to other approximate solutions such as E-difference schemes
Detection of Edges in Spectral Data II. Nonlinear Enhancement
We discuss a general framework for recovering edges in piecewise smooth
functions with finitely many jump discontinuities, where . Our approach is based on two main aspects--localization using
appropriate concentration kernels and separation of scales by nonlinear
enhancement.
To detect such edges, one employs concentration kernels, ,
depending on the small scale . It is shown that odd kernels, properly
scaled, and admissible (in the sense of having small -moments of
order ) satisfy , thus recovering both the location and amplitudes of all edges.As
an example we consider general concentration kernels of the form
to detect edges from the first
spectral modes of piecewise smooth f's. Here we improve in
generality and simplicity over our previous study in [A. Gelb and E. Tadmor,
Appl. Comput. Harmon. Anal., 7 (1999), pp. 101-135]. Both periodic and
nonperiodic spectral projections are considered. We identify, in particular, a
new family of exponential factors, , with superior
localization properties.
The other aspect of our edge detection involves a nonlinear enhancement
procedure which is based on separation of scales between the edges, where
, and the smooth regions where . Numerical examples demonstrate that by coupling
concentration kernels with nonlinear enhancement one arrives at effective edge
detectors
Critical Thresholds in 2D Restricted Euler-Poisson Equations
We provide a complete description of the critical threshold phenomena for the
two-dimensional localized Euler-Poisson equations, introduced by the authors in
[Liu & Tadmor, Comm. Math Phys., To appear]. Here, the questions of global
regularity vs. finite-time breakdown for the 2D Restricted Euler-Poisson
solutions are classified in terms of precise explicit formulae, describing a
remarkable variety of critical threshold surfaces of initial configurations. In
particular, it is shown that the 2D critical thresholds depend on the relative
size of three quantities: the initial density, the initial divergence as well
as the initial spectral gap, that is, the difference between the two
eigenvalues of the initial velocity gradient
Recovery of edges from spectral data with noise -- a new perspective
We consider the problem of detecting edges in piecewise smooth functions from
their N-degree spectral content, which is assumed to be corrupted by noise.
There are three scales involved: the "smoothness" scale of order 1/N, the noise
scale of order and the O(1) scale of the jump discontinuities. We use
concentration factors which are adjusted to the noise variance, >> 1/N,
in order to detect the underlying O(1)-edges, which are separated from the
noise scale, << 1
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