On the Gibbs phenomenon V: recovering exponential accuracy from collocation point values of a piecewise analytic function

1995 ◽  
Vol 71 (4) ◽  
pp. 511-526 ◽  
Author(s):  
David Gottlieb ◽  
Chi-Wang Shu
Keyword(s):  
Analytic Function ◽  
Collocation Point ◽  
Gibbs Phenomenon ◽  
Exponential Accuracy ◽  
Piecewise Analytic Function

Filters, mollifiers and the computation of the Gibbs phenomenon

Acta Numerica ◽  
2007 ◽  
Vol 16 ◽  
pp. 305-378 ◽  
Author(s):  
Eitan Tadmor
Keyword(s):  
Physical Space ◽  
Gibbs Phenomenon ◽  
Delta Function ◽  
High Accuracy ◽  
Piecewise Smooth ◽  
Spectral Information ◽  
Smooth Functions ◽  
Exponential Accuracy ◽  
Local Loss ◽  
Edge Detectors

We are concerned here with processing discontinuous functions from their spectral information. We focus on two main aspects of processing such piecewise smooth data: detecting the edges of a piecewise smooth f, namely, the location and amplitudes of its discontinuities; and recovering with high accuracy the underlying function in between those edges. If f is a smooth function, say analytic, then classical Fourier projections recover f with exponential accuracy. However, if f contains one or more discontinuities, its global Fourier projections produce spurious Gibbs oscillations which spread throughout the smooth regions, enforcing local loss of resolution and global loss of accuracy. Our aim in the computation of the Gibbs phenomenon is to detect edges and to reconstruct piecewise smooth functions, while regaining the high accuracy encoded in the spectral data.To detect edges, we utilize a general family of edge detectors based on concentration kernels. Each kernel forms an approximate derivative of the delta function, which detects edges by separation of scales. We show how such kernels can be adapted to detect edges with one- and two-dimensional discrete data, with noisy data, and with incomplete spectral information. The main feature is concentration kernels which enable us to convert global spectral moments into local information in physical space. To reconstruct f with high accuracy we discuss novel families of mollifiers and filters. The main feature here is making these mollifiers and filters adapted to the local region of smoothness while increasing their accuracy together with the dimension of the data. These mollifiers and filters form approximate delta functions which are properly parametrized to recover f with (root-) exponential accuracy.

Almost limiting configurations of steady interfacial overhanging gravity waves

2018 ◽  
Vol 856 ◽  
pp. 673-708 ◽  
Author(s):  
Dmitri V. Maklakov ◽  
Ruslan R. Sharipov
Keyword(s):  
Analytic Function ◽  
Numerical Method ◽  
Gravity Waves ◽  
Tangent Line ◽  
Horizontal Direction ◽  
Main Concern ◽  
Interfacial Waves ◽  
New Concepts ◽  
Piecewise Analytic Function ◽  
The Waves

We study progressive gravity waves at the interface between two unbounded fluids of different densities. The main concern is to find almost limiting configurations for the so-called overhanging waves. The latter were first computed by Meiron & Saffman (J. Fluid Mech., vol. 129, 1983, pp. 213–218). By means of the Hopf lemma, we rigorously prove that, if$\unicode[STIX]{x1D703}$is the angle between the tangent line to the interfacial curve and the horizontal direction, then$-\unicode[STIX]{x03C0}<\unicode[STIX]{x1D703}<\unicode[STIX]{x03C0}$. This inequality allows us to put forward a criterion of proximity of the interface to the limiting configuration, namely, the angle$|\unicode[STIX]{x1D703}|_{max}$must be close to $\unicode[STIX]{x03C0}$but may not exceed$\unicode[STIX]{x03C0}$. We develop a new numerical method of computing interfacial waves based on the representation of a piecewise-analytic function to be found in such a manner that only the shape of the interface is unknown. All other hydrodynamic quantities can be expressed analytically in terms of functions describing this shape. Using this method, we compute almost limiting configurations of interfacial waves with$|\unicode[STIX]{x1D703}|_{max}>179.98^{\circ }$. Analysing the results of computations, we introduce two new concepts: an inner crest, and an inner solution near the inner crest. These concepts allow us to make a well-grounded prediction for the shapes of limiting interfacial configurations and confirm Saffman & Yuen’s (J. Fluid Mech., vol. 123, 1982, pp. 459–476) conjecture that the waves are geometrically limited.

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