Abstract:
Spontaneous evolution of different scales leads to challenging
difficulties of stable computations with unresolved small scales. We
discuss how modern algorithms address these difficulties detection of
edges, high-resolution reconstruction of piecewise-smooth data between
edges, and the interplay between the theory and computational aspects of
high-resolution in low regularity spaces.
We focus our attention on two particular examples. We discuss
non-oscillatory central schemes for computing peicewise smooth solutions
of hyperbolic conservation laws, Hamilton-Jacobi equations and related
nonlinear problems. The high-resolution of these locally based algorithms
is gained by coupling nonlinear edge detectors, and turning the piecewise
polynomials resconstructions into the direction of smoothness. The second
part is an example for global methods. Here we discuss the reconstruction
of piecewise smooth data from (pseudo-) spectral global information. To
avoid spurious Gibbs oscillations and to regain the superior exponential
accuracy, we proceed in two separate steps: a detection procedure which
identifies (the location and amplitude of finitely many) edges, followed
by a family of spectrally accurate mollifiers which recover the data
between those edges. We demonstrate applications from CFD (--formation of
shocks), geometrical optics, MHD problems, image processing, and more.
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