Abstract
This article concerns with incorporating wavelet bases into existing
streamline upwind Petrov-Galerkin (SUPG) methods for the numerical
solution of nonlinear hyperbolic conservation laws which are known
to develop shock solutions. Here, we utilize an SUPG formulation
using continuous Galerkin in space and discontinuous Galerkin in
time. The main motivation for such a combination is that these methods
have good stability properties thanks to adding diffusion in the
direction of streamlines. But they are more expensive than explicit
semidiscrete methods as they have to use space-time formulations.
Using wavelet bases we maintain the stability properties of SUPG
methods while we reduce the cost of these methods significantly through
natural adaptivity of wavelet expansions. In addition, wavelet bases
have a hierarchical structure. We use this property to numerically
investigate the hierarchical addition of an artificial diffusion
for further stabilization in spirit of spectral diffusion. Furthermore,
we add the hierarchical diffusion only in the vicinity of discontinuities
using the feature of wavelet bases in detection of location of discontinuities.
Also, we again use the last feature of the wavelet bases to perform
a postprocessing using a denosing technique based on a minimization
formulation to reduce Gibbs oscillations near discontinuities while
keeping other regions intact. Finally, we show the performance of
the proposed combination through some numerical examples including
Burgers�, transport, and wave equations as well as systems of shallow
water equations.� 2017 Wiley Periodicals, Inc. Numer Methods Partial
Differential Eq 33: 2062�2089, 2017
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