@Article{CiCP-16-1201,
author = {Langhua Hu, Siyang Yang and Guo-Wei Wei},
title = {Simulation of Inviscid Compressible Flows Using PDE Transform},
journal = {Communications in Computational Physics},
year = {2014},
volume = {16},
number = {5},
pages = {1201--1238},
abstract = {<p style="text-align: justify;">The solution of systems of hyperbolic conservation laws remains an interesting and challenging task due to the diversity of physical origins and complexity of
the physical situations. The present work introduces the use of the partial differential
equation (PDE) transform, paired with the Fourier pseudospectral method (FPM), as
a new approach for hyperbolic conservation law problems. The PDE transform, based
on the scheme of adaptive high order evolution PDEs, has recently been applied to decompose signals, images, surfaces and data to various target functional mode functions
such as trend, edge, texture, feature, trait, noise, etc. Like wavelet transform, the PDE
transform has controllable time-frequency localization and perfect reconstruction. A
fast PDE transform implemented by the fast Fourier Transform (FFT) is introduced to
avoid stability constraint of integrating high order PDEs. The parameters of the PDE
transform are adaptively computed to optimize the weighted total variation during
the time integration of conservation law equations. A variety of standard benchmark
problems of hyperbolic conservation laws is employed to systematically validate the
performance of the present PDE transform based FPM. The impact of two PDE transform parameters, i.e., the highest order and the propagation time, is carefully studied
to deliver the best effect of suppressing Gibbs&#39; oscillations. The PDE orders of 2-6
are used for hyperbolic conservation laws of low oscillatory solutions, while the PDE
orders of 8-12 are often required for problems involving highly oscillatory solutions,
such as shock-entropy wave interactions. The present results are compared with those
in the literature. It is found that the present approach not only works well for problems that favor low order shock capturing schemes, but also exhibits superb behavior
for problems that require the use of high order shock capturing methods.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.031113.160514a},
url = {http://global-sci.org/intro/article_detail/cicp/7078.html}
}