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Simulation of Inviscid Compressible Flows Using PDE Transform

Published online by Cambridge University Press:  03 June 2015

Langhua Hu ,
Siyang Yang  and
Guo-Wei Wei
Langhua Hu*
Affiliation:
Department of Mathematics, Michigan State University, MI 48824, USA
Siyang Yang*
Affiliation:
Department of Mathematics, Michigan State University, MI 48824, USA
Guo-Wei Wei*
Affiliation:
Department of Mathematics, Michigan State University, MI 48824, USA Department of Electrical and Computer Engineering, Michigan State University, MI 48824, USA Center for Mathematical Molecular Biosciences, Michigan State University, MI 48824, USA
*
Email:hulanghu.at.msu.edu@gmail.com
Email:yangsiyang@gmail.com
Corresponding author.Email:wei@math.msu.edu
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Abstract

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’ 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.

Keywords

35K4165N9976L0576N15Partial differential equation transformhyperbolic conservation lawsFourier pseudospectral methodadaptive lowpass filtersGibbs’ oscillations
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 5 , November 2014 , pp. 1201 - 1238
Copyright
Copyright © Global Science Press Limited 2014

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