Shock capturing and high-order methods for hyperbolic conservation laws

10.30819/5084 ◽  
2020 ◽  
Author(s):  
Jan Glaubitz
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
High Order ◽  
Shock Capturing ◽  
High Order Methods ◽  
Hyperbolic Conservation

Simulation of Inviscid Compressible Flows Using PDE Transform

2014 ◽  
Vol 16 (5) ◽  
pp. 1201-1238
Author(s):  
Langhua Hu ◽  
Siyang Yang ◽  
Guo-Wei Wei
Keyword(s):  
Conservation Laws ◽  
Conservation Law ◽  
Hyperbolic Conservation Laws ◽  
High Order ◽  
Shock Capturing ◽  
Propagation Time ◽  
Oscillatory Solutions ◽  
Hyperbolic Conservation ◽  
Time Frequency ◽  
The Impact

AbstractThe 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.

scholarly journals High order explicit local time stepping methods for hyperbolic conservation laws

2020 ◽  
Vol 89 (324) ◽  
pp. 1807-1842
Author(s):  
Thi-Thao-Phuong Hoang ◽  
Lili Ju ◽  
Wei Leng ◽  
Zhu Wang
Keyword(s):  
Conservation Laws ◽  
Local Time ◽  
Hyperbolic Conservation Laws ◽  
High Order ◽  
Hyperbolic Conservation ◽  
Time Stepping ◽  
Local Time Stepping

scholarly journals A HIGH ORDER CENTRAL-UPWIND SCHEME FOR HYPERBOLIC CONSERVATION LAWS

2015 ◽  
Vol 5 (3) ◽  
pp. 453-464
Author(s):  
Xiaohan Cheng ◽  
◽  
Yufeng Nie ◽  
Jianhu Feng ◽  
Li Cai ◽  
...  
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
High Order ◽  
Upwind Scheme ◽  
Hyperbolic Conservation
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