Simulation of Inviscid Compressible Flows Using PDE Transform (Communications in Computational Physics, 2014)
The solution of systems of hyperbolic conservation laws remains an interesting and challenging task due to the diversity of physical origins and complexity ofthe physical situations. The present work introduces the use of the partial differentialequation (PDE) transform, paired with the Fourier pseudospectral method (FPM), asa new approach for hyperbolic conservation law problems. The PDE transform, basedon the scheme of adaptive high order evolution PDEs, has recently been applied to decompose signals, images, surfaces and data to various target functional mode functionssuch as trend, edge, texture, feature, trait, noise, etc. Like wavelet transform, the PDEtransform has controllable time-frequency localization and perfect reconstruction. Afast PDE transform implemented by the fast Fourier Transform (FFT) is introduced toavoid stability constraint of integrating high order PDEs. The parameters of the PDEtransform are adaptively computed to optimize the weighted total variation duringthe time integration of conservation law equations. A variety of standard benchmarkproblems of hyperbolic conservation laws is employed to systematically validate theperformance 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 studiedto deliver the best effect of suppressing Gibbs' oscillations. The PDE orders of 2-6are used for hyperbolic conservation laws of low oscillatory solutions, while the PDEorders of 8-12 are often required for problems involving highly oscillatory solutions,such as shock-entropy wave interactions. The present results are compared with thosein 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 behaviorfor problems that require the use of high order shock capturing methods.