A windowed Fourier pseudospectral method for hyperbolic conservation laws

A class of local spectral wavelet filters, discrete singular convolution (DSC) filters, is utilized to facilitate the Fourier pseudospectral method for the solution of hyperbolic conservation law systems. The DSC lowpass filters are adaptively implemented directly in the Fourier domain (i.e., windowed Fourier pseudospectral method), while a physical domain algorithm is also given to enable the treatment of some special boundary conditions. By adjusting the effective wavenumber region of the DSC filter, Gibbs oscillations can be removed effectively while the high resolution feature of the spectral method can be retained for a wide class of problems with various boundary conditions. The utility and effectiveness of the present approach are validated by extensive numerical experiments. The proposed method could operate at a resolution as high as only five points per wavelength (PPW) for the interaction of shocks and physical high frequency waves, which is some of the best for this class of problems. This high resolution, together with the low complexity of the fast Fourier transform (FFT), endows the proposed method considerable potential for solving large scale problems in hyperbolic conservation laws.