A practical spectral method for hyperbolic conservation laws
Y. H. Sun, Y. C. Zhou, G. W. Wei
TL;DR
This work addresses Gibbs oscillations in Fourier pseudospectral methods for hyperbolic conservation laws by introducing discrete singular convolution (DSC) lowpass filters, especially the DSC-RSK variant with an adjustable effective wavenumber range governed by $r=\sigma/\Delta$. The filters are implemented in both Fourier and physical domains, with an adaptive TVD sensor triggering filtering during RK-4 time integration, and a post-processing option to further reduce oscillations. Extensive 1D and 2D tests on scalar equations and Euler systems demonstrate strong shock-capturing capability, high spectral resolution, and long-time stability, including problems with reflective and nonperiodic boundaries and complex wave–shock interactions. The approach yields a flexible, efficient, and robust framework for high-fidelity simulations of hyperbolic problems, with potential for automatic parameter selection and extension to higher dimensions.
Abstract
A class of high-order lowpass filters, the discrete singular convolution (DSC) filters, is utilized to facilitate the Fourier pseudospectral method for the solution of hyperbolic conservation law systems. The DSC filters are 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, the Gibbs oscillations can be removed effectively while the high resolution feature of the spectral method can be retained. The utility and effectiveness of the present approach is validated by extensive numerical experiments.