High-Order Structure-Preserving Spectral Difference Methods for Hyperbolic Conservation Laws

This paper presents a study focused on solving hyperbolic conservation laws using arbitrary-order spectral difference (SD) methods. The study is structured around several crucial aspects. Firstly, in order to ensure the maximum principle for scalar equations and positivity preservation for Euler systems, we adopt the concept of flux limiters. This adoption leads to the development of the structure-preserving SD scheme with a flux limiter (SDFL), which is proven to preserve the original high-order accuracy. However, the SDFL scheme with lower order might lack conservational properties, despite its strong performance in short-term simulations. Consequently, we have developed a specific variant of the SDFL scheme with conservational properties, referred to as the CSDFL scheme. Secondly, we introduce a modified WENO-ZQ (MWENO-ZQ) reconstruction to suppress spurious oscillations when simulating problems with strong discontinuities. Finally, we conduct extensive numerical experiments to validate the effectiveness of the proposed high-order SD (SDFL,CSDFL) methods with MWENO-ZQ reconstruction. The results demonstrate the robustness and efficiency of these techniques in solving problems involving strong discontinuities, low pressure, and low density.