An improved fifth-order nonlinear spectral difference scheme for hyperbolic conservation laws

In previous work (Lin et al., 2021), a fifth-order nonlinear spectral difference scheme was proposed for hyperbolic conservation laws. Since the mentioned nonlinear shock-capturing scheme is cell-based, it keeps the compactness of the original linear one and hence the advantage in terms of parallel efficiency. However, it was shown that the scheme encountered defects in stability and computational cost. The current work attempts to address these two defects by employing a positivity preserving method originally developed for discontinuous Galerkin schemes and a trouble cell detecting method based on edge detection. Some numerical results are also presented to demonstrate the effectiveness of the positivity preserving method and the shock detecting method. To be more specific, this work focuses on one-and two-dimensional inviscid Euler equations to validate the algorithms. It is shown that the two aforementioned defects are effectively solved by the current improved scheme.