An unconditionally optimal high-order nonlinear spectral difference scheme for hyperbolic conservation laws

In the previous work [Lin et al., “A fifth-order nonlinear spectral difference scheme for hyperbolic conservation laws,” Comput. Fluids 221, 104928 (2021)], a fifth-order nonlinear spectral difference (NSD) scheme is proposed for hyperbolic conservation laws. However, it is noted that the scheme would lose accuracy at critical points, where the derivatives become zero. Since critical points are often encountered in applications, it is desirable for a numerical scheme to maintain its optimal convergence rate at these points. In this paper, we develop a new fifth-order NSD scheme that has the so-called unconditionally optimal high-order property, meaning that the convergence rate is independent of the order of critical points. Additionally, we construct new smoothness indicators to improve the computational efficiency of the algorithm. Several numerical examples are also presented to demonstrate the superior performance of the new NSD scheme compared to the original one.