Positivity-preserving strategies and their comparision based on staggered CPR methods

High-order accurate algorithms often encounter numerical instabilities, such as negative density or pressure, when simulating flow fields with strong discontinuities, extremely low density, or low pressure. To maintain numerical robustness while preserving high-order accuracy, significant research efforts have been devoted to positivity-preserving limiters, particularly for high-order finite element algorithms. However, existing positivity-preserving limiters predominantly focus on ensuring positivity at solution points instead of flux points. In the hybrid correction procedure via reconstruction/compact non-uniform nonlinear weighted (CPR/CNNW) scheme, there is still a lack of in-depth research regarding the positivity preservation at flux points interpolated from solution points. In this paper, we proposed two positivity-preserving strategies for the hybrid CPR/CNNW scheme using positivity-preserving limiters and a first-order upwind method to enforce positivity constraints at flux points. Furthermore, we developed a positivity-preserving method for cell averages within the hybrid CPR/CNNW scheme, incorporating a multi-stage Runge-Kutta time integration. The two positivity-preserving strategies have been validated by numerical simulations of various problems involving discontinuities, focusing on the resolution, efficiency and robustness of the two strategies. The results show that both strategies can prevent computational crashes. In particular, the fpupwind strategy reduces CPU time by approximately 55% for the Mach 2 000 jet problems, respectively, while allowing for a larger time step size, thereby significantly improving computational efficiency.