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.

Newton-like iteration methods are usually used for implicit time stepping to solve large scale nonlinear systems. Each of the nonlinear iterative step requires solving large linear equations composed of a Jacobian matrix of the nonlinear system, and the error of solving such linear equations can have a significant impact on the convergence of the nonlinear system. However, the convergence criterion of the linear iteration is less studied when the Jacobian matrix error exists in the Newton method. Aiming at the above problem, this work first presents the Newton iterative formula with both the Jacobian matrix error and the linear iteration error, and verifies the significant influence of the Jacobian matrix error on the iteration through numerical tests. Then, two different types of linear iterative convergence criteria are tested numerically, and the produced over-solving problem is investigated at the presence of Jacobian matrix error. Finally, a new convergence criterion of the linear iteration is developed based on two of the previous convergence criteria. Numerical tests suggest that the proposed method is effective to relieve over-solving problems, thus improves the computational efficiency.

Correction Procedure via Reconstruction (CPR) is a compact and efficient numerical method, but it still has drawbacks in capturing strong shocks. However, Weighted Compact Nonlinear Scheme (WCNS) can capture shocks well. By introducing second-order scheme based on high-order interpolation of WCNS into high-order CPR method, an efficient hybrid shock-capturing method with high resolution property is proposed. Firstly, shock detection indicator based on the deviation of nonlinear weights from linear weights is used to judge troubled cells. Buffer cells are introduced in the neighbors of troubled cells, and other cells are marked as smooth cells. Therefore we use the second-order scheme to calculate in troubled cells and buffer cells, and CPR to calculate in smooth cells, which is the main idea of the proposed hybrid method. By calculating several numerical problems including isentropic vortex, shock and shock-vortex interaction, the accuracy, shock capturing ability and efficiency of hybrid method are validated with reference results. Numerical results show that the proposed hybrid method has satisfying shock-capturing ability, and has high-resolution property in smooth areas, which can be efficiently applied to numerical simulations of supersonic flows. Moreover, the method is more efficient compared with the second-order scheme based on high-order interpolation of WCNS.