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.

The correction procedure via reconstruction (CPR) method is a compact and efficient high-order method suitable for unstructured grids. However, when discretizing the nonlinear convection term, it is easy to cause numerical instability due to the accumulation of aliasing errors. In the present work, we study the stability of the split form CPR method based on LG (Legendre-Gauss) points in under-resolved flows, and combine the method with the subcell limiting technique to solve under-resolved flows with shock waves. First, numerical tests are carried out to verify that the split form CPR method based on LG points with the boundary flux correction can satisfy the discrete conservation law, and such conservation is still preserved under subcell limiting. In the simulation of under-resolved flows without shock waves, compared to the divergence form CPR method, the split form significantly improves the stability of the calculation, and has smaller numerical errors than the split form CPR method using LGL (Legendre-Gauss-Lobatto) points. When solving under-resolved flows with shock waves, compared to the subcell limiting strategy of discontinuous Galerkin spectral element method based on LGL points, the subcell limiting strategy of split form CPR scheme based on LG points developed in this paper has higher resolution and less oscillation.