Yanqun Jiang

This paper develops two classes of third-order trigonometric weighted compact nonlinear schemes (i.e., TWCNS-ENZ and TWCNS-ENP) for solving compressible Euler equations. The TWCNS-ENZ scheme builds upon a fourth-order central difference scheme for cell-node flux derivatives coupled with nonlinear weighting of two trigonometric polynomial interpolations for unknown cell-edge fluxes. Smoothness indicators used to determine nonlinear weights of sub-stencils are measured by using the undivided difference with exponential vanishing moments to improve resolution. Taylor series analyses are conducted on smoothness indicators to maintain the optimal convergence order of the designed scheme in regions containing first- and second-order critical points. To reduce the local numerical errors, the TWCNS-ENP scheme is established based on the perturbed trigonometric polynomial interpolation applied in the smooth regions that are filtered out by a monotone polynomial interpolation method. In non-smooth regions, the TWCNS-ENZ scheme is activated. Numerical results show that third-order accuracy can be attained in the smooth regions by the designed TWCNS-ENZ and TWCNS-ENP schemes. Compared to the classical third-order WCNS‑JS and WCNS-Z schemes, the proposed TWCNS schemes exhibit superior resolution and shock-capturing capability, resolving finer-scale vortex structures.

Robust and accurate numerical methods for numerical simulations of compressible fluid flows featuring discontinuities around complex geometries are challenging and of increasing practical interest. In this paper, a third-order weighted compact nonlinear scheme (WCNS) for solving inviscid compressible flows on Cartesian grids and a third-order constrained weighted least squares immersed boundary (CWLS-IB) method for dealing with irregular boundaries are developed. The numerical fluxes at the cell edges are calculated by a nonlinear combination of an optimal high-order polynomial on a global stencil and two low-order polynomials on two sub-stencils. A new global smoothness indicator of the global stencil is introduced to improve the accuracy of the scheme at both the first- and second-order critical points. The third-order CWLS method is employed to extrapolate the values at ghost cells outside the physical domain and a robust WENO-type extrapolation is adopted to eliminate oscillations when shocks occur near boundaries by introducing a nonlinear weight that can switch automatically between the high-order and the low-order extrapolation. Several numerical experiments are provided to verify the performance of the present WCNS scheme and the CWLS-IB method in terms of the accuracy, robustness and spatial resolution.

This paper presents two enhanced high-order weighted compact nonlinear schemes (WCNSs) for solving the compressible Euler equations. Based on the framework of the WENO–Z nonlinear interpolation technique, a novel global smoothness indicator of a three-point global stencil is designed to deal with the issue of order reduction at critical points. The WCNS scheme with the enhanced global smoothness indicator is called the WCNS–ZM scheme that ensures third-order convergence accuracy even in the presence of second-order critical points. To enhance the local accuracy, perturbation terms with a free parameter are introduced to the linear interpolation based on a three-point smooth stencil. The free parameter can be optimized to achieve the optimal fourth-order accuracy. The WCNS scheme with perturbation terms is called the WCNS–PM scheme that is adopted in a smooth stencil. To ensure the numerical stability, a monotone polynomial interpolation method is applied to filter out the non-smooth stencils where the WCNS–ZM scheme is applied. Numerical simulation results indicate the remarkable advantages of the two new schemes in terms of accuracy, resolution and computational efficiency, compared to the classical WCNS–JS and WCNS–Z schemes.

Two enhanced fifth-order weighted compact nonlinear schemes (WCNSs), namely the WCNS scheme with the modified ZN-type weights (WCNS-ZNM) and the perturbed WCNS scheme (WCNS-P), were proposed for compressible Euler equations. The WCNS-ZNM scheme incorporates an enhanced WENO-ZN-type nonlinear interpolation technique to obtain unknown flow values at cell edges. The WCNS-P scheme introduces perturbation terms containing a free parameter into the linear interpolations, effectively reducing numerical errors in smooth regions. A monotone polynomial interpolation approach was designed to enhance numerical stability by filtering out non-smooth regions and realizing the automatic switching between the WCNS-P and WCNS-ZNM schemes in smooth and non-smooth regions. Numerical results exhibit better accuracy and performance in resolving small-scale structures and shock waves, compared to the WCNS scheme with the WENO-Z and WENO-ZN weights.