A Robust Weighted Compact Nonlinear Scheme for Compressible Euler Equations

We present a high-resolution and robust numerical scheme for the compressible Euler equations that employs a nonlinear weighted compact approach. This method effectively integrates sixth-order central compact reconstruction with third-order upwind compact and non-compact reconstructions. The resulting scheme retains the high resolution characteristic of compact methods in smooth flow regions, benefiting from reduced dispersion errors typical of such schemes. Moreover, the scheme exhibits non-oscillatory behavior near discontinuities, enabled by the use of nonlinear weighting, which is essential for accurately resolving the Euler equations. The robustness of the method is further enhanced by the interplay between the symmetric sixth-order central compact scheme and the third-order upwind non-compact reconstructions. The compact boundary scheme is meticulously designed to align with interior truncation errors, thereby improving the overall stability of the numerical solution. Extensive benchmark tests validate the scheme’s ability to maintain high resolution with minimal dissipation, effectively capturing fine-scale features and shocks without introducing numerical oscillations, even in the absence of positivity-preserving limiters for some extreme problems.