Compact Reconstruction Schemes with Weighted ENO Limiting for Hyperbolic Conservation Laws

Abstract

The simulation of turbulent compressible flows requires an algorithm with high accuracy and spectral resolution to capture different length scales, as well as nonoscillatory behavior across discontinuities like shock waves. Compact schemes have the desired resolution properties and thus, coupled with a nonoscillatory limiter, are ideal candidates for the numerical solution of such flows. A class of compact-reconstruction weighted essentially non-oscillatory CRWENO schemes is presented in this paper where lower order compact stencils are identified at each interface and combined using the WENO weights. This yields a higher order compact scheme for smooth solutions with superior resolution and lower truncation errors, compared to the WENO schemes. Across discontinuities, the scheme reduces to a lower order nonoscillatory compact scheme by excluding stencils containing the discontinuity. The schemes are analyzed for scalar conservation laws in terms of accuracy, convergence, and computational expense, and extended to the Euler equations of fluid dynamics. The scalar reconstruction is applied to the conserved and characteristic variables. Numerical test cases are presented that show the benefits of these schemes over the traditional WENO schemes.