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.

A new hybrid scheme for computing discontinuous solutions of hyperbolic equations

“…Particularly, the Courant number for the scheme A is equal to 2κ, not κ. (18). It can be seen well on Fig.…”

“…The classical idea of artificial viscosity [9] is developed in [10][11][12][13]. One often employs a scheme, in which compact Hermit interpolations on candidate stencils are used to compute fluxes at cell faces, and then either ENO algorithm [14] is used to choose a proper stencil or WENO algorithm [15][16][17][18][19][20] is used to compute weighting coefficients of compact interpolations on candidate stencils.…”

An optimized dispersion–relation-preserving combined compact difference scheme to solve advection equations

“…The numerical flux can be reconstructed using a left or right biased interpolation [20]. The appropriate interpolation is chosen based on the sign of the wave speed, which in the case of a scalar PDE is given bŷ…”

“…It's well known that the ENO and WENO schemes may be too dissipative for compressible turbulence simulations and aero-acoustics problems. Hence, the compact-reconstruction weighted essentially non-oscillatory (CRWENO) scheme [20] has been presented, in which compact sub-stencils are identified at each interface and combined using the WENO weights. WENO schemes have been intensively used for problems containing both shocks and complicated smooth solution structures [21,22].…”

Comparison of Finite Volume High-Order Schemes for the Wo-Dimensional Euler Equations

“…For periodic problems these schemes are solved using the Thomas Algorithm for periodic tridiagonal systems. Otherwise the standard Thomas Algorithm is used by applying different boundary closures to IU32EU32 (compactBC), which can be found in Pirozzoli [16] and Ghosh and Baeder [17]. Also the impact to the accuracy and stability of the application of an explicit central filter is investigated to reduce non-physical oscillations.…”