An efficient class of WENO schemes with adaptive order

Abstract

Finite difference WENO schemes have established themselves as very worthy performers for entire classes of applications that involve hyperbolic conservation laws. In this paper we report on two major advances that make finite difference WENO schemes more efficient. The first advance consists of realizing that WENO schemes require us to carry out stencil operations very efficiently. In this paper we show that the reconstructed polynomials for any one-dimensional stencil can be expressed most efficiently and economically in Legendre polynomials. By using Legendre basis, we show that the reconstruction polynomials and their corresponding smoothness indicators can be written very compactly. The smoothness indicators are written as a sum of perfect squares. Since this is a computationally expensive step, the efficiency of finite difference WENO schemes is enhanced by the innovation which is reported here. The second advance consists of realizing that one can make a non-linear hybridization between a large, centered, very high accuracy stencil and a lower order WENO scheme that is nevertheless very stable and capable of capturing physically meaningful extrema. This yields a class of adaptive order WENO schemes, which we call WENO-AO (for adaptive order). Thus we arrive at a WENO-AO(5,3) scheme that is at best fifth order accurate by virtue of its centered stencil with five zones and at worst third order accurate by virtue of being non-linearly hybridized with an r=3 CWENO scheme. The process can be extended to arrive at a WENO-AO(7,3) scheme that is at best seventh order accurate by virtue of its centered stencil with seven zones and at worst third order

A MUSCL-type segregated – explicit staggered scheme for the Euler equations

“…Results are comparable to those presented in the recent literature (see e.g. [13,22,44,12,16,15,2]). The scheme seems to accurately capture discontinuities; however, the Kevin-Helmoltz instability which is often observed at the contact discontinuity issued from the Mach triple point is not present here, and adding some numerical diffusion (so artificial shear stresses) does not destabilize this slip line.…”

A stabilized Powell–Sabin finite-element method for the 2D Euler equations in supersonic regime

“…A zoom of the solution in the back of the domain on figure 10 shows the very well resolved Mach stem roll-up, with very good agreement with literature solutions, see for example in Ref. [3]. Also, the position of the head of the jet matches with the value usually reported in the literature which 290 is estimated between 2.71 and 2.74, see [21].…”

“…Most of the works focus on finite volumes (WENO techniques [3]) or discontinuous 5 Galerkin (DG) methods [4,5,6,7]. However in many physical applications, such as magnetohydrodynamics, shells analysis and vibrations, numerical methods with high degrees of regularity are of interest.…”

Resolution and Energy Dissipation Characteristics of Implicit LES and Explicit Filtering Models for Compressible Turbulence

“…The different variants of the WENO reconstructions have all been developed as a response to demands of either different orders of accuracy [65,66] or reduced computational expense. The interested reader is directed to [33] for an excellent discussion on the effect of the order of reconstruction on a classical two-dimensional configuration utilizing the Euler equations.…”