High order weighted essentially nonoscillatory WENO-η schemes for hyperbolic conservation laws

In [8], the authors have designed a new fifth-order WENO finite-difference scheme (named WENO-η) by introducing a new local smoothness indicator which is defined based on the Lagrangian interpolation polynomials and has a more succinct form compared with the classical one proposed by Jiang and Shu [12]. With this new local smoothness indicator, higher order global smoothness indicators were able to be devised and the corresponding scheme (named WENO-ZηZη) displayed less numerical dissipations than the classic fifth-order WENO schemes, including WENO-JS [12] and WENO-Z [5] and [6]. In this paper, a close look is taken at Taylor expansions of the Lagrangian interpolation polynomials of the WENO sub-stencils and the related inherited symmetries of the local smoothness indicators, which allows the extension of the WENO-η   scheme to higher orders of accuracy. Furthermore, general formulae for the global smoothness indicators are derived with which the WENO-ZηZη schemes can be extended to all odd-orders of accuracy. Numerical experiments are conducted to demonstrate the performance of the proposed schemes.