  Higher order finite difference Weighted Essentially Non-Oscillatory (WENO)
schemes for conservation laws are extremely popular because, for
multidimensional problems, they offer high order accuracy at a fraction of the
cost of finite volume WENO or DG schemes. Such schemes come in two
formulations. The very popular classical finite difference WENO (FD-WENO)
method (Shu and Osher, J. Comput. Phys., 83 (1989) 32-78) relies two
reconstruction steps applied to two split fluxes. However, the method cannot
accommodate different types of Riemann solvers and cannot preserve free stream
boundary conditions on curvilinear meshes. This limits its utility. The
alternative finite difference WENO (AFD-WENO) method can overcome these
deficiencies, however, much less work has been done on this method. The reasons
are three-fold. First, it is difficult for the casual reader to understand the
intricate logic that requires higher order derivatives of the fluxes to be
evaluated at zone boundaries. The analytical methods for deriving the update
equation for AFD-WENO schemes are somewhat recondite. To overcome that
difficulty, we provide an easily accessible script that is based on a computer
algebra system in Appendix A of this paper. Second, the method relies on
interpolation rather than reconstruction, and WENO interpolation formulae have
not been documented in the literature as thoroughly as WENO reconstruction
formulae. In this paper, we explicitly provide all necessary WENO interpolation
formulae that are needed for implementing AFD-WENO up to ninth order. The third
reason is that AFD-WENO requires higher order derivatives of the fluxes to be
available at zone boundaries. Since those derivatives are usually obtained by
finite differencing the zone-centered fluxes, they become susceptible to a
Gibbs phenomenon when the solution ...
