  Higher order finite difference Weighted Essentially Non-Oscillatory (WENO)
schemes for conservation laws represent a technology that has been reasonably
consolidated. They are extremely popular because, when applied to
multidimensional problems, they offer high order accuracy at a fraction of the
cost of finite volume WENO or DG schemes. They come in two flavors. There is
the classical finite difference WENO (FD-WENO) method (Shu and Osher, J.
Comput. Phys., 83 (1989) 32-78). However, in recent years there is also an
alternative finite difference WENO (AFD-WENO) method which has recently been
formalized into a very useful general-purpose algorithm for conservation laws
(Balsara et al., Efficient Alternative Finite Difference WENO Schemes for
Hyperbolic Conservation Laws, submitted to CAMC (2023)). However, the FD-WENO
algorithm has only very recently been formulated for hyperbolic systems with
non-conservative products (Balsara et al., Efficient Finite Difference WENO
Scheme for Hyperbolic Systems with Non-Conservative Products, to appear CAMC
(2023)). In this paper we show that there are substantial advantages in
obtaining an AFD-WENO algorithm for hyperbolic systems with non-conservative
products. Such an algorithm is documented in this paper. We present an AFD-WENO
formulation in fluctuation form that is carefully engineered to retrieve the
flux form when that is warranted and nevertheless extends to non-conservative
products. The method is flexible because it allows any Riemann solver to be
used. The formulation we arrive at is such that when non-conservative products
are absent it reverts exactly to the formulation in the second citation above
which is in exact flux conservation form. The ability to transition to a
precise conservation form when non-conservative products are absent ensures,
via the Lax-Wendroff theorem, that shock locations will be exactly ...
