  Higher order finite difference Weighted Essentially Non-Oscillatory (WENO)
schemes have been constructed for conservation laws. For multidimensional
problems, they offer high order accuracy at a fraction of the cost of a finite
volume WENO or DG scheme of comparable accuracy. This makes them quite
attractive for several science and engineering applications. But, to the best
of our knowledge, such schemes have not been extended to non-linear hyperbolic
systems with non-conservative products. In this paper, we perform such an
extension which improves the domain of applicability of such schemes. The
extension is carried out by writing the scheme in fluctuation form. We use the
HLLI Riemann solver of Dumbser and Balsara (2016) as a building block for
carrying out this extension. Because of the use of an HLL building block, the
resulting scheme has a proper supersonic limit. The use of anti-diffusive
fluxes ensures that stationary discontinuities can be preserved by the scheme,
thus expanding its domain of applicability. Our new finite difference WENO
formulation uses the same WENO reconstruction that was used in classical
versions, making it very easy for users to transition over to the present
formulation.
  For conservation laws, the new finite difference WENO is shown to perform as
well as the classical version of finite difference WENO, with two major
advantages:- 1) It can capture jumps in stationary linearly degenerate wave
families exactly. 2) It only requires the reconstruction to be applied once.
Several examples from hyperbolic PDE systems with non-conservative products are
shown which indicate that the scheme works and achieves its design order of
accuracy for smooth multidimensional flows. Stringent Riemann ... *Abstract
truncated, see PDF*
