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.

翻译：守恒律的高阶有限差分加权本质无振荡（WENO）格式已被成功构建。对于多维问题，该格式能以远低于同等精度有限体积WENO或DG格式的计算成本实现高阶精度，因此在科学和工程应用中极具吸引力。然而，据我们所知，此类格式尚未被推广至含非守恒乘积的非线性双曲系统。本文首次完成这一推广，从而拓展了此类格式的适用范围。该推广通过将格式改写为波动形式来实现，并以Dumbser与Balsara（2016）提出的HLLI黎曼求解器为构建基础。由于采用了HLL基础模块，所得格式具备恰当的超音速极限。反扩散通量的引入使格式能保持定常间断，进一步扩展了其适用范围。本文提出的新型有限差分WENO格式采用与经典版本相同的WENO重构过程，使用户能轻松迁移至当前格式。