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*

翻译：针对守恒律方程，高阶有限差分加权本质无振荡（WENO）格式已被构建。对于多维问题，该格式能以远低于同精度有限体积WENO或间断伽辽金（DG）格式的计算成本实现高阶精度，因而在多个科学与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。本文完成了这一推广，从而拓展了该类格式的适用范围。该推广通过将格式写成波动形式实现，并以Dumbser与Balsara（2016）提出的HLLI黎曼求解器为基础模块。由于采用HLL型基础模块，所得格式具有正确的超声速极限。反扩散通量的使用确保格式能够精确维持静止间断，进一步扩大了其适用范围。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重构方法，便于用户向当前格式过渡。针对守恒律，新型有限差分WENO格式的性能与经典版本相当，但具有两大优势：1）能精确捕捉静止线性退化波族中的间断；2）仅需应用一次重构。文中展示了多个含非守恒乘积双曲偏微分方程系统的算例，结果表明该格式有效，且对光滑多维流动能达到设计精度阶数。严苛的黎曼... *摘要截断，详见PDF*