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 ...

翻译：守恒律的高阶有限差分加权本质无振荡（WENO）格式是一种已相当成熟的技术。该类格式因在多维问题中以远低于有限体积WENO或间断伽辽金（DG）格式的计算成本实现高阶精度而广受欢迎。现有两种主要形式：经典有限差分WENO（FD-WENO）方法（Shu与Osher，J. Comput. Phys., 83 (1989) 32-78），以及近年来被形式化为通用守恒律算法的替代有限差分WENO（AFD-WENO）方法（Balsara等，《守恒律的高效替代有限差分WENO格式》，已提交至CAMC (2023)）。然而，FD-WENO算法直至最近才被拓展至含非守恒乘积的双曲系统（Balsara等，《含非守恒乘积双曲系统的高效有限差分WENO格式》，待发表于CAMC (2023)）。本文证明，针对此类系统构建AFD-WENO算法具有显著优势，并系统阐述了该算法。我们提出一种以波动形式表达的AFD-WENO格式，其设计精妙：在适用场景下可恢复通量形式，同时兼顾非守恒乘积的扩展性。该方法灵活性高，允许使用任意黎曼求解器。当非守恒乘积不存在时，该格式精确退化为上述第二种文献中的通量守恒形式。借助Lax-Wendroff定理，这种从非守恒乘积跃迁至精确守恒形式的能力，确保了激波位置的精确捕捉……