Title: Efficient Alternative Finite Difference WENO Schemes for Hyperbolic Systems with Non-Conservative Products Show Chinese title

Title: 高效的非守恒乘积双曲系统有限差分WENO格式的替代方法

Abstract: 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 ... Show Chinese abstract

Abstract: 高阶有限差分加权本质无振荡（WENO）格式用于守恒定律是一种已经相对稳固的技术。 它们非常受欢迎，因为当应用于多维问题时，它们以有限体积WENO或DG方案成本的一小部分提供高阶精度。 它们有两种类型。 有一种是经典的有限差分WENO（FD-WENO）方法（Shu和Osher，J. Comput. Phys., 83 (1989) 32-78）。 然而，近年来也出现了一种替代的有限差分WENO（AFD-WENO）方法，该方法最近被形式化为一种非常有用的适用于守恒定律的通用算法 (Balsara等人，Efficient Alternative Finite Difference WENO Schemes for Hyperbolic Conservation Laws, 提交给CAMC (2023))。 然而，FD-WENO算法仅在最近才被用于具有非守恒乘积的双曲系统（Balsara等人，Efficient Finite Difference WENO Scheme for Hyperbolic Systems with Non-Conservative Products, 将发表于CAMC (2023))。 在本文中，我们展示了为具有非守恒乘积的双曲系统获得AFD-WENO算法的显著优势。 该算法在本文中进行了记录。 我们提出了一种以波动形式呈现的AFD-WENO公式，该公式经过精心设计，在需要时可以恢复通量形式，并且仍然可以扩展到非守恒乘积。 该方法具有灵活性，因为它允许使用任何黎曼求解器。 我们得出的公式是这样的，当不存在非守恒乘积时，它会精确地回到上述第二个引用中的公式，该公式是精确通量守恒形式。 当不存在非守恒乘积时能够转换为精确守恒形式的能力，通过Lax-Wendroff定理确保了激波位置将被精确地...