This work aims to extend the well-known high-order WENO finite-difference methods for systems of conservation laws to nonconservative hyperbolic systems. The main difficulty of these systems both from the theoretical and the numerical points of view comes from the fact that the definition of weak solution is not unique: according to the theory developed by Dal Maso, LeFloch, and Murat in 1995, it depends on the choice of a family of paths. A general strategy is proposed here in which WENO operators are not only used to reconstruct fluxes but also the nonconservative products of the system. Moreover, if a Roe linearization is available, the nonconservative products can be computed through matrix-vector operations instead of path-integrals. The methods are extended to problems with source terms and two different strategies are introduced to obtain well-balanced schemes. These numerical schemes will be then applied to the two-layer shallow water equations in one- and two- dimensions to obtain high-order methods that preserve water-at-rest steady states.

翻译：本研究旨在将著名的高阶WENO有限差分法从守恒律系统推广至非守恒双曲系统。这类系统在理论和数值层面的主要困难在于弱解定义的非唯一性：根据Dal Maso、LeFloch和Murat于1995年建立的理论，其定义依赖于路径族的选择。本文提出一种通用策略，其中WENO算子不仅用于重构通量，还用于重构系统的非守恒乘积项。此外，若存在Roe线性化，非守恒乘积项可通过矩阵-向量运算而非路径积分进行计算。该方法进一步推广至含源项问题，并引入两种不同策略以获得均衡格式。这些数值格式将应用于一维和二维双层浅水方程，从而构建能保持静水稳态的高阶数值方法。