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 new strategy is introduced here that allows non-conservative products to be written as the derivative of a generalized flux function that is defined locally on the basis of the selected family of paths. WENO reconstructions are then applied to this generalized flux. Moreover, if a Roe linearization is available, the generalized flux function can be evaluated through matrix vector operations instead of path-integrals. Two different known techniques are used to extend the methods to problems with source terms and the well-balanced properties of the resulting schemes are studied. These numerical schemes are applied to a coupled Burgers system and 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方程组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，其解依赖于路径族的选择。本文提出一种新策略，可将非守恒乘积表示为广义通量函数的导数，该广义通量函数基于所选路径族在局部定义。随后将WENO重构技术应用于该广义通量。此外，若存在Roe线性化，则可通过矩阵向量运算而非路径积分来求解广义通量函数。研究采用两种不同的已知技术将方法推广至含源项问题，并分析了所得格式的保平衡特性。将这些数值格式应用于耦合Burgers系统及一维/二维双层浅水方程，获得了能够保持静水稳态的高阶数值方法。