This paper introduces well-balanced path-conservative discontinuous Galerkin (DG) methods for two-layer shallow water equations, ensuring exactness for both still water and moving water equilibrium steady states. The approach involves approximating the equilibrium variables within the DG piecewise polynomial space, while expressing the DG scheme in the form of path-conservative schemes. To robustly handle the nonconservative products governing momentum exchange between the layers, we incorporate the theory of Dal Maso, LeFloch, and Murat (DLM) within the DG method. Additionally, linear segment paths connecting the equilibrium functions are chosen to guarantee the well-balanced property of the resulting scheme. The simple ``lake-at-rest" steady state is naturally satisfied without any modification, while a specialized treatment of the numerical flux is crucial for preserving the moving water steady state. Extensive numerical examples in one and two dimensions validate the exact equilibrium preservation of the steady state solutions and demonstrate its high-order accuracy. The performance of the method and high-resolution results further underscore its potential as a robust approach for nonconservative hyperbolic balance laws.

翻译：本文提出了一种用于两层浅水方程的良平衡路径守恒间断伽辽金（DG）方法，确保在静水和动水平衡稳态下均能精确保持。该方法在DG分段多项式空间内近似平衡变量，同时将DG格式表示为路径守恒形式。为稳健处理控制层间动量交换的非守恒乘积，我们在DG方法中融入了Dal Maso、LeFloch和Murat（DLM）理论。此外，选取连接平衡函数的线性分段路径以保证所得格式的良平衡特性。简单的“静止湖泊”稳态无需任何修改即可自然满足，而数值通量的特殊处理对于保持动水平稳态至关重要。一维和二维的大量数值算例验证了该方法对稳态解的精确平衡保持能力，并展示了其高阶精度。该方法的性能及高分辨率结果进一步凸显了其作为非守恒双曲平衡律稳健方法的潜力。