标题： 平衡路径保守的不连续伽辽金方法，用于双层浅水方程的平衡保持空间 显示英文标题

标题： Well-balanced path-conservative discontinuous Galerkin methods with equilibrium preserving space for two-layer shallow water equations

摘要： 本文介绍了用于双层浅水方程的平衡路径保守不连续伽辽金（DG）方法，确保静水和动水平衡稳态的精确性。 该方法涉及在DG分段多项式空间中逼近平衡变量，同时将DG格式表示为路径保守格式。 为了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）的理论。 此外，选择连接平衡函数的线段路径以保证所得格式的平衡性质。 简单的“静湖”稳态无需任何修改即可自然满足，而对数值通量的专门处理对于保持动水稳态至关重要。 一维和二维的大量数值例子验证了稳态解的精确平衡保持，并展示了其高阶精度。 该方法的性能和高分辨率结果进一步证明了其作为非守恒双曲平衡定律稳健方法的潜力。 显示英文摘要

摘要： 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.