The two-layer problem is defined as the coexistence of two immiscible fluids, separated by an interface surface. Under the shallow-flow hypothesis, 1D models are based on a four equations system accounting for the mass and momentum conservation in each fluid layer. Mathematically, the system of conservation laws modelling 1D two-layer flows has the important drawback of loss of hyperbolicity, causing that numerical schemes based on the eigenvalues of the Jacobian become unstable. In this work, well-balanced FORCE scheme is proposed for 1D two-layer shallow flows. The FORCE scheme combines the first-order Lax–Friedrichs flux and the second-order Lax–Wendroff flux. The scheme is supplemented with a hydrostatic reconstruction procedure in order to ensure the well-balanced behaviour of the model for steady flows even under wet–dry conditions. Additionally, a method to obtain high-accuracy numerical solutions for two-layer steady flows including friction dissipation is proposed to design reference benchmark tests for model validation. The enhanced FORCE scheme is faced to lake-at-rest benchmarking tests and steady flow cases including friction, demonstrating its well-balanced character. Furthermore, the numerical results obtained for highly unsteady two-layer dambreaks are used to analyse the robustness and accuracy of the model under a wide range of flow conditions.

两层问题定义为由界面表面分隔的两种不混溶流体的共存。在浅流假设下，一维模型基于四个方程组，说明了每个流体层的质量和动量守恒。在数学上，对一维两层流进行建模的守恒律系统具有重要的缺点，即双曲率损失，导致基于雅可比特征值的数值方案变得不稳定。在这项工作中，为一维两层浅流提出了一种均衡的FORCE方案。FORCE方案结合了一阶Lax–Friedrichs通量和二阶Lax–Wendroff通量。该方案还附加了静液压重建程序，以确保即使在干湿条件下，模型对于稳定流动也具有良好的平衡性能。此外，提出了一种获取包括摩擦耗散在内的两层稳定流高精度数值解的方法，以设计用于模型验证的参考基准测试。增强的FORCE方案面临静止湖基准测试和包括摩擦在内的稳定流量工况，证明了其良好的平衡特性。此外，对于高度不稳定的两层水坝破坏所获得的数值结果可用于分析在各种流动条件下模型的鲁棒性和准确性。展示了其均衡的特性。此外，对于高度不稳定的两层水坝破坏所获得的数值结果可用于分析在各种流动条件下模型的鲁棒性和准确性。展示了其均衡的特性。此外，对于高度不稳定的两层水坝破坏所获得的数值结果可用于分析在各种流动条件下模型的鲁棒性和准确性。