In this work, we develop an improved shock-capturing and high-resolution Lagrangian–Eulerian method for hyperbolic systems and balance laws. This is a new method to deal with discontinuous flux and complicated source terms having concentrations for a wide range of applications science and engineering, namely, 1D shallow-water equations, sedimentation processes, Geophysical flows in 2D, N-Wave models, and Riccati-type problems with forcing terms. We also include numerical simulations of a 1D two-phase flow model in porous media, with gravity and a nontrivial singular \(\delta \)-source term representing an injection point. Moreover, we present approximate solutions for 2D nonlinear systems (Compressible Euler Flows and Shallow-Water Equations) for distinct benchmark configurations available in the literature aiming to present convincing and robust numerical results. In addition, for the linear advection model in 1D and for a smooth solution of the nonlinear Burgers’ problem, second order approximations were obtained. We also present a high-resolution approximation of the nonlinear non-convex Buckley–Leverett problem. Based on the work of A. Harten, we derive a convergent Lagrangian–Eulerian scheme that is total variation diminishing and second-order accurate, away from local extrema and discontinuous data. Additionally, using a suitable Kružkov’s entropy definition, introduced by K. H. Karlsen and J. D. Towers, we can verify that our improved Lagrangian–Eulerian scheme converges to the unique entropy solution for conservation laws with a discontinuous space-time dependent flux. A key hallmark of our method is the dynamic tracking forward of the no-flow curves, which are locally conservative and preserve the natural setting of weak entropic solutions related to hyperbolic problems that are not reversible systems in general. In the end, we have a general procedure to construct a class of Lagrangian–Eulerian schemes to deal with hyperbolic problems with or without forcing terms. The proposed scheme is free of Riemann problem solutions and no adaptive space-time discretizations are needed. The numerical experiments verify the efficiency and accuracy of our new Lagrangian–Eulerian method.

在这项工作中，我们为双曲系统和平衡定律开发了一种改进的震荡捕获和高分辨率拉格朗日-欧拉方法。这是一种处理不连续通量和具有复杂浓度的复杂源项的新方法，适用于广泛的应用科学和工程，即一维浅水方程，沉积过程，二维中的地球物理流，N波模型和Riccati-用强制术语键入问题。我们还包括在多孔介质中具有重力和非平凡奇异\（\ delta \）的一维两相流模型的数值模拟表示注入点的源术语。此外，我们提出了二维非线性系统（可压缩的欧拉流和浅水方​​程）的近似解决方案，用于文献中提供的各种基准配置，旨在提出令人信服的稳健数值结果。此外，对于一维线性对流模型和非线性Burgers问题的光滑解，获得了二阶近似值。我们还提出了非线性非凸Buckley-Leverett问题的高分辨率近似。根据A. Harten的工作，我们得出了收敛的Lagrangian-Eulerian方案，该方案总变量减小且二阶准确，并且远离局部极值和不连续数据。此外，使用由KH Karlsen和JD Towers引入的合适的Kružkov熵定义，我们可以验证我们改进的Lagrangian-Eulerian方案收敛于具有不连续时空相关通量的守恒律的唯一熵解。我们方法的关键特征是动态跟踪无流量曲线，这些曲线局部保守，并保留了与双曲问题有关的弱熵解的自然设置，而双​​曲问题通常不是可逆系统。最后，我们有一个通用程序来构造一类拉格朗日-欧拉式方案，以处理带或不带强迫项的双曲问题。所提出的方案没有黎曼问题解，并且不需要自适应时空离散化。数值实验验证了我们新的拉格朗日-欧拉方法的效率和准确性。