This work presents a suitable mathematical analysis to understand the properties of convergence and bounded variation of a new { fully discrete locally conservative} Lagrangian--Eulerian {explicit} numerical scheme to the entropy solution in the sense of Kruzhkov via weak asymptotic method. We also make use of the weak asymptotic method to connect the theoretical developments with the computational approach within the practical framework of a solid numerical analysis. This method also serves to address the issue of notions of solutions, and its resulting algorithms have been proven to be effective to study nonlinear wave formations and rarefaction interactions in intricate applications. The weak asymptotic solutions we compute in this study with our novel Lagrangian--Eulerian framework are shown to coincide with classical solutions and Kruzhkov entropy solutions in the scalar case. Moreover, we present and discuss significant computational aspects by means of numerical experiments related to nontrivial problems: a nonlocal traffic model, the $2 \times 2$ symmetric Keyfitz--Kranzer system, and numerical studies via Wasserstein distance to explain shock interaction with the fundamental inviscid Burgers' model for fluids. Therefore, the proposed weak asymptotic analysis, when applied to the Lagrangian--Eulerian framework, fits in properly with the classical theory while optimizing the mathematical computations for the construction of new accurate numerical schemes.
翻译:这项工作提供了一种适当的数学分析, 以了解新的 { 完全离散的当地保守 } Lagrangian- Eulelian {explit} 的趋同性和约束性变异性, 以了解新的 { 完全离散的当地保守} Lagrangian- Eulelian {explit} 数字制成的特性。 我们还利用弱弱的无线化方法, 将理论发展与计算方法联系起来, 在实实在在的数值分析的实用框架内, 这个方法还有助于解决解决方案的概念问题, 其产生的算法已证明有效地研究非线性波形成和复杂应用中稀有的相互作用。 我们在本项研究中与我们的新Lagrangian- Eulerian 的Lagrangang- Eulerian 框架拼写得软弱的无线性解决方案。 此外, 我们介绍和讨论重要的计算方面, 通过与非三角问题相关的数字实验: 非本地交通模式, 其结果计算方法的2\ time 2 $ 2, 用于研究非线性 Keyfitz- kranzervizer 和 的稀变精度 系统 。