High-order approximation techniques for nonlinear hyperbolic PDEs

许多应用是基于非线性偏微分方程组，其中稳定性不是能量估计的结果。这是在非线性守恒律、对流或多相流以及自由边界问题中的情况，在这些问题中，激波锋面和不连续性是重要的特征，给数值方法带来了很大的困难。这些问题的自然背景涉及到熵的物理概念，并需要质量、温度或密度等量的正性。研究人员建议继续开发一种新的非线性逼近技术来求解上述类型的微分方程。这种新的方法包括计算所谓的熵残差，并使用它来设计对手头问题的Galerkin公式的稳定化机制。这是一个与标准稳定技术不同的观点。研究人员建议在热力学第二原理的基础上设计一种非线性粘性，同时考虑相关物理量的正性/有界性。尽管非线性算法比较复杂和难以分析，但在处理粗糙解、复杂几何和强非线性时，它们会带来巨大的好处。在过去的几十年里，人们致力于发展稳健的数值方法来模拟非线性现象。在许多领域已经取得了重大进展，但目前的技术水平远远不能提供复杂物理过程的准确和忠实的数字表示。例如，界面、锋面和激波形成的精确近似仍然是一个巨大的挑战。拟议中的项目在许多领域产生了广泛的影响。在机械和航天工程中，该方法改进了用于模拟高速气体动力学、非线性弹性和相变问题的数值模型。在石油工程中，这套新的方法有利于更准确地模拟复杂几何油藏中的多相流。此外，该项目还将在地球物理、纳米技术和环境问题等其他领域产生重大影响，这些领域需要可靠的模拟来解决冲击、尖锐界面和其他非线性现象。

Many applications are based on nonlinear partial differential equations in which stability is not a result of an energy estimate. This is the case in nonlinear conservation laws, convection-dominated or multiphase flows, and free-boundary problems, where shocks fronts and discontinuities are important features and pose significant difficulties for numerical methods. The natural setting for these problems involves the physical notion of entropy and requires the positivity of quantities like mass, temperature or density. The investigators propose to continue the development of a new nonlinear approximation technique for solving the above class of differential equations. This new approach consists of computing the so-called entropy residual and use it to design a stabilization mechanism to the Galerkin formulation of the problem at hand. This is a different point of view than that of standard stabilization techniques. The investigators propose to design a nonlinear viscosity based on the second principle of thermodynamics and respect positivity/boundedness of the relevant quantities at the same time. Even though the nonlinear algorithms are more complicated and difficult to analyze, they yield great benefits when working with rough solutions, complicated geometry, and strong nonlinearities. In the past several decades, a large amount of work has been dedicated to the development of robust numerical methods modeling nonlinear phenomena. Significant advances have been made in many areas, but the current state of the art is far from providing accurate and faithful numerical representations of the complex physical processes. For instance, accurate approximation of interfaces, sharp fronts, and shock formations is still an enormous challenge. The proposed project has a broad impact in many fields. In mechanical and aerospace engineering, the proposed method improves on numerical models for simulating high velocity gas dynamics, nonlinear elasticity and phase transition problems. In petroleum engineering the new set of methods is beneficial for more accurate simulation of multiphase flows in reservoirs with complicated geometry. Moreover, the project will also have significant impact in other fields such as geophysics, nanotechnology, and environmental problems where reliable simulations for resolving shocks, sharp interfaces, and other nonlinear phenomena are needed.