High-order approximation techniques for nonlinear hyperbolic PDEs

许多应用是基于非线性偏微分方程，其中稳定性不是能量估计的结果。这是在非线性守恒律，对流为主或多相流，和自由边界问题的情况下，激波阵面和不连续性是重要的功能，并提出了显着的困难，数值方法。这些问题的自然背景涉及熵的物理概念，并需要质量、温度或密度等量的正性。研究人员建议继续发展一种新的非线性近似技术来求解上述一类微分方程。这种新方法包括计算所谓的熵残差，并使用它来设计一个稳定机制的Galerkin制定的问题。这是与标准稳定技术不同的观点。研究人员建议根据热力学第二原理设计非线性粘度，同时考虑相关量的正性/有界性。 尽管非线性算法更复杂，更难以分析，但在处理粗糙解、复杂几何和强非线性时，它们会产生很大的好处。在过去的几十年里，大量的工作一直致力于发展强大的数值方法来模拟非线性现象。 在许多领域已经取得了显着的进步，但目前的技术水平是远远不能提供准确和忠实的数值表示的复杂的物理过程。例如，界面、尖锋和激波形成的精确近似仍然是一个巨大的挑战。拟议的项目在许多领域具有广泛的影响。在机械和航空航天工程中，所提出的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精确地模拟复杂几何形状油藏中的多相流。 此外，该项目还将对其他领域产生重大影响，如电子物理学，纳米技术和环境问题，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.