Collaborative Research: Development of High-Resolution Finite-Volume Methods for Systems of Nonlinear Time-Dependent PDEs

该项目旨在为非线性时变偏微分方程系统开发高精度、高效和稳健的数值方法，特别是涉及多维双曲守恒/平衡律系统和相关问题。拟议的研究的主要部分将集中在新的有限体积方法的发展，将提供一个线性接触波的分辨率提高，并纳入新的技术来解决问题，涉及复杂的非线性波现象和爆破/尖峰的解决方案。所提出的方法将被应用到各种非线性问题，其中包括系统的气体动力学，非线性弹性和声学系统，现代交通流模型，几个趋化性和生物对流模型，以及其他。这些问题将在最具挑战性的高空间维度，复杂的几何形状和移动接口的情况下进行研究。对于每个问题，高分辨率的有限体积计划将系统地推导出的方式，基本系统的偏微分方程所满足的主要属性也将满足离散水平。新方案的关键特征之一将是它们的非线性稳定性，这将通过方案保持密度等物理量的正性的能力来确保。为了实现这一目标，几个高阶正性保持技术将explored. Although提供的例子，证实了分析方法，上述应用程序是一个实质性的独立的价值，在今天的科学，包括天体物理学，气象学，天体物理学，半导体，交通流量，图像处理，金融和生物建模和许多其他领域中出现的广泛的一类问题。现代高分辨率有限体积法以及补充技术的发展对于解决许多实际重要的问题是必不可少的，其中一些是目前无法实现的，因为现有的数值方法要么效率低/不准确，要么根本不适用。

The project is aimed at developing highly accurate, efficient and robust numerical methods for systems of nonlinear time-dependent PDEs, with particular reference to multidimensional hyperbolic systems of conservation/balance laws and related problems. The principal part of the proposed research will be focused on the development of new finite-volume methods that will provide an improved resolution of linear contact waves and incorporate new techniques for solving problems involving complicated nonlinear wave phenomena and blowing up/spiky solutions. The proposed methods will be applied to a variety of nonlinear problems, among which are systems of gas dynamics, nonlinear elasticity and acoustics systems, modern traffic flow models, several chemotaxis and bioconvection models, and others. These problems will be studied in the most challenging cases of high space dimensions, complex geometries and moving interfaces. For each problem, a high-resolution finite-volume scheme will be systematically derived in a way that the main properties satisfied by the underlying system of PDEs will be also satisfied on the discrete level. One of the key features of the new schemes will be their nonlinear stability, which will be ensured by ability of the scheme to preserve positivity of such physical quantities as density. To achieve this goal, several high-order positivity preserving techniques will be explored.Besides providing the examples that corroborate the analytical approach, the foregoing applications are of a substantial independent value for a broad class of problems arising in today's science including geophysics, meteorology, astrophysics, semiconductors, traffic flows, image processing, financial and biological modeling and many other areas. Development of modern high-resolution finite-volume methods as well as of supplementary techniques is essential for solving many practically important problems, some of which are currently out of reach because the existing numerical methods are either inefficient/inaccurate or not applicable at all.