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

该项目的目的是为非线性时变偏微分方程系统发展高度精确、有效和可靠的数值方法，特别是关于守恒/平衡定律的多维双曲系统和有关问题。所提出的研究的主要部分将集中在发展新的有限体积方法上，这些方法将提供线性接触波的改进分辨率，并纳入解决涉及复杂非线性波现象和爆破/尖解的问题的新技术。所提出的方法将应用于各种非线性问题，其中包括气体动力学系统、非线性弹性和声学系统、现代交通流模型、几种趋化性和生物对流模型等。这些问题将在高空间维度，复杂几何形状和移动界面的最具挑战性的情况下进行研究。对于每个问题，将系统地推导出一个高分辨率有限体积方案，该方案的基本性质将在离散水平上得到满足。新格式的关键特征之一将是它们的非线性稳定性，这将通过格式保持密度等物理量的正性的能力来保证。为了实现这一目标，我们将探索几种高阶正性保持技术。除了提供证实分析方法的例子外，上述应用对于当今科学中出现的广泛问题具有重要的独立价值，这些问题包括地球物理学、气象学、天体物理学、半导体、交通流量、图像处理、金融和生物建模以及许多其他领域。现代高分辨率有限体积方法及其辅助技术的发展对于解决许多实际重要问题是必不可少的，其中一些问题目前无法实现，因为现有的数值方法要么效率低下/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.