Development of Robust, Efficient and Highly Accurate Numerical Methods Based on Godunov-Type Central Schemes

现代科学技术的发展需要强大、高效和高精度的数值方法来求解含时偏微分方程，包括多维双曲守恒律、平衡律、对流反应扩散方程及其相关问题。近年来，人们提出了一族简单、通用、高分辨率的中心有限体积格式，作为对更复杂、更面向问题的迎风方法的一种有吸引力的替代方案，本项目的主要目标是将中心格式应用于各种多相多流体流动模型，即Saint-Venant浅水方程组（描述河流和海岸地区的水流），多层浅水方程（海洋学中出现），浅水中污染物输运模型，几种化学分类模型，反应流（特别是描述刚性爆轰波的模型）、旋转球体上的浅水方程、非均匀弹性、颗粒物质流、含尘气体模型（描述火山爆发）等。自然地，这些应用，特别是在高空间维度、复杂几何形状和移动边界/界面的情况下，需要开发和实现附加的数值技术，例如不同的自适应策略、与拉格朗日型方法的杂交、精确和有效的算子分裂、在原始偏微分方程系统中平衡的项之间的数值平衡中心格式已被证明是求解多维偏微分方程组的可靠和鲁棒的工具，这些方程组描述了流体力学、气体动力学、物理学、气象学、磁流体力学、天体物理学、多组分流、颗粒流、反应流、半导体、非牛顿流、几何光学、交通流、图像处理、金融和生物建模、微分博弈、最优控制以及许多其他领域。然而，在大多数实际应用中使用的模型比仅仅双曲守恒律系统更复杂，因此中心格式只能作为设计鲁棒、高效和高精度数值方法的基础。本项目旨在开发一系列的基本技术，这些技术对于将中心格式的适用性扩展到许多实际重要的问题是必不可少的，其中一些问题目前还无法实现，因为现有的数值方法要么效率太低/不准确，要么根本不适用。

Development of modern technology requires robust, efficient and highlyaccurate numerical methods for solving time-dependent partialdifferential equations, including multidimensional systems of hyperbolicconservation laws, balance laws, convection-reaction-diffusion equationsand related problems. A family of simple, universal and high-resolutionfinite-volume central schemes has been recently offered as an appealingalternative to more complicated and problem oriented upwind methods.The main goal of the project is applying the central schemes to variousmulti-phase and multi-fluid flow models, the Saint-Venant system ofshallow water equations (which describes flows in rivers and coastalareas), multi-layer shallow water equations (arising in oceanology),models of transport of pollutant in shallow water, several chemotaxismodels, reactive flows (in particular, the models describing stiffdetonation waves), shallow water equations on a rotating sphere,heterogeneous elasticity, granular material flows, dusty gas models(which describe volcanic eruptions), and others. Naturally, theseapplications, especially in the cases of high space dimensions, complexgeometries and moving boundaries/interfaces, require development andimplementation of additional numerical techniques such as differentadaption strategies, hybridization with Lagrangian-type methods,accurate and efficient operator splitting, numerical balancing betweenthe terms that are balanced in the original system of partialdifferential equations (development of well-balanced schemes), andothers that will be in the focus of the proposed research project.Central schemes have proved to be a reliable and robust tool for solvingmultidimensional systems of partial differential equations that describea variety of fundamental conservation laws in fluid mechanics, gasdynamics, geophysics, meteorology, magnetohydrodynamics, astrophysics,multi-component flows, granular flows, reactive flows, semiconductors,non-Newtonian flows, geometric optics, traffic flow, image processing,financial and biological modeling, differential games, optimal control,and many other areas. However, the models used in most practicalapplications are more complicated than just hyperbolic systems ofconservation laws, and therefore central schemes may only serve as abasis in designing robust, efficient and highly accurate numericalmethods. This project is aimed at developing a series of supplementarytechniques that are essential for the extension of applicability ofcentral schemes to many practically important problems, some of themare currently out of reach because the existing numerical methods areeither too inefficient/inaccurate or not applicable at all.