Innovative Numerical Methods for Nonlinear Time-Dependent PDEs

该项目的目标是开发精确、高效和稳健的非线性偏微分方程组的数值方法，特别是考虑允许非光滑(不连续)解的问题和涉及高度不同尺度的问题，因此很难用数值方法求解。拟议研究的主要部分将集中于开发新的技术来解决涉及复杂的非线性波动现象的问题，具有复杂计算区域的问题，移动边界和材料/层界面的问题，以及包括不确定现象的问题。新技术将基于粒子法和有限体积法，以及它们的杂交。后一种方法将利用粒子方法的主要优点，如无网格方法和激波捕捉有限体积方法，特别是在复杂几何、自由边界和具有结构相互作用的流动问题中。还将结合使用随机和数值工具来解决具有不确定性的问题和问题，其中应考虑多个尺度。所设计的方法将应用于各种非线性问题，其中包括Euler-Poincare方程、多相和多流体流动模型、不可压缩湍流中的污染物输运模型、趋化性模型、描述刚性爆轰波的反应性Euler方程、守恒律的零扩散-弥散极限等。随机初值问题，如随机摄动的KdV方程和具有随机力的Burgers型方程也可以用所提出的方法求解。重要的是，除了提供证实分析方法的例子外，上述应用对于现代科学中出现的广泛类别的问题具有实质性的独立价值。近年来，求解偏微分方程组的数值方法已经发展成为一种重要且极其有效的工具，用于定量和定性地研究不同应用领域中的许多现象，否则根本无法研究这些现象。拟议的项目将对计算方法的发展做出重大贡献，并将提供更强大的工具来分析计算机上的应用问题。在这一建议中，着重于设计复杂问题的数值方法，如多相和多流体模型、污染传播模型、聚合物体系、趋化性模型、主动流体传输模型、多尺度和随机初值问题等，这些问题出现在流体和气体动力学、地球物理、气象、天体物理、多组分流、颗粒流、反应流、聚合物流等领域的各种科学应用中。所研究方法的广泛应用也反映了拟议项目的跨学科特点。

The project is aimed at developing accurate, efficient, and robust numerical methods for nonlinear PDEs, with particular reference to problems that admit nonsmooth (discontinuous) solutions and problems that involve highly disparate scales, and therefore, are difficult to solve numerically. The principal part of the proposed research will be focused on the development of new techniques for solving problems involving complicated nonlinear wave phenomena, problems with complex computational domains, moving boundaries and material/layer interfaces, as well as problems that include uncertain phenomena. The new techniques will be based on particle methodsand finite-volume methods, as well as their hybridization. The latter approach will utilize major advantages of particle methods, as mesh-free methods, and shock-capturing finite-volume methods, especially in problems with complex geometries, free boundaries, and flows with structural interactions. A combination of stochastic and numerical tools will also be used for solving problems with uncertainties and problems, in which multiple scales should be taken into account. The designed methods will be applied to a variety of nonlinear problems, among which are the Euler-Poincare equations, multi-phase and multi-fluid flow models, models of transport of pollutant in turbulent incompressible flow, chemotaxis models, reactive Euler equations describing stiff detonation waves, zero diffusion-dispersion limits for conservation laws, and others. Stochastic initial-value problems such as the randomly perturbed KdV equation and the Burgers equation with random force will also be solved by the proposed methods. It is significant that, besides providing the examples that corroborate the analytical approach, the foregoing applications are of a substantial independent value for a broad class of problems arisingin modern science. In recent years, numerical methods for solving partial differential equations have evolved into an important and extremely efficient tool for the quantitative and qualitative study of many phenomena in different applied ares that otherwise could not have been studied at all. The proposed project will contribute significantly toward development of computational methods and will provide considerably more powerful tools for analyzing applied problems on the computer. In this proposal, a strong accent is put on designing numerical methods for complicated problems such as multi-phase and multi-fluid models, models of pollution propagation, polymer systems, chemotaxis models, active fluid transport models, multi-scale and stochastic initial-value problems, etc. These problems arise in a variety of scientific applicationsin fluid and gas dynamics, geophysics, meteorology, astrophysics, multi-component flows, granular flows, reactive flows, polymer flows, and other fields. A wide spectrum of applications of the studied methods reflects also the interdisciplinary character of the proposed project.