Innovative Numerical Methods for Nonlinear Time-Dependent PDEs

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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.