Development of modern technology requires robust, efficient and highly accurate numerical methods for solving time-dependent partial differential equations, including multidimensional systems of hyperbolic conservation laws, balance laws, convection-reaction-diffusion equations and related problems. A family of simple, universal and high-resolution finite-volume central schemes has been recently offered as an appealing alternative to more complicated and problem oriented upwind methods. The main goal of the project is applying the central schemes to various multi-phase and multi-fluid flow models, the Saint-Venant system of shallow water equations (which describes flows in rivers and coastal areas), multi-layer shallow water equations (arising in oceanology), models of transport of pollutant in shallow water, several chemotaxis models, reactive flows (in particular, the models describing stiff detonation waves), shallow water equations on a rotating sphere, heterogeneous elasticity, granular material flows, dusty gas models (which describe volcanic eruptions), and others. Naturally, these applications, especially in the cases of high space dimensions, complex geometries and moving boundaries/interfaces, require development and implementation of additional numerical techniques such as different adaption strategies, hybridization with Lagrangian-type methods, accurate and efficient operator splitting, numerical balancing between the terms that are balanced in the original system of partial differential equations (development of well-balanced schemes), and others that will be in the focus of the proposed research project.
Central schemes have proved to be a reliable and robust tool for solving multidimensional systems of partial differential equations that describe a variety of fundamental conservation laws in fluid mechanics, gas dynamics, 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 practical applications are more complicated than just hyperbolic systems of conservation laws, and therefore central schemes may only serve as a basis in designing robust, efficient and highly accurate numerical methods. This project is aimed at developing a series of supplementary techniques that are essential for the extension of applicability of central schemes to many practically important problems, some of them are currently out of reach because the existing numerical methods are either too inefficient/inaccurate or not applicable at all.
