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

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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.