Collaborative Research: Development of High-Resolution Finite-Volume Methods for Systems of Nonlinear Time-Dependent PDEs

The project is aimed at developing highly accurate, efficient and robust numerical methods for systems of nonlinear time-dependent PDEs, with particular reference to multidimensional hyperbolic systems of conservation/balance laws and related problems. The principal part of the proposed research will be focused on the development of new finite-volume methods that will provide an improved resolution of linear contact waves and incorporate new techniques for solving problems involving complicated nonlinear wave phenomena and blowing up/spiky solutions. The proposed methods will be applied to a variety of nonlinear problems, among which are systems of gas dynamics, nonlinear elasticity and acoustics systems, modern traffic flow models, several chemotaxis and bioconvection models, and others. These problems will be studied in the most challenging cases of high space dimensions, complex geometries and moving interfaces. For each problem, a high-resolution finite-volume scheme will be systematically derived in a way that the main properties satisfied by the underlying system of PDEs will be also satisfied on the discrete level. One of the key features of the new schemes will be their nonlinear stability, which will be ensured by ability of the scheme to preserve positivity of such physical quantities as density. To achieve this goal, several high-order positivity preserving techniques will be explored.

Besides providing the examples that corroborate the analytical approach, the foregoing applications are of a substantial independent value for a broad class of problems arising in today's science including geophysics, meteorology, astrophysics, semiconductors, traffic flows, image processing, financial and biological modeling and many other areas. Development of modern high-resolution finite-volume methods as well as of supplementary techniques is essential for solving many practically important problems, some of which are currently out of reach because the existing numerical methods are either inefficient/inaccurate or not applicable at all.