Nonlinear transport equations model a variety of physical phenomena which often occur in real life. Examples of such phenomena are fluid mechanics, meteorology, reactive flows, multi-component flows, image processing, financial and biological modeling and others. Computing the solutions to these equations remains an important and challenging problem that requires development of fast, reliable and accurate numerical methods. At the same time, understanding the analytical properties of their solutions is not only useful for developing fast solvers, but remains imperative for real world applications.
On the analytic side, we propose to utilize techniques from approximation theory and Harmonic Analysis to prove analytical results for nonlinear equations and to use these techniques to develop numerical methods. The techniques to be employed include Littlewood-Paley theory, wavelet decompositions, maximal functions, interpolation and K-functionals. Emphasis will be placed on the understanding the relation between micro- and macroscopic models for transport, where a main vehicle in moving from the micro to the macro level are averaging lemmas. On the numerical side, the project aims to further develop the Godunov type central schemes for multidimensional systems of conservation laws and related problems. These schemes do not employ Riemann problem solvers and characteristic decomposition. Their high resolution and simplicity turns them into a universal tool for solving a wide range of problems. The application of these schemes to multi-phase and multi-fluid flow models, the Saint-Venant systems of shallow water equations, multi-layer shallow water systems, shallow water equations on a rotating sphere, granular material flows is another goal of the proposed research, which requires further development of central schemes and incorporation of various adaptive techniques into the central framework.
