Geometric, multiscale, and stabilized numerical methods for dissipative and highly oscillatory problems

Differential equations are a powerful tool in modeling real life phenomena, and have important applications in Physics, Chemistry, Biology, Medicine, economy, and other domains. This research project aims to construct numerical schemes for different types of problems involving differential equations. The construction of such schemes is an adaptive process that takes into account the nature and the characteristics of the problem under study. For stiff problems, one might need to design multiscale and/or stabilized methods, while for problems with special geometric structure to be preserved, some attention will be given to structure-preserving (or geometric) integrators.The overall objectives are to design new numerical methods for some new classes of problems in the areas of deterministic, possibly kinetic, and stochastic (partial) differential equations (both dissipative and oscillatory), as well as in the domain of optimal control. Another objective is to combine some existing methods (using a splitting approach for example) in order to solve some interesting equations (such as Vlasov-Fokker-Planck equation in plasma physics) more efficiently with respect to accuracy and computational cost. The nature of the new schemes depends on the type of the problem. Each of these approaches has its own challenges and difficulties.The mentioned mathematical problems have plenty of applications in many domains as mentioned at the beginning, including diffusion phenomena, wave propagation, fluid dynamics, molecular dynamics, aerodynamics, stock prices, etc. Deriving the desired numerical integrators will rely on many mathematical approaches and tools to come up with robust and efficient schemes with accuracy and computational cost are as independent as possible of any variable parameter in the problem of interest. All the obtained results will be published in high rank international peer-reviewed journals of numerical analysis and related fields.Achieving the desired objectives will have a great positive impact on the field for many reasons. On the one hand, the accuracy of the solutions of many very applied equations will be improved, and on the other hand, the computational time of such solutions will be decreased by significant factors. Moreover, the new methods will be designed in a way that makes them easy to implement, and the corresponding software will be made publicly available.