Waves in fluids and solids

The project is concerted effort between research groups at NTNU, UiO, and UiB working in nonlinear partial differential equations. The aim is to use recent progress in the theory of partial differential equations to gain a better understanding of existing numerical methods as well as developing new methods. The work will be focused on waves in fluids, nonlinear wave equations of variational form, and high-precision time integration schemes. Concerning waves in fluids, a new aspect is that the project wil l take into account compressibility effects. Another issue to be treated is the effect of a free surface, giving special consideration to the interplay of surface tension and viscosity. For the numerical treatment of certain nonlinear wave equations, the variational structure of the problem will be exploited. In some cases the variational formulation can be used to formulate conservative numerical schemes. In other cases the problem contains a variational inequality that can be treated using special nume rical techniques, such as discontinuous Galerkin methods. For some tasks it is imperative to have numerical schemes able to perform long term integration. In particular, this applies to partial differential equations of Hamiltonian form, where symplectic schemes can be used. Here the focus will be on further development of such methods, building on recent advances in the theory of multi-symplectic schemes. The project will graduate 2 PhD candidates and will hire 2 postdoctoral fellows (one for one year a nd one for two years), and organize a workshop. The project will establish close colloboration with groups with similar aims at KTH, Uppsala and Chalmers in Sweden.