The aim of the proposed project is further development and analysis of adaptive large timestep finite volume evolution Galerkin (FVEG) methods for two- and three-dimensional systems of hyperbolic balance laws. These particularly accurate and efficient schemes are based on exact and approximate evolution operators, derived from bicharacteristic theory of hyperbolic conservation laws. The goal is to make the FVEG schemes
suitable for a range of practical multi-dimensional applications in hydraulic, geophysics and meteorology. A key step towards efficiency is to build a large timestep algorithm, where the CFL condition is dictated entirely by the flow velocities, not by the fast gravitational or acoustic waves. While first steps towards this goal have been made both for macro/micro explicit timesteps and macro implicit timesteps, a main theoretical challenge is to design macro/macro explicit timesteps. For this, we intend to develop a multiscale analysis which is is based upon the splitting of fast and slow waves in the wave cones of the EG operator. Essential for stability are also the following applied mathematics / mathematical modelling issues: anisotropic wave cones (due variable topography); multidimensional open boundary conditions; stable treatment of dry states; well-balancing of convective and gravitational forces. Numerical analysis and computer science issues decisive for efficiency are: adaptivity and error control; data structures in 3D. These issues require a thorough analysis of this new generation of FVEG schemes.