Adaptive large time step finite volume schemes for geophysical flows

A characteristic feature of many geophysical flows is their multiscale behaviour with wave speeds differing by orders of magnitude. If explicit time-discretization is used for numerical approximation to a governing system that supports multiscale waves, the maximum stable time step will be limited by wave speed of the most rapidly propagating waves. For geophysical flows this means that the waves that carry the least energy and are of little physical significance enforce a severe stability constraint. In order to obtain a reasonably efficient numerical model for simulation of geophysical flows (e.g. atmospheric circulation), it is necessary to circumvent the stability constraint associated with acoustic waves and put the stability limit into closer agreement with the time step limitations arising from accuracy considerations. The aim of the proposed project is a development of adaptive large time step finite volume evolution Galerkin (FVEG) methods for multiscale geophysical flows. These schemes are based on approximate evolution operators, derived from bicharacteristic theory of multidimensional hyperbolic balance laws. For a number of standard test problems conducted over several years, these schemes have proven to be particularly accurate and efficient. The proposal intends to develop the schemes further to make them suitable for a range of geophysical and meteorological applications in two and three space dimensions. The key techniques to be developed during the one-year project are large time step discretizations using both semi-implicit as well as explicit time approximations. Problems arising from the special features of geophysical/meteorological flows will be solved in the cooperation with Prof. Dr. Volkmar Wirth, Institute for Atmospheric Physics, University of Mainz.