Mathematics of Planet Earth

In this talk an overview is given on multirate time integration methods developed over the years for the integration of the compressible Euler equation and advection-diffusion-reaction equations. Physical and chemical processes in the atmosphere occur on different time scales ranging from seconds to hours and days. Let us mention waves of different speeds in the atmosphere, turbulent diffusion, condensation of water vapor, temporal emission patterns, or photo chemistry. Further time scales come in to play due to the shallow nature of the atmosphere and the huge computational area in case of simulations of the whole globe. To handle these problems anisotropic grids with different grid sizes are used in the vertical and horizontal direction and additional local grid refinement is applied in the horizontal direction. In our solution strategy we follow the method of lines, first discretize in space and then solve in a second stage a huge system of ordinary differential equations (ODE) in time. Most of the proposed time integration methods originate from the idea of source splitting and the recursive use of Runge-Kutta like methods. At each stage of these Runge-Kutta methods again an ODE has to be solved but now only a part of the original right hand side is taken into account whereas the rest act as an additional constant source term. This idea can than be applied again to the subproblems. For these methods order conditions are derived and different strategies for determining reliable schemes with good stability properties will be explained. Two special applications are multirate methods for the advection equation on locally refined grids and generalized split-explicit methods for the compressible Euler equation.