reponsible: Stefan Jebens
Duration: 1 July 2007 – 30 June 2009
The main problem which arises from the integration of the compressible Euler equations are the acoustic modes. Because they are faster than the physically interesting modes they restrict the maximal time step size due to the CFL condition. Common strategies to avoid this problem are the use of split-explicit methods where the differential equation is split and the acoustic modes are integrated with a smaller time step size and the use of semi-implicit methods. For the spatial discretization finite volume methods have proven to be a good choice because they allow an easy implementation of orography and conserving methods.
Oswald Knoth has developed the All-Scale Atmospheric Model ASAM at the IfT. It uses finite volumes with cut cells for the representation of orography because terrain-following coordinates point out to be inaccurate in some situations. For the time discretization a Rosenbrock method of order 3 is implemented. Because of the semi-impliciteness of such a method the appearance of small cells due to the use of cut cells is no problem.
Furthermore the implementation of explicit peer methods to the compressible Euler equations was investigated in line with the Metström project. Peer methods are general linear methods with the same order in every stage and therefore they do not suffer from order reduction for stiff problems as methods with low stage order like Runge-Kutta methods do. Another advantage is the fact that they need no artificial damping term to run stable in opposite to other widely used methods.
Further work will done on the development of implicit peer methods and the improvement of the solvers for the linear systems of equations which arise from the use of semi-implicit methods. Implicit peer methods have been applied to many other problems, e.g. shallow water equations, and might be an attractive alternative to existing implicit solvers. ASAM uses domain decomposition in all 3 dimensions for parallelization and adaptive grids to produce more accurate solutions in regions of interest, e.g. in the near of mountains. But the fine-course interfaces cause problems for the iterative solver especially when the grid is partitioned in vertical direction which results in an increasing number of iterations to solve the linear systems of equations with a Krylov technique. Therefore the development of a more sophisticated parallel preconditioner is necessary for an efficient use of implicit methods together with a block adaptive mesh refinement algorithm.