Finite Volume Schemes for the Shallow Water Equations on the Sphere

Short Description

The simulation of global atmospheric flows (e. g. weather forecast, propagation of tsunamis) on the rotating Earth can be addressed by solving the shallow water equations on the sphere.


Numerical results

In order to obtain a numerical solution a well-balanced first order Finite Volume scheme for surfaces was designed. The following movie shows the first 15 hours of the propagation of tsunami waves caused by an asteroid impact in the middle of the Atlantic Ocean. The overall time maximum of the wave height above sea level is colored, where red depicts a wave height of at least 10m, and blue depicts 0m. The initial values assume an impactor radius of 500m, velocity of 20km/s and density of 3gm/cm^3 (initial values taken from [Ward, Asphaug 00]).

Shallow Water Equations on the Sphere Partitioning of the sphere

© 2011 Thomas Mueller

The simulation was performed on an unstructured surface grid with more than 3 million elements in a parallel run on 16 cores using DUNE-FEM. The picture shows the partitioning of the sphere.



This project is supported by the Sonderforschungsbereich / Transregio 71 "Geometric Partial Differential Equations" Sonderforschungsbereich / Transregio 71 Geometric Partial Differential Equations



  • T. Müller and A. Pfeiffer. Well-balanced simulation of geophysical flows via the Shallow Water Equations with bottom topography: Consistency and Numerical Computation. to appear in: Proceedings of Hyp2012 - the 14th International Conference on Hyperbolic Problems held in Padova, Italy.
  • T. Müller and D. Kröner. Related Problems for TV-Estimates for Conservation Laws on Surfaces.
    Proc. of the 13th International Conference on Hyperbolic Problems: Theory, Numerics, Applications, Beijing, China, 2010. In: Tatsien Li; Song Jiang (eds.) Hyperbolic Problems: Theory, Numerics and Applications (2), 584--592, Ser. Contemp. Appl. Math. CAM 18, Higher Ed. Press, Beijing 2012.
  • S.N. Ward, E. Asphaug: "Asteroid Impact Tsunami: a Probalistic Hazard Assessment", Icarus, 145(1), 64--78 (2010)
  • Amante, C. and B. W. Eakins, ETOPO1 1 Arc-Minute Global Relief Model: Procedures, Data Sources and Analysis. NOAA Technical Memorandum NESDIS NGDC-24, 19 pp, March 2009.
  • A. Dedner, R. Klöfkorn, M. Nolte, M. Ohlberger. A generic interface for parallel and adaptive scientific computing: Abstraction principles and the DUNE-FEM module. Computing Vol. 90, No. 3, pp. 165--196, 2011