Mathematics Colloquium 10/21/22

Abstract:  The discipline of numerical weather prediction was formulated as a problem of mathematical physics by Richardson in the 1920s, although the first three-dimensional implementations did not appear until 50 years later. The corresponding models are built on the basis of a set of partial differential equations (PDE) for the rotating geophysical fluid in the external gravity field. From a mathematical point of view, it is convenient to separate the problem of spatial discretization of the underlying PDEs and their time integration. The first task is adequately solved while the second is still the subject of ongoing debates, as shown by the numerous articles published over the last decade.

The talk will address the derivation of exponential time integration schemes for weather models in spherical and Cartesian geometries. The stability of these schemes is ensured by the approximation of the oscillatory term using an absolutely stable exponential formula. Therefore, the schemes are not subject to the CFL condition imposed by the linear oscillatory part of the system and they also eliminate the phase errors associated with all known semi-implicit time stepping algorithms.

Theoretical considerations will be illustrated by the simulation of a large scale atmospheric flow on the sphere. It is remarkable that the proposed methods allow very significant increase of the length of a time step while increasing the accuracy of calculations. The same conclusion is achieved for the smaller scale flow down to the scale of the separate convective clouds.

Based on the evidence from the theoretical and experimental computer studies it is likely that the new time integration methods will lead to o a more accurate weather prediction model in the future.

Biosketch: Dr. Pudykiewicz holds a M.Sc. degree in Atmosheric Sciences from the University of Warsaw and a Ph.D. degree in Technical Sciences from the Polish Academy of Sciences. He worked as a research scientist in the Weather and Meteorology Research Division of Environment Canada from August 1982 to July 2022. His main activities in this position included the development of numerical models for weather prediction and environmental transport of atmospheric pollutants. After retiring from this position, he continues work on fundamental problems related to the predictability of geophysical systems.

For more information, please contact:  Dr. Vu Thai Luan (luan@math.msstate.edu).