Partial Differential Equations with multiple time scales: Theory and Computational Techniques
Rupert Klein, Juliane Rosemeier
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Content:
Many problems in the sciences are determined by processes on several scales. Such problems are denoted as multi-scale problems. One example for a multi-scale problem is the partial differential equations (PDEs), which govern geophysical fluid flows. Averaging methods can be used for the analytical description of the slow scales. These descriptions are beneficial for numerical time-stepping schemes, as they allow for taking bigger time-steps than the unaveraged problem. The focus of this course will be on averaging methods for PDEs describing fluid flows and the design of parallelisable numerical time stepping methods, which are versions of the Parareal method and incorporate averaging techniques.
Requirements: basic courses in analysis, basic course numerical mathematics
Literature:
Wingate, B.A.; Rosemeier, J.; Haut, T., Mean Flow from Phase Averages in the 2D Boussinesq Equations. Atmosphere 2023, 14, 1523.
https://doi.org/10.3390/atmos14101523
T. Haut, B. Wingate, An asymptotic parallel-in-time method for highly oscillatory pde's, SIAM Journal on Scientific Computing, 36 (2014), pp. A693-A713
J.-L. Lions, G. Turinici, A "parareal" in time discretization of PDE's, Comptes Rendus de l'Academie des Sciences - Series I - Mathematics, 332 (2001), pp. 661-668
Sanders, F. Verhulst, J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, Springer New York, NY, 2ed., 2000
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