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数学大讲堂

Uniformly accurate numerical schemes for some atmospheric flows

Abstract
In typical atmospheric flows the flow velocity is much smaller than sound or gravity wave speeds. This yields the so-called weakly compressible flows that belong to the class of stiff problems.
In our talk we will present new IMplicite-EXplicite (IMEX) finite volume schemes for the Euler equations with a gravity source term that are based on the acoustic/advection splitting strategy. More precisely, we split the whole nonlinear system of the Euler equations into a stiff linear part governing fast acoustic and gravity waves and a non-stiff nonlinear part that models slow nonlinear advection effects.  For time discretization we have used higher order globally stiffly accurate IMEX schemes and approximate stiff linear operator implicitly and the non-stiff nonlinear operator explicitly.  Consequently, we can efficiently resolve slow nonlinear dynamics due to advection effects.
We have proved that the proposed schemes are uniformly stable and accurate with respect to the singular parameter, the Mach number. Theoretical results will be illustrated also by numerical experiments.

If time permits we will also report on new innovative concepts of  generalized solutions to the Euler equations of gas dynamics and K-convergence.


Bio

Professor Lukácová-Medvidová graduated from Charles University in Prague in 1994. From 1995-1998 she was an assistant professor at the University of Technology in Brno, Czech Republic  and from 1998-2002 she worked there as associated professor. She also hold a postdoctoral position at the University of Magdeburg.
From 2002 till 2010 she was Professor at the University of Technology in Hamburg, Germany and since 2010 she is Chair Professor for Applied Mathematics, Numerics of Partial Differential Equations at the University in Mainz, Germany. She was a visiting guest professor in Necas Center in Prague, CSCAMM in Maryland, Capital Normal University in Beijing, Indian Institute of Science in Bangalore, Olga Taussky-Pauli Fellow in Vienna and Simons Visiting Professor at the Academy of Sciences in Warsaw. 
In 2013 she was awarded by the Bronze Medal of the University of Kosice in Slovakia.
Prof. Lukácová-Medvidová’s research focuses on mathematical modeling and numerical simulations of complex flows. She deals with multidimensional hyperbolic conservation laws and multiscale problems arising in soft matters. She made contributions to the analysis and numerics of non-Newtonian, polymeric flows and fluid-structure interaction problems. Recently, she is developing innovative concepts of generalized solutions for the Euler equations of gas dynamics both from the analytical as well as numerical point of view.