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| Titel |
Variational methods for direct/inverse problems of atmospheric dynamics and chemistry |
| VerfasserIn |
Vladimir Penenko, Alexey Penenko, Elena Tsvetova |
| Konferenz |
EGU General Assembly 2013
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 15 (2013) |
| Datensatznummer |
250076047
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| Zusammenfassung |
| We present a variational approach for solving direct and inverse problems of atmospheric
hydrodynamics and chemistry. It is important that the accurate matching of numerical
schemes has to be provided in the chain of objects: direct/adjoint problems – sensitivity
relations – inverse problems, including assimilation of all available measurement data.
To solve the problems we have developed a new enhanced set of cost-effective
algorithms.
The matched description of the multi-scale processes is provided by a specific choice of
the variational principle functionals for the whole set of integrated models. Then all
functionals of variational principle are approximated in space and time by splitting and
decomposition methods.
Such approach allows us to separately consider, for example, the space-time problems of
atmospheric chemistry in the frames of decomposition schemes for the integral
identity sum analogs of the variational principle at each time step and in each of 3D
finite-volumes. To enhance the realization efficiency, the set of chemical reactions is
divided on the subsets related to the operators of production and destruction. Then the
idea of the Euler’s integrating factors is applied in the frames of the local adjoint
problem technique [1]-[3]. The analytical solutions of such adjoint problems play
the role of integrating factors for differential equations describing atmospheric
chemistry. With their help, the system of differential equations is transformed to the
equivalent system of integral equations. As a result we avoid the construction and
inversion of preconditioning operators containing the Jacobi matrixes which arise in
traditional implicit schemes for ODE solution. This is the main advantage of our
schemes.
At the same time step but on the different stages of the “global” splitting scheme, the
system of atmospheric dynamic equations is solved. For convection – diffusion equations for
all state functions in the integrated models we have developed the monotone and stable
discrete-analytical numerical schemes [1]-[3] conserving the positivity of the chemical
substance concentrations and possessing the properties of energy and mass balance that are
postulated in the general variational principle for integrated models. All algorithms for
solution of transport, diffusion and transformation problems are direct (without
iterations).
The work is partially supported by the Programs No 4 of Presidium RAS and No 3 of
Mathematical Department of RAS, by RFBR project 11-01-00187 and Integrating projects of
SD RAS No 8 and 35. Our studies are in the line with the goals of COST Action
ES1004.
References
Penenko V., Tsvetova E. Discrete-analytical methods for the implementation of
variational principles in environmental applications// Journal of computational
and applied mathematics, 2009, v. 226, 319-330.
Penenko A.V. Discrete-analytic schemes for solving an inverse coefficient heat
conduction problem in a layered medium with gradient methods// Numerical
Analysis and Applications, 2012, V. 5, pp 326-341.
V. Penenko, E. Tsvetova. Variational methods for constructing the monotone
approximations for atmospheric chemistry models //Numerical Analysis and
Applications, 2013 (in press). |
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