Bergische Universität Wuppertal Fachbereich Mathematik und Naturwissenschaften Angewandte Mathematik - Numerische Analysis (AMNA)

People Research Publications Teaching

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Summary

The goal of our research project is the study of the iterative parabolic approximations for the nonlinear Helmholtz equation (NHE). This study is a natural continuation of the previous joint work where the iterative parabolic approximations for the linear Helmholtz equation were studied and the transparent boundary conditions (TBCs) for the coupled system of the parabolic equations (PEs) were developed.

This research study will be a natural combination of the ideas emerging from the recent papers of the Vladivostok research group and by M. Ehrhardt and significantly improved in his follow-up works. This combination of ideas can be successfully developed from the close collaboration in the framework of the proposed project.

Scientific Objectives

• derive the iterative high-order parabolic equations for the NHE (outlined in the previous section, general case of the inhomogeneous medium requires some additional work)
• derive the initial conditions for the iterative PE system; derive the interface conditions at media boundaries
• using simple analytical solutions of the NHE to check, whether the nonparaxial effects in these solutions can be caught using the iterative PEs
• study the problem of the energy conservation in the case of the NHE equation and check, whether the energy conservation properties are maintained (at least asymptotically) by the nonlinear iterative PEs
• develop an efficient numerical technique for the solution of the iterative PEs, e.g. a generalization of the Crank-Nicholson finite-difference scheme or an ETD-type scheme
• validate the developed numerical scheme using the analytical solutions of the NHE for the comparison
• illustrate the application of the developed numerical scheme on some physically meaningful examples

German team:

• Matthias Ehrhardt (Leader)

Russian team:

• Pavel Petrov (Leader)
• M. Yu Trofimov
• A.D. Zakharenko

German institution:

• Department of Mathematics and Informatics, Faculty of Mathematics and Natural Sciences, University of Wuppertal.

Russian institution:

• V.I. Il'ichev Pacific Oceanological Institute, Far Eastern Federal University, Vladivostok.

Publications related to the Project

[PeEh15a]	P. Petrov, M. Ehrhardt,
	On Mayfield's stability proof for the discretized transparent boundary condition for the parabolic equation, accepted: Applied Mathematics Letters, 2015
[PeEh15b]	P. Petrov, M. Ehrhardt,
	Transparent boundary conditions for the hierarchies of high-order parabolic approximations , J. Comput. Phys. Vol. 313 (May 2016), 144-158.

Talks related to the Project

• P.S. Petrov, O.S. Zaikin, High-performance computing in the study on the accuracy of the bottom parameters reconstruction using two different fitness functions, UACE2017 - international conference and exhibition on Underwater Acoustics, Koukounariés, Skiathos island, Greece, September 3-8, 2017.
• S.A. Sergeev, A.A. Tolchennikov, P.S. Petrov, Application of the Maslov canonical operator technique to the development of a ray-based time-domain sound propagation model for the deep-water acoustics, UACE2017 - international conference and exhibition on Underwater Acoustics, Koukounariés, Skiathos island, Greece, September 3-8, 2017.
• B.G. Katsnelson, P.S. Petrov, Whispering gallery waves near the curved isobaths in shallow water, UACE2017 - international conference and exhibition on Underwater Acoustics, Koukounariés, Skiathos island, Greece, September 3-8, 2017.

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