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Numerical detection of complex singularities for functions of two or more variables. Presenter:Alexandr Virodov Additional Authors: Prof. Michael Siegel.

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Presentation on theme: "Numerical detection of complex singularities for functions of two or more variables. Presenter:Alexandr Virodov Additional Authors: Prof. Michael Siegel."— Presentation transcript:

1 Numerical detection of complex singularities for functions of two or more variables. Presenter:Alexandr Virodov Additional Authors: Prof. Michael Siegel Kamyar Malakuti Nan Maung

2 Outline Motivation 1D – Well known result 2D – Our generalization 2D – Application examples 3D – Theory and example

3 Motivation Kelvin-Helmholtz Instability

4 Motivation Rayleigh-Taylor instability

5 Theory – 1D C. Sulem, P.L. Sulem, and H. Frisch. Tracing complex singularities with spectral methods. J. of Comp. Phys., 50: , Asymptotic relation Im(x) Re(x)

6 Example – 1D Inviscid Burgers Equation

7 Theory – 2D For it can be shown that Im(x) Re(x)

8 Synthetic Data in 2D

9 Burgers Equation Traveling Wave solution

10 Burgers Equation I

11 Burgers Equation II

12 3 dimensions Most general form Again, it can be shown that

13 Synthetic Data in 3D

14 Further research Application of the method to 3D Burgers equation Application of the method to the Eulers equation Accuracy and stability of the method for specific cases

15 Questions? References: C. Sulem, P.L. Sulem, and H. Frisch. Tracing complex singularities with spectral methods. J. of Comp. Phys., 50: , K. Malakuti. Numerical detection of complex singularities in two and three dimensions S. Li, H. Li. Parallel AMR Code for Compressible MHD or HD Equations. M. Paperin. consult.de/CloudStructures/images/kelvin-helmholtz-instab/k-w-system.gif Brockmann Consult, Geesthacht, 2009.http://www.brockmann- consult.de/CloudStructures/images/kelvin-helmholtz-instab/k-w-system.gif


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