Numerical detection of complex singularities for functions of two or more variables....
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Transcript of Numerical detection of complex singularities for functions of two or more variables....
![Page 1: Numerical detection of complex singularities for functions of two or more variables. Presenter:Alexandr Virodov Additional Authors: Prof. Michael Siegel.](https://reader036.fdocuments.in/reader036/viewer/2022062404/5519d1085503468b0c8b47ee/html5/thumbnails/1.jpg)
Numerical detection of complex singularities for functions of two or more variables.
Presenter:Alexandr Virodov
Additional Authors:Prof. Michael SiegelKamyar MalakutiNan Maung
![Page 2: Numerical detection of complex singularities for functions of two or more variables. Presenter:Alexandr Virodov Additional Authors: Prof. Michael Siegel.](https://reader036.fdocuments.in/reader036/viewer/2022062404/5519d1085503468b0c8b47ee/html5/thumbnails/2.jpg)
Outline Motivation 1D – Well known result 2D – Our generalization 2D – Application examples 3D – Theory and example
![Page 3: Numerical detection of complex singularities for functions of two or more variables. Presenter:Alexandr Virodov Additional Authors: Prof. Michael Siegel.](https://reader036.fdocuments.in/reader036/viewer/2022062404/5519d1085503468b0c8b47ee/html5/thumbnails/3.jpg)
Motivation Kelvin-Helmholtz Instability
![Page 4: Numerical detection of complex singularities for functions of two or more variables. Presenter:Alexandr Virodov Additional Authors: Prof. Michael Siegel.](https://reader036.fdocuments.in/reader036/viewer/2022062404/5519d1085503468b0c8b47ee/html5/thumbnails/4.jpg)
Motivation Rayleigh-Taylor instability
![Page 5: Numerical detection of complex singularities for functions of two or more variables. Presenter:Alexandr Virodov Additional Authors: Prof. Michael Siegel.](https://reader036.fdocuments.in/reader036/viewer/2022062404/5519d1085503468b0c8b47ee/html5/thumbnails/5.jpg)
Theory – 1D C. Sulem, P.L. Sulem, and H. Frisch.
Tracing complex singularities with spectral methods. J. of Comp. Phys., 50:138-161, 1983.
Asymptotic relationIm(x)
Re(x)
![Page 6: Numerical detection of complex singularities for functions of two or more variables. Presenter:Alexandr Virodov Additional Authors: Prof. Michael Siegel.](https://reader036.fdocuments.in/reader036/viewer/2022062404/5519d1085503468b0c8b47ee/html5/thumbnails/6.jpg)
Example – 1D Inviscid Burger’s Equation
![Page 7: Numerical detection of complex singularities for functions of two or more variables. Presenter:Alexandr Virodov Additional Authors: Prof. Michael Siegel.](https://reader036.fdocuments.in/reader036/viewer/2022062404/5519d1085503468b0c8b47ee/html5/thumbnails/7.jpg)
Theory – 2D
For
it can be shown that
Im(x)
Re(x)
![Page 8: Numerical detection of complex singularities for functions of two or more variables. Presenter:Alexandr Virodov Additional Authors: Prof. Michael Siegel.](https://reader036.fdocuments.in/reader036/viewer/2022062404/5519d1085503468b0c8b47ee/html5/thumbnails/8.jpg)
Synthetic Data in 2D
![Page 9: Numerical detection of complex singularities for functions of two or more variables. Presenter:Alexandr Virodov Additional Authors: Prof. Michael Siegel.](https://reader036.fdocuments.in/reader036/viewer/2022062404/5519d1085503468b0c8b47ee/html5/thumbnails/9.jpg)
Burger’s EquationTraveling Wave solution
![Page 10: Numerical detection of complex singularities for functions of two or more variables. Presenter:Alexandr Virodov Additional Authors: Prof. Michael Siegel.](https://reader036.fdocuments.in/reader036/viewer/2022062404/5519d1085503468b0c8b47ee/html5/thumbnails/10.jpg)
Burger’s Equation I
![Page 11: Numerical detection of complex singularities for functions of two or more variables. Presenter:Alexandr Virodov Additional Authors: Prof. Michael Siegel.](https://reader036.fdocuments.in/reader036/viewer/2022062404/5519d1085503468b0c8b47ee/html5/thumbnails/11.jpg)
Burger’s Equation II
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3 dimensions
Most general form
Again, it can be shown that
![Page 13: Numerical detection of complex singularities for functions of two or more variables. Presenter:Alexandr Virodov Additional Authors: Prof. Michael Siegel.](https://reader036.fdocuments.in/reader036/viewer/2022062404/5519d1085503468b0c8b47ee/html5/thumbnails/13.jpg)
Synthetic Data in 3D
![Page 14: Numerical detection of complex singularities for functions of two or more variables. Presenter:Alexandr Virodov Additional Authors: Prof. Michael Siegel.](https://reader036.fdocuments.in/reader036/viewer/2022062404/5519d1085503468b0c8b47ee/html5/thumbnails/14.jpg)
Further research Application of the method to 3D
Burger’s equation
Application of the method to the Euler’s equation
Accuracy and stability of the method for specific cases
![Page 15: Numerical detection of complex singularities for functions of two or more variables. Presenter:Alexandr Virodov Additional Authors: Prof. Michael Siegel.](https://reader036.fdocuments.in/reader036/viewer/2022062404/5519d1085503468b0c8b47ee/html5/thumbnails/15.jpg)
Questions? References:
C. Sulem, P.L. Sulem, and H. Frisch. Tracing complex singularities with spectral methods. J. of Comp. Phys., 50:138-161, 1983.
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. http://math.lanl.gov/Research/Highlights/amrmhd.shtml
M. Paperin. http://www.brockmann-consult.de/CloudStructures/images/kelvin-helmholtz-instab/k-w-system.gif Brockmann Consult, Geesthacht, 2009.