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Numerical detection of complex singularities for functions of two or more variables.

Numerical detection of complex singularities for functions of two or more variables. Presenter: Alexandr Virodov Additional Authors: Prof. Michael Siegel Kamyar Malakuti Nan Maung. Outline. Motivation 1D – Well known result 2D – Our generalization 2D – Application examples

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Numerical detection of complex singularities for functions of two or more variables.

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  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., 50:138-161, 1983. • Asymptotic relation Im(x) Re(x)

  6. Example – 1D • Inviscid Burger’s Equation

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

  8. Synthetic Data in 2D

  9. Burger’s EquationTraveling Wave solution

  10. Burger’s Equation I

  11. Burger’s 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 Burger’s equation • Application of the method to the Euler’s 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., 50:138-161, 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.

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