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Numerical detection of complex singularities for functions of two or more variables.
Presenter:Alexandr Virodov
Additional Authors:Prof. Michael SiegelKamyar MalakutiNan Maung
Outline Motivation 1D – Well known result 2D – Our generalization 2D – Application examples 3D – Theory and example
Motivation Kelvin-Helmholtz Instability
Motivation Rayleigh-Taylor instability
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)
Example – 1D Inviscid Burger’s Equation
Theory – 2D
For
it can be shown that
Im(x)
Re(x)
Synthetic Data in 2D
Burger’s EquationTraveling Wave solution
Burger’s Equation I
Burger’s Equation II
3 dimensions
Most general form
Again, it can be shown that
Synthetic Data in 3D
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
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.