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

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

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Motivation Kelvin-Helmholtz Instability

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Motivation Rayleigh-Taylor instability

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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)

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Example – 1D Inviscid Burgers Equation

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

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Synthetic Data in 2D

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Burgers Equation Traveling Wave solution

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Burgers Equation I

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Burgers Equation II

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

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Synthetic Data in 3D

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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

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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 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.http://www.brockmann- consult.de/CloudStructures/images/kelvin-helmholtz-instab/k-w-system.gif

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