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.