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
DO I HAVE TIME TO GO THROUGH OUTLINE???

3. Motivation Kelvin-Helmholtz Instability Kelvin-Helmholtz
Irrotational fluids
Velocity jump
Analytic continuation
Moving singularity in Lower complex plane
Curvature singularity

4. Motivation Rayleigh-Taylor instability Rayleigh-Taylor instability
Heavy fluid over lighter fluid
Inviscid and irrotational

5. Theory – 1D C. Sulem, P.L. Sulem, and H. Frisch.
Tracing complex singularities with spectral methods.
J. of Comp. Phys., 50: , 1983.
Asymptotic relation
Im(x)
Start from a well known result by Sulem, Sulem and Frisch
Given that form of singularity
Exponential decay of Fourier coefficients
Algebraic decay dependent on power of singularity
Exponential decay in delta
Delta – distance to closest singularity (analyticity strip)
Easy to derive invertible linear system
As far as we know, the 1d result was never generalized to higher dimensions
But before that, an example of application of the method in 1d
(higher dimensions employ similar concepts)
Re(x)

6. Example – 1D Inviscid Burger’s Equation [REMOVE??? TOO LITTLE TIME???]
Infinite slope in finite time
See the decay of forier coefficients
Becomes slower as singularity moves closer
Expected value of singularity
Square root of x or 0.5
Good numerical convergence

7. Theory – 2D For it can be shown that Im(x) Re(x)
TODO: Fix u formula (take from Tex/PDF), Fix beta, lambda instead of alpha, m.
Approach – Do analytic continuation in X only, keeping Y real
Singularity curve (more general surface)
The zero set of the denominator is a parabula in lower complex half-plane of x
We can solve for the other parameters of the complex curve
Those additional parameters are the advantage in using 1d result over function parametrized by Y.

8. Synthetic Data in 2D
We start with the known form of the function, transformed into periodic function in x, y
We plug in specific sets of parameters and expect to get them back
Indeed, the fits are very nice and converge fast
The noise at the end of the fit is due to machine accuracy limits

9. Burger’s Equation Traveling Wave solution
TODO: Explain forcing & Traveling waves
Consult Kamyar about the best way to explain in 2-3 minutes.

10. Burger’s Equation I
Here we work with an equation form which is very similar to the 1D Burger’s equation
[IS IT TRUE FOR THIS ONE? I THINK NOT, CONFIRM] in fact, with some transformations, it can be shown to be 1d.
The red triangle dots are fits for the solution with 128 modes [MODES OR NODES? WHAT’S THE PROPER TERM?]
The blue lines are fits for the solution with 256 modes
The fits are not as nice as for synthetic data, but improve as the number of modes grows
WHY DELTA PASSES ZERO HERE?

11. Burger’s Equation II
WHY WE CHOSE THIS INITIAL COND.? WHAT’S SPECIAL ABOUT IT?
Again, we can see that with increase of modes, the result gains accuracy.

12. 3 dimensions
Most general form
Again, it can be shown that
ZETA!!!

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
IS IT TRUE FOR EULER? JUST A THOUGHT I HAD.

15. Questions? References:
C. Sulem, P.L. Sulem, and H. Frisch. Tracing complex singularities with spectral methods. J. of Comp. Phys., 50: , 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.
M. Paperin. Brockmann Consult, Geesthacht, 2009.
QUOTE KAMYAR’S THESIS PROPERLY