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 3D – Theory and example DO I HAVE TIME TO GO THROUGH OUTLINE???

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

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

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) 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)

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

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.

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

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

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?

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.

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

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

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. QUOTE KAMYAR’S THESIS PROPERLY