4Motivation Rayleigh-Taylor instability Rayleigh-Taylor instability Heavy fluid over lighter fluidInviscid and irrotational
5Theory – 1D C. Sulem, P.L. Sulem, and H. Frisch. Tracing complex singularities with spectral methods.J. of Comp. Phys., 50: , 1983.Asymptotic relationIm(x)Start from a well known result by Sulem, Sulem and FrischGiven that form of singularityExponential decay of Fourier coefficientsAlgebraic decay dependent on power of singularityExponential decay in deltaDelta – distance to closest singularity (analyticity strip)Easy to derive invertible linear systemAs far as we know, the 1d result was never generalized to higher dimensionsBut before that, an example of application of the method in 1d(higher dimensions employ similar concepts)Re(x)
6Example – 1D Inviscid Burger’s Equation [REMOVE??? TOO LITTLE TIME???] Infinite slope in finite timeSee the decay of forier coefficientsBecomes slower as singularity moves closerExpected value of singularitySquare root of x or 0.5Good numerical convergence
7Theory – 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 realSingularity curve (more general surface)The zero set of the denominator is a parabula in lower complex half-plane of xWe can solve for the other parameters of the complex curveThose additional parameters are the advantage in using 1d result over function parametrized by Y.
8Synthetic Data in 2DWe start with the known form of the function, transformed into periodic function in x, yWe plug in specific sets of parameters and expect to get them backIndeed, the fits are very nice and converge fastThe noise at the end of the fit is due to machine accuracy limits
9Burger’s Equation Traveling Wave solution TODO: Explain forcing & Traveling wavesConsult Kamyar about the best way to explain in 2-3 minutes.
10Burger’s Equation IHere 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 modesThe fits are not as nice as for synthetic data, but improve as the number of modes growsWHY DELTA PASSES ZERO HERE?
11Burger’s Equation IIWHY WE CHOSE THIS INITIAL COND.? WHAT’S SPECIAL ABOUT IT?Again, we can see that with increase of modes, the result gains accuracy.
123 dimensionsMost general formAgain, it can be shown thatZETA!!!
14Further research Application of the method to 3D Burger’s equation Application of the method to the Euler’s equationAccuracy and stability of the method for specific casesIS IT TRUE FOR EULER? JUST A THOUGHT I HAD.
15Questions? 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 dimensionsS. Li, H. Li. Parallel AMR Code for Compressible MHD or HD Equations.M. Paperin. Brockmann Consult, Geesthacht, 2009.QUOTE KAMYAR’S THESIS PROPERLY