Jon Wilkening, University of California Berkley

I will describe a spectrally accurate numerical method for finding time-periodic solutions of nonlinear PDE. We minimize a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We use adjoint methods (originally developed for shape optimization) to compute the gradient of this functional with respect to the initial condition. We then minimize the functional using a quasi-Newton gradient descent algorithm, BFGS. We use our method to compute families of time-periodic solutions of the vortex sheet with surface tension and the gravity-driven water wave. If time permits, I will also talk about the Benjamin-Ono equation and the 1d compressible Euler equations.