@TechReport{	  it:2013-007,
  author	= {Emil Kieri},
  title		= {Accelerated Convergence for {S}chr{\"o}dinger Equations
		  with Non-Smooth Potentials},
  institution	= {Department of Information Technology, Uppsala University},
  department	= {Division of Scientific Computing},
  year		= {2013},
  number	= {2013-007},
  month		= apr,
  abstract	= {When numerically solving the time-dependent
		  Schr{\"o}dinger equation for the electrons in an atom or
		  molecule, the Coulomb singularity poses a challenge. The
		  solution will have limited regularity, and high-order
		  spatial discretisations, which are much favoured in the
		  chemical physics community, are not performing to their
		  full potential. By exploiting knowledge about the jumps in
		  the derivatives of the solution we construct a correction,
		  and show how this improves the convergence rate of Fourier
		  collocation from second to fourth order. This allows for a
		  substantial reduction in the number of grid points. The new
		  method is applied to the higher harmonic generation from
		  atomic hydrogen.}
}