Accelerated convergence for Schrödinger equations with non-smooth potentials
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When numerically solving the time-dependent Schrödinger 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.
KeywordsTime-dependent Schrödinger equation Spectral methods Non-smooth coefficients Higher harmonic generation
Mathematics Subject Classification (2010)35Q40 65M12 65M70
The author thanks Sverker Holmgren for helpful discussions, and Markus Kowalewski for providing access to his software for quantum dynamics computations.