Hybrid adaptive grid algorithm for wave packet propagation

A three-stage hybrid adaptive algorithm is applied to wave packet scattering from a repulsive barrier. As the packet approaches the barrier during the first stage, quantum hydrodynamic equations (QHE) are integrated on a moving equally-spaced grid. In the second stage, when nodes form in the reflected wave packet, the wavefunction is integrated on a fixed grid while the transmitted part is still treated using the QHE. In the third stage, when ripples have replaced the nodes, the QHE are integrated separately for the reflected and transmitted components. The method is stable for long propagation times (1 ns).