Pseudo-spectral solution of nonlinear Schro¨dinger equations
Journal of Computational Physics
Fourth-order symplectic integration
Physica D
On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime
Journal of Computational Physics
Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime
Journal of Computational Physics
Computing Semiclassical Quantum Dynamics with Hagedorn Wavepackets
SIAM Journal on Scientific Computing
Fast Gaussian wavepacket transforms and Gaussian beams for the Schrödinger equation
Journal of Computational Physics
Fast multiscale Gaussian beam methods for wave equations in bounded convex domains
Journal of Computational Physics
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In this paper an efficient and accurate method for simulating the propagation of a localized solution of the Schrodinger equation with a smooth potential near the semiclassical limit is presented. We are interested computing arbitrarily accurate solutions when the non dimensional Planck's constant, @e, is small, but not negligible. The method is based on a time dependent transformation of the independent variables, closely related to Gaussian wave packets. A rescaled wave function, w, satisfies a new Schrodinger equation with a time dependent potential which is a perturbation of the harmonic oscillator, the perturbation being O(@e), so that all stiffness (in space and time) is greatly reduced. In fact, for integration in a fixed time interval, the number of modes required to fully resolve the problem decreases when @e is decreased. The original wave function may be reconstructed by Fourier interpolation, although expectation values of the observables can be computed directly from the function w itself. If the initial condition is a Gaussian wave packet, very few modes are necessary to fully resolve the w variable, so for short time very accurate solutions can be obtained at low computational cost. Initial conditions other than Gaussians wave packets can also be used. In this paper, the Gaussian wave packet transform is carefully outlined and applied to the Schrodinger equation in one dimension. Detailed numerical tests show the efficiency and accuracy of the approach. In the sequel of this paper, this approach has been extended to the multidimensional case.