Abstract
The spectral consistency of the finite-difference theta-method for the unsteady Schrödinger equation is investigated. Optimal sampling parameters providing a minimum error for a given spectral range are obtained. It is shown that the op ti mized scheme provides a reduction (by a factor of 5–6) in the error of the approximate solution in comparison with the 4th order accuracy scheme. It is shown that the 4th order scheme provides the best spectral consistency only in the case if the spectral range length tends to zero. The conditions for equivalence between the finite-difference scheme and the scheme in the form of two first-order conjugated IIR filters are found. The obtained scheme is the best scheme in the class of conservative finite difference schemes for solving the Schrödinger equation. Practical issues arising in the process of implementing a numerical solution are considered. The obtained results can be efficiently used for solving linear and non-linear Schrödinger equations.
Highlights
The spectral consistency of the finite-difference theta-method for the unsteady Schrödinger equation is investi gated
Optimal sampling parameters providing a minimum error for a given spectral range are obtained
It is shown that the optimized scheme provides a reduction in the error of the approximate solution in comparison with the 4th order accuracy scheme
Summary
The spectral consistency of the finite-difference theta-method for the unsteady Schrödinger equation is investi gated. The obtained scheme is the best scheme in the class of conservative finite difference schemes for solving the Schrödinger equation. В работе [4] было показано, что оценки погрешности разностных схем могут быть получены на основе изучения спектральной согласованности амплитудных и фазово-частотных характеристик дифференциальной и разностной задач.
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