11 research outputs found
Gradient Symplectic Algorithms for Solving the Radial Schrodinger Equation
The radial Schrodinger equation for a spherically symmetric potential can be
regarded as a one dimensional classical harmonic oscillator with a
time-dependent spring constant. For solving classical dynamics problems,
symplectic integrators are well known for their excellent conservation
properties. The class of {\it gradient} symplectic algorithms is particularly
suited for solving harmonic oscillator dynamics. By use of Suzuki's rule for
decomposing time-ordered operators, these algorithms can be easily applied to
the Schrodinger equation. We demonstrate the power of this class of gradient
algorithms by solving the spectrum of highly singular radial potentials using
Killingbeck's method of backward Newton-Ralphson iterations.Comment: 19 pages, 10 figure
Cost-reduction implicit exponential Runge-Kutta methods for highly oscillatory systems
In this paper, two novel classes of implicit exponential Runge-Kutta (ERK)
methods are studied for solving highly oscillatory systems. First of all, we
analyze the symplectic conditions of two kinds of exponential integrators, and
present a first-order symplectic method. In order to solve highly oscillatory
problems, the highly accurate implicit ERK integrators (up to order four) are
formulated by comparing the Taylor expansions of numerical and exact solutions,
it is shown that the order conditions of two new kinds of exponential methods
are identical to the order conditions of classical Runge-Kutta (RK) methods.
Moreover, we investigate the linear stability properties of these exponential
methods. Finally, numerical results not only present the long time energy
preservation of the first-order symplectic method, but also illustrate the
accuracy and efficiency of these formulated methods in comparison with standard
ERK methods
Phase-Fitted and Amplification-Fitted Higher Order Two-Derivative Runge-Kutta Method for the Numerical Solution of Orbital and Related Periodical IVPs
A phase-fitted and amplification-fitted two-derivative Runge-Kutta (PFAFTDRK) method of high algebraic order for the numerical solution of first-order Initial Value Problems (IVPs) which possesses oscillatory solutions is derived. We present a sixth-order four-stage two-derivative Runge-Kutta (TDRK) method designed using the phase-fitted and amplification-fitted property. The stability of the new method is analyzed. The numerical experiments are carried out to show the efficiency of the derived methods in comparison with other existing Runge-Kutta (RK) methods