6,886 research outputs found
Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients
open access articleMotivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge-Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which increases the efficiency of our pair and all other pairs included in the numerical experiments. The numerical results on the nonlinear Schrödinger equation with periodic solution verified the superiority of the new algorithm in terms of efficiency. The new method also presents a good behaviour of the maximum absolute error and the global norm in time, even after a high number of oscillations
Optimized explicit Runge-Kutta schemes for the spectral difference method applied to wave propagation problems
Explicit Runge-Kutta schemes with large stable step sizes are developed for
integration of high order spectral difference spatial discretization on
quadrilateral grids. The new schemes permit an effective time step that is
substantially larger than the maximum admissible time step of standard explicit
Runge-Kutta schemes available in literature. Furthermore, they have a small
principal error norm and admit a low-storage implementation. The advantages of
the new schemes are demonstrated through application to the Euler equations and
the linearized Euler equations.Comment: 37 pages, 3 pages of appendi
Nystrom Methods in the RKQ Algorithm for Initial-value Problems
We incorporate explicit Nystrom methods into the RKQ algorithm for stepwise
global error control in numerical solutions of initial-value problems. The
initial-value problem is transformed into an explicitly second-order problem,
so as to be suitable for Nystrom integration. The Nystrom methods used are
fourth-order, fifth-order and 10th-order. Two examples demonstrate the
effectiveness of the algorithm.Comment: This is an extension of ideas published in J. Math. Res. (open
access); see refs [1] and [2
Optimal stability polynomials for numerical integration of initial value problems
We consider the problem of finding optimally stable polynomial approximations
to the exponential for application to one-step integration of initial value
ordinary and partial differential equations. The objective is to find the
largest stable step size and corresponding method for a given problem when the
spectrum of the initial value problem is known. The problem is expressed in
terms of a general least deviation feasibility problem. Its solution is
obtained by a new fast, accurate, and robust algorithm based on convex
optimization techniques. Global convergence of the algorithm is proven in the
case that the order of approximation is one and in the case that the spectrum
encloses a starlike region. Examples demonstrate the effectiveness of the
proposed algorithm even when these conditions are not satisfied
An asynchronous leapfrog method II
A second order explicit one-step numerical method for the initial value
problem of the general ordinary differential equation is proposed. It is
obtained by natural modifications of the well-known leapfrog method, which is a
second order, two-step, explicit method. According to the latter method, the
input data for an integration step are two system states, which refer to
different times. The usage of two states instead of a single one can be seen as
the reason for the robustness of the method. Since the time step size thus is
part of the step input data, it is complicated to change this size during the
computation of a discrete trajectory. This is a serious drawback when one needs
to implement automatic time step control.
The proposed modification transforms one of the two input states into a
velocity and thus gets rid of the time step dependency in the step input data.
For these new step input data, the leapfrog method gives a unique prescription
how to evolve them stepwise.
The stability properties of this modified method are the same as for the
original one: the set of absolute stability is the interval [-i,+i] on the
imaginary axis. This implies exponential growth of trajectories in situations
where the exact trajectory has an asymptote.
By considering new evolution steps that are composed of two consecutive old
evolution steps we can average over the velocities of the sub-steps and get an
integrator with a much larger set of absolute stability, which is immune to the
asymptote problem.
The method is exemplified with the equation of motion of a one-dimensional
non-linear oscillator describing the radial motion in the Kepler problem.Comment: 41 pages, 25 figure
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