2,468 research outputs found
High order three part split symplectic integrators: Efficient techniques for the long time simulation of the disordered discrete nonlinear Schroedinger equation
While symplectic integration methods based on operator splitting are well
established in many branches of science, high order methods for Hamiltonian
systems that split in more than two parts have not been studied in great
detail. Here, we present several high order symplectic integrators for
Hamiltonian systems that can be split in exactly three integrable parts. We
apply these techniques, as a practical case, for the integration of the
disordered, discrete nonlinear Schroedinger equation (DDNLS) and compare their
efficiencies. Three part split algorithms provide effective means to
numerically study the asymptotic behavior of wave packet spreading in the DDNLS
- a hotly debated subject in current scientific literature.Comment: 5 Figures, Physics Letters A (accepted
Splitting and composition methods in the numerical integration of differential equations
We provide a comprehensive survey of splitting and composition methods for
the numerical integration of ordinary differential equations (ODEs). Splitting
methods constitute an appropriate choice when the vector field associated with
the ODE can be decomposed into several pieces and each of them is integrable.
This class of integrators are explicit, simple to implement and preserve
structural properties of the system. In consequence, they are specially useful
in geometric numerical integration. In addition, the numerical solution
obtained by splitting schemes can be seen as the exact solution to a perturbed
system of ODEs possessing the same geometric properties as the original system.
This backward error interpretation has direct implications for the qualitative
behavior of the numerical solution as well as for the error propagation along
time. Closely connected with splitting integrators are composition methods. We
analyze the order conditions required by a method to achieve a given order and
summarize the different families of schemes one can find in the literature.
Finally, we illustrate the main features of splitting and composition methods
on several numerical examples arising from applications.Comment: Review paper; 56 pages, 6 figures, 8 table
FAST: A Fully Asynchronous Split Time-Integrator for Self-Gravitating Fluid
We describe a new algorithm for the integration of self-gravitating fluid
systems using SPH method. We split the Hamiltonian of a self-gravitating fluid
system to the gravitational potential and others (kinetic and internal
energies) and use different time-steps for their integrations. The time
integration is done in the way similar to that used in the mixed variable or
multiple stepsize symplectic schemes. We performed three test calculations. One
was the spherical collapse and the other was an explosion. We also performed a
realistic test, in which the initial model was taken from a simulation of
merging galaxies. In all test calculations, we found that the number of
time-steps for gravitational interaction were reduced by nearly an order of
magnitude when we adopted our integration method. In the case of the realistic
test, in which the dark matter potential dominates the total system, the total
calculation time was significantly reduced. Simulation results were almost the
same with those of simulations with the ordinary individual time-step method.
Our new method achieves good performance without sacrificing the accuracy of
the time integration.Comment: 14 pages, 8 figures, accepted for publication in PAS
Particle-Particle Particle-Tree: A Direct-Tree Hybrid Scheme for Collisional N-Body Simulations
In this paper, we present a new hybrid algorithm for the time integration of
collisional N-body systems. In this algorithm, gravitational force between two
particles is divided into short-range and long-range terms, using a
distance-dependent cutoff function. The long-range interaction is calculated
using the tree algorithm and integrated with the constant-timestep leapfrog
integrator. The short-range term is calculated directly and integrated with the
high-order Hermite scheme. We can reduce the calculation cost per orbital
period from O(N^2) to O(N log N), without significantly increasing the
long-term integration error. The results of our test simulations show that
close encounters are integrated accurately. Long-term errors of the total
energy shows random-walk behaviour, because it is dominated by the error caused
by tree approximation.Comment: 22 pages, 15 figure
Palindromic 3-stage splitting integrators, a roadmap
The implementation of multi-stage splitting integrators is essentially the
same as the implementation of the familiar Strang/Verlet method. Therefore
multi-stage formulas may be easily incorporated into software that now uses the
Strang/Verlet integrator. We study in detail the two-parameter family of
palindromic, three-stage splitting formulas and identify choices of parameters
that may outperform the Strang/Verlet method. One of these choices leads to a
method of effective order four suitable to integrate in time some partial
differential equations. Other choices may be seen as perturbations of the
Strang method that increase efficiency in molecular dynamics simulations and in
Hybrid Monte Carlo sampling.Comment: 20 pages, 8 figures, 2 table
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