202 research outputs found
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
One-stage exponential integrators for nonlinear Schrödinger equations over long times
Near-conservation over long times of the actions, of the energy, of the mass and of the momentum along the numerical solution of the cubic Schrödinger equation with small initial data is shown. Spectral discretization in space and one-stage exponential integrators in time are used. The proofs use modulated Fourier expansion
A symmetric low-regularity integrator for the nonlinear Schr\"odinger equation
We introduce and analyze a symmetric low-regularity scheme for the nonlinear
Schr\"odinger (NLS) equation beyond classical Fourier-based techniques. We show
fractional convergence of the scheme in -norm, from first up to second
order, both on the torus and on a smooth bounded domain , , equipped with homogeneous Dirichlet boundary
condition. The new scheme allows for a symmetric approximation to the NLS
equation in a more general setting than classical splitting, exponential
integrators, and low-regularity schemes (i.e. under lower regularity
assumptions, on more general domains, and with fractional rates). We motivate
and illustrate our findings through numerical experiments, where we witness
better structure preserving properties and an improved error-constant in
low-regularity regimes
Exponential integrators for the stochastic Manakov equation
This article presents and analyses an exponential integrator for the
stochastic Manakov equation, a system arising in the study of pulse propagation
in randomly birefringent optical fibers. We first prove that the strong order
of the numerical approximation is if the nonlinear term in the system is
globally Lipschitz-continuous. Then, we use this fact to prove that the
exponential integrator has convergence order in probability and almost
sure order , in the case of the cubic nonlinear coupling which is relevant
in optical fibers. Finally, we present several numerical experiments in order
to support our theoretical findings and to illustrate the efficiency of the
exponential integrator as well as a modified version of it
Fourth-order time-stepping for stiff PDEs on the sphere
We present in this paper algorithms for solving stiff PDEs on the unit sphere
with spectral accuracy in space and fourth-order accuracy in time. These are
based on a variant of the double Fourier sphere method in coefficient space
with multiplication matrices that differ from the usual ones, and
implicit-explicit time-stepping schemes. Operating in coefficient space with
these new matrices allows one to use a sparse direct solver, avoids the
coordinate singularity and maintains smoothness at the poles, while
implicit-explicit schemes circumvent severe restrictions on the time-steps due
to stiffness. A comparison is made against exponential integrators and it is
found that implicit-explicit schemes perform best. Implementations in MATLAB
and Chebfun make it possible to compute the solution of many PDEs to high
accuracy in a very convenient fashion
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