1,474 research outputs found
Convergence rates for dispersive approximation schemes to nonlinear Schrödinger equations
This article is devoted to the analysis of the convergence rates of several numerical approximation schemes for linear and nonlinear Schrödinger equations on the real line. Recently, the authors have introduced viscous and two-grid numerical approximation schemes that mimic at the discrete level the so-called Strichartz dispersive estimates of the continuous Schrödinger equation. This allows to guarantee the convergence of numerical approximations for initial data in L2(R), a fact that cannot be proved in the nonlinear setting for standard conservative schemes unless more regularity of the initial data is assumed. In the present article we obtain explicit convergence rates and prove that dispersive schemes fulfilling the Strichartz estimates are better behaved for Hs(R) data if 0 < s< 1/2. Indeed, while dispersive schemes ensure a polynomial convergence rate, non-dispersive ones only yield logarithmic ones
Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators
Dozens of exponential integration formulas have been proposed for the
high-accuracy solution of stiff PDEs such as the Allen-Cahn, Korteweg-de Vries
and Ginzburg-Landau equations. We report the results of extensive comparisons
in MATLAB and Chebfun of such formulas in 1D, 2D and 3D, focusing on fourth and
higher order methods, and periodic semilinear stiff PDEs with constant
coefficients. Our conclusion is that it is hard to do much better than one of
the simplest of these formulas, the ETDRK4 scheme of Cox and Matthews
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
Macroscopic dynamics of incoherent soliton ensembles: soliton-gas kinetics and direct numerical modeling
We undertake a detailed comparison of the results of direct numerical
simulations of the integrable soliton gas dynamics with the analytical
predictions inferred from the exact solutions of the relevant kinetic equation
for solitons. We use the KdV soliton gas as a simplest analytically accessible
model yielding major insight into the general properties of soliton gases in
integrable systems. Two model problems are considered: (i) the propagation of a
`trial' soliton through a one-component `cold' soliton gas consisting of
randomly distributed solitons of approximately the same amplitude; and (ii)
collision of two cold soliton gases of different amplitudes (soliton gas shock
tube problem) leading to the formation of an incoherend dispersive shock wave.
In both cases excellent agreement is observed between the analytical
predictions of the soliton gas kinetics and the direct numerical simulations.
Our results confirm relevance of the kinetic equation for solitons as a
quantitatively accurate model for macroscopic non-equilibrium dynamics of
incoherent soliton ensembles.Comment: 20 pages, 8 figures, 34 references. Other author's papers can be
downloaded at http://www.denys-dutykh.com
Controllability of the 1D Schrodinger equation by the flatness approach
We derive in a straightforward way the exact controllability of the 1-D
Schrodinger equation with a Dirichlet boundary control. We use the so-called
flatness approach, which consists in parameterizing the solution and the
control by the derivatives of a "flat output". This provides an explicit
control input achieving the exact controllability in the energy space. As an
application, we derive an explicit pair of control inputs achieving the exact
steering to zero for a simply-supported beam
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