160 research outputs found
Asymptotic behavior of splitting schemes involving time-subcycling techniques
This paper deals with the numerical integration of well-posed multiscale
systems of ODEs or evolutionary PDEs. As these systems appear naturally in
engineering problems, time-subcycling techniques are widely used every day to
improve computational efficiency. These methods rely on a decomposition of the
vector field in a fast part and a slow part and take advantage of that
decomposition. This way, if an unconditionnally stable (semi-)implicit scheme
cannot be easily implemented, one can integrate the fast equations with a much
smaller time step than that of the slow equations, instead of having to
integrate the whole system with a very small time-step to ensure stability.
Then, one can build a numerical integrator using a standard composition method,
such as a Lie or a Strang formula for example. Such methods are primarily
designed to be convergent in short-time to the solution of the original
problems. However, their longtime behavior rises interesting questions, the
answers to which are not very well known. In particular, when the solutions of
the problems converge in time to an asymptotic equilibrium state, the question
of the asymptotic accuracy of the numerical longtime limit of the schemes as
well as that of the rate of convergence is certainly of interest. In this
context, the asymptotic error is defined as the difference between the exact
and numerical asymptotic states. The goal of this paper is to apply that kind
of numerical methods based on splitting schemes with subcycling to some simple
examples of evolutionary ODEs and PDEs that have attractive equilibrium states,
to address the aforementioned questions of asymptotic accuracy, to perform a
rigorous analysis, and to compare them with their counterparts without
subcycling. Our analysis is developed on simple linear ODE and PDE toy-models
and is illustrated with several numerical experiments on these toy-models as
well as on more complex systems. Lie andComment: IMA Journal of Numerical Analysis, Oxford University Press (OUP):
Policy A - Oxford Open Option A, 201
Propagation of Gevrey regularity over long times for the fully discrete Lie Trotter splitting scheme applied to the linear Schroedinger equation
International audienceIn this paper, we study the linear Schroedinger equation over the d-dimensional torus, with small values of the perturbing potential. We consider numerical approximations of the associated solutions obtained by a symplectic splitting method (to discretize the time variable) in combination with the Fast Fourier Transform algorithm (to discretize the space variable). In this fully discrete setting, we prove that the regularity of the initial datum is preserved over long times, i.e. times that are exponentially long with the time discretization parameter. We here refer to Gevrey regularity, and our estimates turn out to be uniform in the space discretization parameter. This paper extends a previous text by Dujardin and Faou, where a similar result has been obtained in the semi-discrete situation, i.e. when the mere time variable is discretized and space is kept a continuous variable
Normal form and long time analysis of splitting schemes for the linear Schrödinger equation.
We consider the linear Schrödinger equation on a one dimensional torus and its time-discretization by splitting methods. Assuming a non-resonance condition on the stepsize and a small size of the potential, we show that the numerical dynamics can be reduced over exponentially long time to a collection of two dimensional symplectic systems for asymptotically large modes. For the numerical solution, this implies the long time conservation of the energies associated with the double eigenvalues of the free Schrödinger operator. The method is close to standard techniques used in finite dimensional perturbation theory, but extended here to infinite dimensional operators
High order linearly implicit methods for semilinear evolution PDEs
This paper considers the numerical integration of semilinear evolution PDEs
using the high order linearly implicit methods developped in a previous paper
in the ODE setting. These methods use a collocation Runge--Kutta method as a
basis, and additional variables that are updated explicitly and make the
implicit part of the collocation Runge--Kutta method only linearly implicit. In
this paper, we introduce several notions of stability for the underlying
Runge--Kutta methods as well as for the explicit step on the additional
variables necessary to fit the context of evolution PDE. We prove a main
theorem about the high order of convergence of these linearly implicit methods
in this PDE setting, using the stability hypotheses introduced before. We use
nonlinear Schr\''odinger equations and heat equations as main examples but our
results extend beyond these two classes of evolution PDEs. We illustrate our
main result numerically in dimensions 1 and 2, and we compare the efficiency of
the linearly implicit methods with other methods from the litterature. We also
illustrate numerically the necessity of the stability conditions of our main
result
\^A-and \^I-stability of collocation Runge-Kutta methods
This paper deals with stability of classical Runge-Kutta collocation methods.
When such methods are embedded in linearly implicit methods as developed in
[12] and used in [13] for the time integration of nonlinear evolution PDEs, the
stability of these methods has to be adapted to this context. For this reason,
we develop in this paper several notions of stability, that we analyze. We
provide sufficient conditions that can be checked algorithmically to ensure
that these stability notions are fulfilled by a given Runge-Kutta collocation
method. We also introduce examples and counterexamples used in [13] to
highlight the necessity of these stability conditions in this context
Peregrine comb: multiple compression points for Peregrine rogue waves in periodically modulated nonlinear Schr{\"o}dinger equations
It is shown that sufficiently large periodic modulations in the coefficients
of a nonlinear Schr{\"o}dinger equation can drastically impact the spatial
shape of the Peregrine soliton solutions: they can develop multiple compression
points of the same amplitude, rather than only a single one, as in the
spatially homogeneous focusing nonlinear Schr{\"o}dinger equation. The
additional compression points are generated in pairs forming a comb-like
structure. The number of additional pairs depends on the amplitude of the
modulation but not on its wavelength, which controls their separation distance.
The dynamics and characteristics of these generalized Peregrine soliton are
analytically described in the case of a completely integrable modulation. A
numerical investigation shows that their main properties persist in
nonintegrable situations, where no exact analytical expression of the
generalized Peregrine soliton is available. Our predictions are in good
agreement with numerical findings for an interesting specific case of an
experimentally realizable periodically dispersion modulated photonic crystal
fiber. Our results therefore pave the way for the experimental control and
manipulation of the formation of generalized Peregrine rogue waves in the wide
class of physical systems modeled by the nonlinear Schr{\"o}dinger equation
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
Exponential RungeâKutta methods for the SchroÌdinger equation
Abstract We consider exponential Runge-Kutta methods of collocation type, and use them to solve linear and semi-linear Schrödinger Cauchy problems on the ddimensional torus. We prove that in both cases (linear and non-linear) and with suitable assumptions, s-stage methods are of order s and we give sufficient conditions to achieve orders s + 1 and s + 2. We show and explain the effects of resonant time steps that occur when solving linear Schrödinger problems on a finite time interval with such methods. This work is inspired b
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