76,504 research outputs found
An explicit and symmetric exponential wave integrator for the nonlinear Schr\"{o}dinger equation with low regularity potential and nonlinearity
We propose and analyze a novel symmetric exponential wave integrator (sEWI)
for the nonlinear Schr\"odinger equation (NLSE) with low regularity potential
and typical power-type nonlinearity of the form ,
where is the density with the wave function and is the exponent of the nonlinearity. The sEWI is explicit and
stable under a time step size restriction independent of the mesh size. We
rigorously establish error estimates of the sEWI under various regularity
assumptions on potential and nonlinearity. For "good" potential and
nonlinearity (-potential and ), we establish an optimal
second-order error bound in -norm. For low regularity potential and
nonlinearity (-potential and ), we obtain a first-order
-norm error bound accompanied with a uniform -norm bound of the
numerical solution. Moreover, adopting a new technique of \textit{regularity
compensation oscillation} (RCO) to analyze error cancellation, for some
non-resonant time steps, the optimal second-order -norm error bound is
proved under a weaker assumption on the nonlinearity: . For
all the cases, we also present corresponding fractional order error bounds in
-norm, which is the natural norm in terms of energy. Extensive numerical
results are reported to confirm our error estimates and to demonstrate the
superiority of the sEWI, including much weaker regularity requirements on
potential and nonlinearity, and excellent long-time behavior with
near-conservation of mass and energy.Comment: 35 pages, 10 figure
Error analysis of trigonometric integrators for semilinear wave equations
An error analysis of trigonometric integrators (or exponential integrators)
applied to spatial semi-discretizations of semilinear wave equations with
periodic boundary conditions in one space dimension is given. In particular,
optimal second-order convergence is shown requiring only that the exact
solution is of finite energy. The analysis is uniform in the spatial
discretization parameter. It covers the impulse method which coincides with the
method of Deuflhard and the mollified impulse method of Garc\'ia-Archilla,
Sanz-Serna & Skeel as well as the trigonometric methods proposed by Hairer &
Lubich and by Grimm & Hochbruck. The analysis can also be used to explain the
convergence behaviour of the St\"ormer-Verlet/leapfrog discretization in time.Comment: 25 page
The NLS approximation for two dimensional deep gravity waves
This article is concerned with infinite depth gravity water waves in two
space dimensions. We consider this system expressed in position-velocity
potential holomorphic coordinates. Our goal is to study this problem with small
wave packet data, and to show that this is well approximated by the cubic
nonlinear Schr\"odinger equation (NLS) on the natural cubic time scale.Comment: 23 page
Two dimensional water waves in holomorphic coordinates II: global solutions
This article is concerned with the infinite depth water wave equation in two
space dimensions. We consider this problem expressed in position-velocity
potential holomorphic coordinates,and prove that small localized data leads to
global solutions. This article is a continuation of authors' earlier paper
arXiv:1401.1252.Comment: 21 pages. We have updated the authors' inf
Strichartz estimates on Schwarzschild black hole backgrounds
We study dispersive properties for the wave equation in the Schwarzschild
space-time. The first result we obtain is a local energy estimate. This is then
used, following the spirit of earlier work of Metcalfe-Tataru, in order to
establish global-in-time Strichartz estimates. A considerable part of the paper
is devoted to a precise analysis of solutions near the trapping region, namely
the photon sphere.Comment: 44 pages; typos fixed, minor modifications in several place
Local decay of waves on asymptotically flat stationary space-times
In this article we study the pointwise decay properties of solutions to the
wave equation on a class of stationary asymptotically flat backgrounds in three
space dimensions. Under the assumption that uniform energy bounds and a weak
form of local energy decay hold forward in time we establish a local
uniform decay rate for linear waves. This work was motivated by open problems
concerning decay rates for linear waves on Schwarzschild and Kerr backgrounds,
where such a decay rate has been conjectured by R. Price. Our results apply to
both of these cases.Comment: 33 pages; minor corrections, updated reference
Exponentially accurate Hamiltonian embeddings of symplectic A-stable Runge--Kutta methods for Hamiltonian semilinear evolution equations
We prove that a class of A-stable symplectic Runge--Kutta time
semidiscretizations (including the Gauss--Legendre methods) applied to a class
of semilinear Hamiltonian PDEs which are well-posed on spaces of analytic
functions with analytic initial data can be embedded into a modified
Hamiltonian flow up to an exponentially small error. As a consequence, such
time-semidiscretizations conserve the modified Hamiltonian up to an
exponentially small error. The modified Hamiltonian is -close to the
original energy where is the order of the method and the time
step-size. Examples of such systems are the semilinear wave equation or the
nonlinear Schr\"odinger equation with analytic nonlinearity and periodic
boundary conditions. Standard Hamiltonian interpolation results do not apply
here because of the occurrence of unbounded operators in the construction of
the modified vector field. This loss of regularity in the construction can be
taken care of by projecting the PDE to a subspace where the operators occurring
in the evolution equation are bounded and by coupling the number of excited
modes as well as the number of terms in the expansion of the modified vector
field with the step size. This way we obtain exponential estimates of the form
with and ; for the semilinear wave
equation, , and for the nonlinear Schr\"odinger equation, . We give
an example which shows that analyticity of the initial data is necessary to
obtain exponential estimates
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