62 research outputs found
Trigonometric time integrators for the Zakharov system
The main challenge in the analysis of numerical schemes for the Zakharov
system originates from the presence of derivatives in the nonlinearity. In this
paper a new trigonometric time-integration scheme for the Zakharov system is
constructed and convergence is proved. The time-step restriction is independent
from a spatial discretization. Numerical experiments confirm the findings
Randomized exponential integrators for modulated nonlinear Schr\"odinger equations
We consider the nonlinear Schr\"odinger equation with dispersion modulated by
a (formal) derivative of a time-dependent function with fractional Sobolev
regularity of class for some . Due to the loss
of smoothness in the problem classical numerical methods face severe order
reduction. In this work, we develop and analyze a new randomized exponential
integrator based on a stratified Monte Carlo approximation. The new
discretization technique averages the high oscillations in the solution
allowing for improved convergence rates of order . In addition, the
new approach allows us to treat a far more general class of modulations than
the available literature. Numerical results underline our theoretical findings
and show the favorable error behavior of our new scheme compared to classical
methods
Uniformly accurate oscillatory integrators for the Klein-Gordon-Zakharov system from low- to high-plasma frequency regimes
We present a novel class of oscillatory integrators for the
Klein-Gordon-Zakharov system which are uniformly accurate with respect to the
plasma frequency . Convergence holds from the slowly-varying low-plasma up
to the highly oscillatory high-plasma frequency regimes without any step size
restriction and, especially, uniformly in . The introduced schemes are
moreover asymptotic consistent and approximates the solutions of the
corresponding Zakharov limit system in the high-plasma frequency limit (). We in particular present the construction of the first- and
second-order uniformly accurate oscillatory integrators and establish rigorous,
uniform error estimates. Numerical experiments underline our theoretical
convergence results
Asymptotic preserving schemes for the Klein-Gordon equation in the non-relativistic limit regime
International audienceWe consider the Klein-Gordon equation in the non-relativistic limit regime, i.e. the speed of light c tending to infinity. We construct an asymptotic expansion for the solution with respect to the small parameter depending on the inverse of the square of the speed of light. As the first terms of this asymptotic can easily be simulated our approach allows us to construct numerical algorithms that are robust with respect to the large parameter c producing high oscillations in the exact solution
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