46 research outputs found
On the comparison of asymptotic expansion techniques for the nonlinear Klein–Gordon equation in the nonrelativistic limit regime
This work concerns the time averaging techniques for the nonlinear Klein-Gordon (KG) equation in the nonrelativistic limit regime which have recently gained a lot attention in numerical analysis. This is due to the fact that the solution becomes highly-oscillatory in time in this regime which causes the breakdown of classical integration schemes. To overcome this numerical burden various novel numerical methods with excellent efficiency were derived in recent years. The construction of each method thereby requests essentially the averaged model of the problem. However, the averaged model of each approach is found by different kinds of asymptotic approximation techniques reaching from the modulated Fourier expansion over the multiscale expansion by frequency up to the Chapman-Enskog expansion. In this work we give a first comparison of these recently introduced asymptotic series, reviewing their approximation validity to the KG in the asymptotic limit, their smoothness assumptions as well as their geometric properties, e.g., energy conservation
and long-time behaviour of the remainder
A UNIFORMLY AND OPTIMALLY ACCURATE METHOD FOR THE KLEIN-GORDON-ZAKHAROV SYSTEM IN SIMULTANEOUS HIGH-PLASMA-FREQUENCY AND SUBSONIC LIMIT REGIME *
We present a uniformly and optimally accurate numerical method for solving the Klein-Gordon-Zakharov (KGZ) system with two dimensionless parameters 0 < ε ≤ 1 and 0 < γ ≤ 1, which are inversely proportional to the plasma frequency and the acoustic speed, respectively. In the simultaneous high-plasma-frequency and subsonic limit regime, i.e. ε < γ → 0 + , the KGZ system collapses to a cubic Schrödinger equation, and the solution propagates waves with O(ε 2)-wavelength in time and meanwhile contains rapid outgoing initial layers with speed O(1/γ) in space due to the incompatibility of the initial data. By presenting a multiscale decomposition of the KGZ system, we propose a multiscale time integrator Fourier pseduospectral method which is explicit, efficient and uniformly accurate for solving the KGZ system for all 0 < ε < γ ≤ 1. Numerical results are reported to show the efficiency and accuracy of scheme. Finally, the method is applied to investigate the convergence rates of the KGZ system to its limiting models when ε < γ → 0 +
Uniformly Accurate Methods for Klein-Gordon type Equations
The main contribution of this thesis is the development of a novel class of uniformly accurate methods for Klein-Gordon type equations.
Klein-Gordon type equations in the non-relativistic limit regime, i.e., , are numerically very challenging to treat, since the solutions are highly oscillatory in time. Standard Gautschi-type methods suffer from severe time step restrictions as they require a CFL-condition with time step size to resolve the oscillations. Within this thesis we overcome this difficulty by introducing limit integrators, which allows us to reduce the highly oscillatory problem to the integration of a non-oscillatory limit system. This procedure allows error bounds of order without any step size restrictions. Thus, these integrators are very efficient in the regime . However, limit integrators fail for small values of .
In order to derive numerical schemes that work well for small as well as for large , we use the ansatz of "twisted variables", which allows us to develop uniformly accurate methods with respect to . In particular, we introduce efficient and robust uniformly accurate exponential-type integrators which resolve the solution in the relativistic regime as well as in the highly oscillatory non-relativistic regime without any step size restriction. In contrast to previous works, we do not employ any asymptotic nor multiscale expansion of the solution. Compared to classical methods our new schemes allow us to reduce the regularity assumptions as they converge under the same regularity assumptions required for the integration of the corresponding limit system. In addition, the newly derived first- and second-order exponential-type integrators converge to the classical Lie and Strang splitting schemes for the limit system.
Moreover, we present uniformly accurate schemes for the Klein-Gordon-Schrödinger and the Klein-Gordon-Zakharov system. For all uniformly accurate integrators we establish rigorous error estimates and underline their uniform convergence property numerically
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
Uniformly accurate exponential-type integrators for Klein-Gordon equations with asymptotic convergence to classical splitting schemes in the nonlinear Schrödinger limit
International audienceWe introduce efficient and robust exponential-type integrators for Klein-Gordon equations which resolve the solution in the relativistic regime as well as in the highly-oscillatory non-relativistic regime without any step-size restriction, and under the same regularity assumptions on the initial data required for the integration of the corresponding limit system. In contrast to previous works we do not employ any asymptotic/multiscale expansion of the solution. This allows us derive uniform convergent schemes under far weaker regularity assumptions on the exact solution. In particular, the newly derived exponential-type integrators of first-, respectively, second-order converge in the non-relativistic limit to the classical Lie, respectively, Strang splitting in the nonlinear Schrödinger limit
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
Nonlinear Evolution Equations: Analysis and Numerics
The qualitative theory of nonlinear evolution equations is an
important tool for studying the dynamical behavior of systems in
science and technology. A thorough understanding of the complex
behavior of such systems requires detailed analytical and numerical
investigations of the underlying partial differential equations