46 research outputs found
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
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 +
A uniformly accurate (UA) multiscale time integrator pseudospectral method for the Dirac equation in the nonrelativistic limit regime
We propose and rigourously analyze a multiscale time integrator Fourier
pseudospectral (MTI-FP) method for the Dirac equation with a dimensionless
parameter which is inversely proportional to the speed of
light. In the nonrelativistic limit regime, i.e. , the
solution exhibits highly oscillatory propagating waves with wavelength
and in time and space, respectively. Due to the rapid
temporal oscillation, it is quite challenging in designing and analyzing
numerical methods with uniform error bounds in . We
present the MTI-FP method based on properly adopting a multiscale decomposition
of the solution of the Dirac equation and applying the exponential wave
integrator with appropriate numerical quadratures. By a careful study of the
error propagation and using the energy method, we establish two independent
error estimates via two different mathematical approaches as
and ,
where is the mesh size, is the time step and depends on the
regularity of the solution. These two error bounds immediately imply that the
MTI-FP method converges uniformly and optimally in space with exponential
convergence rate if the solution is smooth, and uniformly in time with linear
convergence rate at for all and optimally with
quadratic convergence rate at in the regimes when either
or . Numerical results are
reported to demonstrate that our error estimates are optimal and sharp.
Finally, the MTI-FP method is applied to study numerically the convergence
rates of the solution of the Dirac equation to those of its limiting models
when .Comment: 25 pages, 1 figur
Numerical methods and comparison for the Dirac equation in the nonrelativistic limit regime
We analyze rigorously error estimates and compare numerically
spatial/temporal resolution of various numerical methods for the discretization
of the Dirac equation in the nonrelativistic limit regime, involving a small
dimensionless parameter which is inversely proportional to
the speed of light. In this limit regime, the solution is highly oscillatory in
time, i.e. there are propagating waves with wavelength and
in time and space, respectively. We begin with several frequently used
finite difference time domain (FDTD) methods and obtain rigorously their error
estimates in the nonrelativistic limit regime by paying particular attention to
how error bounds depend explicitly on mesh size and time step as
well as the small parameter . Based on the error bounds, in order
to obtain `correct' numerical solutions in the nonrelativistic limit regime,
i.e. , the FDTD methods share the same
-scalability on time step: . Then we
propose and analyze two numerical methods for the discretization of the Dirac
equation by using the Fourier spectral discretization for spatial derivatives
combined with the exponential wave integrator and time-splitting technique for
temporal derivatives, respectively. Rigorous error bounds for the two numerical
methods show that their -scalability on time step is improved to
when . Extensive numerical results
are reported to support our error estimates.Comment: 34 pages, 2 figures. arXiv admin note: substantial text overlap with
arXiv:1511.0119