714 research outputs found
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
Characteristic Evolution and Matching
I review the development of numerical evolution codes for general relativity
based upon the characteristic initial value problem. Progress in characteristic
evolution is traced from the early stage of 1D feasibility studies to 2D
axisymmetric codes that accurately simulate the oscillations and gravitational
collapse of relativistic stars and to current 3D codes that provide pieces of a
binary black hole spacetime. Cauchy codes have now been successful at
simulating all aspects of the binary black hole problem inside an artificially
constructed outer boundary. A prime application of characteristic evolution is
to extend such simulations to null infinity where the waveform from the binary
inspiral and merger can be unambiguously computed. This has now been
accomplished by Cauchy-characteristic extraction, where data for the
characteristic evolution is supplied by Cauchy data on an extraction worldtube
inside the artificial outer boundary. The ultimate application of
characteristic evolution is to eliminate the role of this outer boundary by
constructing a global solution via Cauchy-characteristic matching. Progress in
this direction is discussed.Comment: New version to appear in Living Reviews 2012. arXiv admin note:
updated version of arXiv:gr-qc/050809
On the comparison of asymptotic expansion techniques for the nonlinear KleinâGordon 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
Splitting methods for nonlinear Dirac equations with thirring type interaction in the nonrelativistic limit regime
Nonlinear Dirac equations describe the motion of relativistic spin- particles in presence of external electromagnetic felds, modelled by an electric and magnetic potential, and taking into account a nonlinear particle self-interaction. In recent years, the construction of numerical splitting schemes for the solution of these systems in the nonrelativistic limit regime, i.e., the speed of light c formally tending to infnity, has gained a lot of attention. In this paper, we consider a nonlinear Dirac equation with Thirring type interaction, where in contrast to the case of the Soler type nonlinearity a classical twoterm splitting scheme cannot be straightforwardly applied. Thus, we propose and analyze a three-term Strang splitting scheme which relies on splitting the full problem into the free Dirac subproblem, a potential subproblem, and a nonlinear subproblem, where each subproblem can be solved exactly in time. Moreover, our analysis shows that the error of our scheme improves from () to () if the magnetic potential in the system vanishes. Furthermore, we propose an effcient limit approximation scheme for solving nonlinear Dirac systems in the nonrelativistic limit regime 1 which allows errors of () without any -dependent time step restriction
Uniformly accurate numerical schemes for the nonlinear Dirac equation in the nonrelativistic limit regime
We apply the two-scale formulation approach to propose uniformly accurate
(UA) schemes for solving the nonlinear Dirac equation in the nonrelativistic
limit regime. The nonlinear Dirac equation involves two small scales
and with in the nonrelativistic
limit regime. The small parameter causes high oscillations in time which brings
severe numerical burden for classical numerical methods. We transform our
original problem as a two-scale formulation and present a general strategy to
tackle a class of highly oscillatory problems involving the two small scales
and . Suitable initial data for the two-scale
formulation is derived to bound the time derivatives of the augmented solution.
Numerical schemes with uniform (with respect to )
spectral accuracy in space and uniform first order or second order accuracy in
time are proposed. Numerical experiments are done to confirm the UA property.Comment: 22 pages, 6 figures. To appear on Communications in Mathematical
Science
Boson Stars: Alternatives to primordial black holes?
The present surge for the astrophysical relevance of boson stars stems from
the speculative possibility that these compact objects could provide a
considerable fraction of the non-baryonic part of dark matter within the halo
of galaxies. For a very light `universal' axion of effective string models,
their total gravitational mass will be in the most likely range of \sim 0.5
M_\odot of MACHOs. According to this framework, gravitational microlensing is
indirectly ``weighing" the axion mass, resulting in \sim 10^{-10} eV/c^2. This
conclusion is not changing much, if we use a dilaton type self-interaction for
the bosons. Moreover, we review their formation, rotation and stability as
likely candidates of astrophysical importance.Comment: 14 pages, uses REVTeX, 1 postscript figur
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