Penalty Methods for the Hyperbolic System Modelling the Wall-Plasma Interaction in a Tokamak

The penalization method is used to take account of obstacles in a tokamak, such as the limiter. We study a non linear hyperbolic system modelling the plasma transport in the area close to the wall. A penalization which cuts the transport term of the momentum is studied. We show numerically that this penalization creates a Dirac measure at the plasma-limiter interface which prevents us from defining the transport term in the usual sense. Hence, a new penalty method is proposed for this hyperbolic system and numerical tests reveal an optimal convergence rate without any spurious boundary layer.Comment: 8 pages; International Symposium FVCA6, Prague : Czech Republic (2011

Co-periodic stability of periodic waves in some Hamiltonian PDEs

International audienceThe stability theory of periodic traveling waves is much less advanced than for solitary waves, which were first studied by Boussinesq and have received a lot of attention in the last decades. In particular, despite recent breakthroughs regarding periodic waves in reaction-diffusion equations and viscous systems of conservation laws [Johnson–Noble–Rodrigues–Zumbrun, Invent math (2014)], the stability of periodic traveling wave solutions to dispersive PDEs with respect to 'arbitrary' perturbations is still widely open in the absence of a dissipation mechanism. The focus is put here on co-periodic stability of periodic waves, that is, stability with respect to perturbations of the same period as the wave, for KdV-like systems of one-dimensional Hamiltonian PDEs. Fairly general nonlinearities are allowed in these systems, so as to include various models of mathematical physics, and this precludes complete integrability techniques. Stability criteria are derived and investigated first in a general abstract framework, and then applied to three basic examples that are very closely related, and ubiquitous in mathematical physics, namely, a quasilinear version of the generalized Korteweg–de Vries equation (qKdV), and the Euler–Korteweg system in both Eulerian coordinates (EKE) and in mass Lagrangian coordinates (EKL). Those criteria consist of a necessary condition for spectral stability , and of a sufficient condition for orbital stability. Both are expressed in terms of a single function, the abbreviated action integral along the orbits of waves in the phase plane, which is the counterpart of the solitary waves moment of instability introduced by Boussinesq. However, the resulting criteria are more complicated for periodic waves because they have more degrees of freedom than solitary waves, so that the action is a function of N + 2 variables for a system of N PDEs, while the moment of instability is a function of the wave speed only once the endstate of the 1 solitary wave is fixed. Regarding solitary waves, the celebrated Grillakis–Shatah– Strauss stability criteria amount to looking for the sign of the second derivative of the moment of instability with respect to the wave speed. For periodic waves, stability criteria involve all the second order, partial derivatives of the action. This had already been pointed out by various authors for some specific equations, in particular the generalized Korteweg–de Vries equation — which is special case of (qKdV) — but not from a general point of view, up to the authors' knowledge. The most striking results obtained here can be summarized as: an odd value for the difference between N and the negative signature of the Hessian of the action implies spectral instability, whereas a negative signature of the same Hessian being equal to N implies orbital stability. Furthermore, it is shown that, when applied to the Euler–Korteweg system, this approach yields several interesting connexions between (EKE), (EKL), and (qKdV). More precisely, (EKE) and (EKL) share the same abbreviated action integral, which is related to that of (qKdV) in a simple way. This basically proves simultaneous stability in both formulations (EKE) and (EKL) — as one may reasonably expect from the physical point view —, which is interesting to know when these models are used for different phenomena — e.g. shallow water waves or nonlinear optics. In addition, stability in (EKE) and (EKL) is found to be linked to stability in the scalar equation (qKdV). Since the relevant stability criteria are merely encoded by the negative signature of (N + 2) × (N + 2) matrices, they can at least be checked numerically. In practice, when N = 1 or 2, this can be done without even requiring an ODE solver. Various numerical experiments are presented, which clearly discriminate between stable cases and unstable cases for (qKdV), (EKE) and (EKL), thus confirming some known results for the generalized KdV equation and the Nonlinear Schrödinger equation, and pointing out some new results for more general (systems of) PDEs

Stiff Stability of the Hydrogen atom in dissipative Fokker electrodynamics

We introduce an ad-hoc electrodynamics with advanced and retarded Lienard-Wiechert interactions plus the dissipative Lorentz-Dirac self-interaction force. We study the covariant dynamical system of the electromagnetic two-body problem, i.e., the hydrogen atom. We perform the linear stability analysis of circular orbits for oscillations perpendicular to the orbital plane. In particular we study the normal modes of the linearized dynamics that have an arbitrarily large imaginary eigenvalue. These large eigenvalues are fast frequencies that introduce a fast (stiff) timescale into the dynamics. As an application, we study the phenomenon of resonant dissipation, i.e., a motion where both particles recoil together in a drifting circular orbit (a bound state), while the atom dissipates center-of-mass energy only. This balancing of the stiff dynamics is established by the existence of a quartic resonant constant that locks the dynamics to the neighborhood of the recoiling circular orbit. The resonance condition quantizes the angular momenta in reasonable agreement with the Bohr atom. The principal result is that the emission lines of quantum electrodynamics (QED) agree with the prediction of our resonance condition within one percent average deviation.Comment: 1 figure, Notice that Eq. (34) of the Phys. Rev. E paper has a typo; it is missing the square Brackets of eq. (33), find here the correct e

The boundary Riemann solver coming from the real vanishing viscosity approximation

We study a family of initial boundary value problems associated to mixed hyperbolic-parabolic systems: v^{\epsilon} _t + A (v^{\epsilon}, \epsilon v^{\epsilon}_x ) v^{\epsilon}_x = \epsilon B (v^{\epsilon} ) v^{\epsilon}_{xx} The conservative case is, in particular, included in the previous formulation. We suppose that the solutions $v^{\epsilon}$ to these problems converge to a unique limit. Also, it is assumed smallness of the total variation and other technical hypotheses and it is provided a complete characterization of the limit. The most interesting points are the following two. First, the boundary characteristic case is considered, i.e. one eigenvalue of $A$ can be $0$. Second, we take into account the possibility that $B$ is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if it is not satisfied, then pathological behaviours may occur.Comment: 84 pages, 6 figures. Text changes in Sections 1 and 3.2.3. Added Section 3.1.2. Minor changes in other section

Strong Deformation Effects in Hot Rotating 46Ti

Exotic-deformation effects in 46Ti nucleus were investigated by analysing the high-energy gamma-ray and the alpha-particle energy spectra. One of the experiments was performed using the charged-particle multi-detector array ICARE together with a large volume (4"x4") BGO detector. The study focused on simultaneous measurement of light charged particles and gamma-rays in coincidence with the evaporation residues. The experimental data show a signature of very large deformations of the compound nucleus in the Jacobi transition region at the highest spins. These results are compared to data from previous experiments performed with the HECTOR array coupled to the EUROBALL array, where it was found that the GDR strength function is highly fragmented, strongly indicating a presence of nuclei with very large deformation.Comment: 10 pages, 6 figures, Proceedings of the Zakopane Conference on Nuclear Physics, to be published in Acta Phys. Pol. B (2007

GDR Feeding of the Highly-Deformed Band in 42Ca

The gamma-ray spectra from the decay of the GDR in the compound nucleus reaction 18O+28Si at bombarding energy of 105 MeV have been measured in an experiment using the EUROBALL IV and HECTOR arrays. The obtained experimental GDR strength function is highly fragmented, with a low energy (10 MeV) component, indicating a presence of a large deformation and Coriolis effects. In addition, the preferential feeding of the highly-deformed band in 42Ca by this GDR low energy component is observed.Comment: 6 pages, 2 figures, Proceedings of the Zakopane2004 Symposium, to be published in Acta Phys. Pol. B36 (2005

Charged particle decay of hot and rotating 88^{88}Mo nuclei in fusion-evaporation reactions

A study of fusion-evaporation and (partly) fusion-fission channels for the $^{88}$Mo compound nucleus, produced at different excitation energies in the reaction $^{48}$Ti + $^{40}$Ca at 300, 450 and 600 MeV beam energies, is presented. Fusion-evaporation and fusion-fission cross sections have been extracted and compared with the existing systematics. Experimental data concerning light charged particles have been compared with the prediction of the statistical model in its implementation in the Gemini++ code, well suited even for high spin systems, in order to tune the main model parameters in a mass region not abundantly covered by exclusive experimental data. Multiplicities for light charged particles emitted in fusion evaporation events are also presented. Some discrepancies with respect to the prediction of the statistical model have been found for forward emitted $\alpha$-particles; they may be due both to pre-equilibrium emission and to reaction channels (such as Deep Inelastic Collisions, QuasiFission/QuasiFusion) different from the compound nucleus formation.Comment: 14 pages, 14 figure

Two-way multi-lane traffic model for pedestrians in corridors

We extend the Aw-Rascle macroscopic model of car traffic into a two-way multi-lane model of pedestrian traffic. Within this model, we propose a technique for the handling of the congestion constraint, i.e. the fact that the pedestrian density cannot exceed a maximal density corresponding to contact between pedestrians. In a first step, we propose a singularly perturbed pressure relation which models the fact that the pedestrian velocity is considerably reduced, if not blocked, at congestion. In a second step, we carry over the singular limit into the model and show that abrupt transitions between compressible flow (in the uncongested regions) to incompressible flow (in congested regions) occur. We also investigate the hyperbolicity of the two-way models and show that they can lose their hyperbolicity in some cases. We study a diffusive correction of these models and discuss the characteristic time and length scales of the instability

Isospin mixing in Zr 80: from finite to zero temperature

S. Ceruti et al.; 5 págs.; 4 figs.; PACS numbers: 24.30.Cz, 24.60.Dr, 24.80.+y, 25.70.GhThe isospin mixing was deduced in the compound nucleus Zr80 at an excitation energy of E∗=54 MeV from the γ decay of the giant dipole resonance. The reaction Ca40+Ca40 at Ebeam=136 MeV was used to form the compound nucleus in the isospin I=0 channel, while the reaction Cl37+Ca44 at Ebeam=95 MeV was used as the reference reaction. The γ rays were detected with the AGATA demonstrator array coupled with LaBr3:Ce detectors. The temperature dependence of the isospin mixing was obtained and the zero-temperature value deduced. The isospin-symmetry-breaking correction δC used for the Fermi superallowed transitions was extracted and found to be consistent with β-decay data.This work was supported by PRIN No. 2001024324_01302, the Polish National Center for Science Grants No. 2013/08/ M/ST2/00591 and No. 2011/03/B/ST2/01894, and the Spanish Grant No. FPA2011-29854-C04-01. German Bundesministerium für Bildung und Forschung (BMBF) under Contract No. 05P12PKFNE TP4.Peer Reviewe