By Albert R. Baswell

Advances in arithmetic study offers unique study effects at the innovative of arithmetic study. every one article has been rigorously chosen in an try and current mammoth examine effects throughout a huge spectrum.

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The longitudinal Lorentz force 1 v y B z = − k 0 v 02 sin(2ψ ) . This drive a longitudinal response at the 2nd harmonic of the 2 light wave. 28 M. Shoucri Figure 21. 8 at −1 t = 60ω pe . Figure 22. Longitudinal electric field. The Method of Characteristics for the Numerical Solution… 29 Figure 23. Charge (ni–ne). 8 at t=60. (23) shows the charge (ni –ne ). We note the system is solved for ions and electrons, but the response of the ions at t=60 is still negligible. In the conclusion of this section 2 , we stress the importance of the cubic spline and the good performance of the interpolation technique, having low numerical diffusion and dispersion and high accuracy.

No small scales filtering or dissipation is added to the present code, as is done in [62,63]. Instead, we use a fine grid of N x xN y = 2048 x512 to resolve small details, and stop the calculation when the small details are of the order of the 50 M. Shoucri Figure 40. G+ at time t=35. Figure 41. G- at time t=30. grid size.. 2 . The equations are integrated is the initial numerically in perturbation. the spatial domain − 2π < x < 2π , 0 < y < 2π . The domain is periodic in the y direction, and we apply Dirichlet boundary conditions in the x direction.

So in this case there is no 2nd harmonic longitudinal response to the leading order. The Method of Characteristics for the Numerical Solution… 45 Figure 33. Frequency. spectrum of the plasma. Figure 34. Wavenumber spectrum of the plasma. 3 becomes valid. 25 . 044 , which is small. 8 at t=60. (21)) . Note again the clear picture of the vortices, in the low density region of the phase-space, with very little numerical noise appearing. Figs(36) and (37) show respectively the electric field and the charge (ni –ne ) across the box.