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cosine packets, a comparative study
Multiscale numerical method for nonlinear Maxwell equations
The aim of this work is to propose an efficient numerical
approximation of high frequency pulses propagating in nonlinear
dispersive optical media. We consider the nonlinear Maxwell's
equations with instantaneous nonlinearity. We first derive a
physically and asymptotically equivalent model that is
semi-linear. Then, for a large class of semi-linear systems, we
describe the solution in terms of profiles. These profiles are
solution of a singular equation involving one more variable
describing the phase of the solution. We introduce a discretization
of this equation using finite differences in space and time and an
appropriate Fourier basis (with few elements) for the phase. The
main point is that accurate solution of the nonlinear Maxwell
equation can be computed with a mesh size of order of the
wave length. This approximation is asymptotic-preserving
in the sense that a multi-scale expansion can be performed on the
discrete solution and the result of this expansion is a
discretization of the continuous limit. In order to improve the
computational delay, computations are performed in a window moving at
the group velocity of the pulse. The second harmonic generation is
used as an example to illustrate the proposed methodology. However,
the numerical method proposed for this benchmark study can be applied to many
other cases of nonlinear optics with high frequency pulses.