In this paper, we consider Maxwell’s equations in linear dispersive media described by a
single-pole Lorentz model for electronic polarization. We study two classes of commonly
used spatial discretizations: finite difference methods (FD) with arbitrary even order
accuracy in space and high spatial order discontinuous Galerkin (DG) finite element
methods. Both types of spatial discretizations are coupled with second order semi-implicit
leap-frog and implicit trapezoidal temporal schemes. By performing detailed dispersion
analysis for the semi-discrete and fully discrete schemes, we obtain rigorous quantification
of the dispersion error for Lorentz dispersive dielectrics. In particular, comparisons of
dispersion error can be made taking into account the model parameters, and mesh sizes
in the design of the two types of schemes. This work is a continuation of our previous
research on energy-stable numerical schemes for nonlinear dispersive optical media [6,7].
The results for the numerical dispersion analysis of the reduced linear model, considered
in the present paper, can guide us in the optimal choice of discretization parameters
for the more complicated and nonlinear models. The numerical dispersion analysis of
the fully discrete FD and DG schemes, for the dispersive Maxwell model considered
in this paper, clearly indicate the dependence of the numerical dispersion errors on
spatial and temporal discretizations, their order of accuracy, mesh discretization parameters
and model parameters. The results obtained here cannot be arrived at by considering
discretizations of Maxwell’s equations in free space. In particular, our results contrast the
advantages and disadvantages of using high order FD or DG schemes and leap-frog or
trapezoidal time integrators over different frequency ranges using a variety of measures
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High Spatial Order Energy Stable FDTD Methods for Maxwell’s Equations in Nonlinear Optical Media in One Dimension
In this paper, we consider electromagnetic (EM) wave propagation in nonlinear
optical media in one spatial dimension. We model the EM wave propagation by the time-
dependent Maxwell’s equations coupled with a system of nonlinear ordinary differential
equations (ODEs) for the response of the medium to the EM waves. The nonlinearity in the
ODEs describes the instantaneous electronic Kerr response and the residual Raman molecular
vibrational response. The ODEs also include the single resonance linear Lorentz dispersion.
For such model, we will design and analyze fully discrete finite difference time domain
(FDTD) methods that have arbitrary (even) order in space and second order in time. It is
challenging to achieve provable stability for fully discrete methods, and this depends on
the choices of temporal discretizations of the nonlinear terms. In Bokil et al. (J Comput
Phys 350:420–452, 2017), we proposed novel modifications of second-order leap-frog and
trapezoidal temporal schemes in the context of discontinuous Galerkin methods to discretize
the nonlinear terms in this Maxwell model. Here, we continue this work by developing
similar time discretizations within the framework of FDTD methods. More specifically, we
design fully discrete modified leap-frog FDTD methods which are proved to be stable under
appropriate CFL conditions. These method can be viewed as an extension of the Yee-FDTD
scheme to this nonlinear Maxwell model. We also design fully discrete trapezoidal FDTD
methods which are proved to be unconditionally stable. The performance of the fully discrete
FDTD methods are demonstrated through numerical experiments involving kink, antikink
waves and third harmonic generation in soliton propagation.
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- Award ID(s):
- 1720116
- NSF-PAR ID:
- 10065567
- Date Published:
- Journal Name:
- Journal of scientific computing
- ISSN:
- 0885-7474
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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