Godunov scheme for Maxwell’s equations with Kerr nonlinearity

Aregba-Driollet, Denise

doi:10.4310/cms.2015.v13.n8.a10

Cited by 8 publications

(10 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A snapshot of the electric field and magnetic induction at the final time T = 10 −7 (number of steps N = 100) using the backward Euler-type method for an Escher mesh is taken. The parameters are: time step size t = 10 −9 , χ (1) = 2.2, and χ (3) directly [15], [18]- [25], [28], [29], [60] or solve the problem only in 1D and 2D, e.g. [26], [37]- [46].…”

Section: Examplementioning

confidence: 99%

See 1 more Smart Citation

Energy-Stable Time-Domain Finite Element Methods for the 3D Nonlinear Maxwell's Equations

Anees

Angermann

2020

IEEE Photonics J.

View full text Add to dashboard Cite

In this paper, time-domain finite element methods for the full system of Maxwell's equations with cubic nonlinearities in 3D are presented, including a selection of computational experiments. The new capabilities of these methods are to efficiently model linear and nonlinear effects of the electrical polarization. The novel strategy has been developed to bring under control the discrete nonlinearity model in space and time. It results in energy stable discretizations both at the semi-discrete and the fully discrete levels, with spatial discretization using edge and face elements (Nédeléc-Raviart-Thomas formulation). In particular, the proposed time discretization schemes are unconditionally stable with respect to a specially defined nonlinear electromagnetic energy, which is an upper bound of the electromagnetic energy commonly used. The approaches presented prove to be robust and allow the modeling of 3D optical problems that can be directly derived from the full system of Maxwell's nonlinear equations, and allow the treatment of complex nonlinearities and geometries of various physical systems coupled with electromagnetic fields.

show abstract

Section: Examplementioning

confidence: 99%

“…Finite volume methods have been applied to nonlinear Kerr media in 1D and 2D cases [28], [29]. For the Maxwell's equations with Kerr-type nonlinearity, a hyperbolic system is derived and approximated by the Godunov method.…”

Section: Introductionmentioning

confidence: 99%

Energy-Stable Time-Domain Finite Element Methods for the 3D Nonlinear Maxwell's Equations

Anees

Angermann

2020

IEEE Photonics J.

View full text Add to dashboard Cite

show abstract

“…with e and h denoting the electric and magnetic field intensities and d and b the corresponding fluxes. For the following discussion, we assume that the relation between fields and fluxes is given by d = d(e) := 0 ( (1) + (3) |e| 2 )e, b = b(h) := 0 h…”

Section: Introductionmentioning

confidence: 99%

On higher order passivity preserving schemes for nonlinear Maxwell's equations

Egger¹,

Shashkov²

2022

Preprint

View full text Add to dashboard Cite

We present two strategies for designing passivity preserving higher order discretization methods for Maxwell's equations in nonlinear Kerr-type media. Both approaches are based on variational approximation schemes in space and time. This allows to rigorously prove energy conservation or dissipation, and thus passivity, on the fully discrete level. For linear media, the proposed methods coincide with certain combinations of mixed finite element and implicit Runge-Kutta schemes. The order optimal convergence rates, which can thus be expected for linear problems, are also observed for nonlinear problems in the numerical tests.

show abstract

Section: Introductionmentioning

confidence: 99%

“…In [42] a pseudospectral spatial domain (PSSD) approach is presented for linear Lorentz dispersion and nonlinear Kerr response, and in [30] optical carrier wave shock is studied using the PSSD technique. FV based methods for nonlinear Kerr media are addressed in [1,14] in which the Maxwell-Kerr model is approached as a hyperbolic system and approximated by a Godunov scheme, and a third order Roe solver, respectively, in one and two spatial dimensions.In this work, we use high order discontinuous Galerkin (DG) methods for the spatial discretization of our nonlinear Maxwell models. This is motivated by various properties of DG methods, including high order accuracy, excellent dispersive and dissipative properties in standard wave simulations, flexibility in adaptive implementation and high parallelization, and suitability for complicated geometry (e.g., [10,24]).…”

mentioning

confidence: 99%

Energy stable discontinuous Galerkin methods for Maxwell's equations in nonlinear optical media

Bokil

Cheng

Jiang

et al. 2017

Journal of Computational Physics

View full text Add to dashboard Cite

The propagation of electromagnetic waves in general media is modeled by the time-dependent Maxwell's partial differential equations (PDEs), coupled with constitutive laws that describe the response of the media. In this work, we focus on nonlinear optical media whose response is modeled by a system of first order nonlinear ordinary differential equations (ODEs), which include a single resonance linear Lorentz dispersion, and the nonlinearity comes from the instantaneous electronic Kerr response and the residual Raman molecular vibrational response. To design efficient, accurate, and stable computational methods, we apply high order discontinuous Galerkin discretizations in space to the hybrid PDE-ODE Maxwell system with several choices of numerical fluxes, and the resulting semi-discrete methods are shown to be energy stable. Under some restrictions on the strength of the nonlinearity, error estimates are also established. When we turn to fully discrete methods, the challenge to achieve provable stability lies in the temporal discretizations of the nonlinear terms. To overcome this, novel strategies are proposed to treat the nonlinearity in our model within the framework of the second-order leap-frog and implicit trapezoidal time integrators. The performance of the overall algorithms are demonstrated through numerical simulations of kink and antikink waves, and third-harmonic generation in soliton propagation.Key words. Maxwell's equations, nonlinear dispersion, discontinuous Galerkin method, energy stability, error estimates. Introduction.Nonlinear optics is the study of the behavior of light in nonlinear media. This field has developed into a significant branch of physics since the introduction of intense lasers with high peak powers. In nonlinear media, the material response depends nonlinearly on the optical field, and many interesting physical phenomena, such as frequency mixing and second/third-harmonic generation have been observed and harnessed for practical applications. We refer to classical textbooks [3,5,37] for a more detailed review of the field of nonlinear optics.Our interest here is in the development of novel numerical schemes for the Maxwell's equations in nonlinear optical media. Relative to the widely used asymptotic and paraxial wave models derived from Maxwell's equations, such as nonlinear Schrödinger equation (NLS) and beam propagation method (BPM) [3,5], simulations of the nonlinear Maxwell's system in the time domain are more computationally intensive. However, these simulations have the advantage of being substantially more robust because they directly solve for fundamental quantities, the electromagnetic fields in space and time. These simulations also avoid the simplifying assumptions that lead to conventional asymptotic and paraxial propagation analyses, and are able to treat interacting waves at different frequencies directly [28]. Recent optics and photonics research has focused on phenomena at smaller and smaller length scales or multiple spatial scales. For such phenomena simula...

show abstract

Godunov scheme for Maxwell’s equations with Kerr nonlinearity

Cited by 8 publications

References 13 publications

Energy-Stable Time-Domain Finite Element Methods for the 3D Nonlinear Maxwell's Equations

Energy-Stable Time-Domain Finite Element Methods for the 3D Nonlinear Maxwell's Equations

On higher order passivity preserving schemes for nonlinear Maxwell's equations

Energy stable discontinuous Galerkin methods for Maxwell's equations in nonlinear optical media

Contact Info

Product

Resources

About