Energy Stable Discontinuous Galerkin Methods for Maxwell’s Equations in Nonlinear Optical Media Yingda Cheng Michigan State University “Computational Aspects of Time Dependent Electromagnetic Wave Problems in Complex Materials”, ICERM, June 2018 Joint work with Vrushali Bokil, Fengyan Li, Yan Jiang Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 1
Introduction Outline Introduction 1 Numerical methods 2 Temporal discretizations Spatial discretizations Numerical results 3 Conclusion 4 Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 2
Introduction Nonlinear optics Nonlinear optics is the study of the behavior of light propagating in optical media where the material response depends on the fields nonlinearly. Nonlinearity is particularly relevant when the intensity of the light is very high (e.g. laser). Examples of nonlinear behavior ◮ The refractive index, and consequently the speed of light in a nonlinear optical medium, depends on light intensity. ◮ The frequency of light is altered as it passes through a nonlinear optical medium. For example, the light can change from red to blue. Applications: laser frequency conversion, second/third-harmonic generation (frequency-mixing), self-phase modulation. References: Bloembergen (96), Boyd (03), New (11). Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 3
Introduction Numerical simulations Common approach: simulate approximate models , such as nonlinear Schr¨ odinger equation (NLS) equation, beam propagation method (BPM) for wavepackets. More costly approach: simulate nonlinear Maxwell models directly. This approach is more robust because it avoids the simplifying assumptions that lead to conventional asymptotic and paraxial propagation analyses, and can treat interacting waves at different frequencies directly. Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 4
Introduction The model under consideration Maxwell’s equations in a non-magnetic nonlinear optical medium ∂ t B + ∇ × E = 0 , in (0 , T ) × Ω , (1a) ∂ t D + J s − ∇ × H = 0 , in (0 , T ) × Ω , (1b) ∇ · B = 0 , ∇ · D = ρ, in (0 , T ) × Ω , (1c) where E , D , H , B are the electric field, electric flux density, magnetic field, magnetic induction, ρ, J s are the charge and source current density. Constitutive relations delay + a (1 − θ ) E | E | 2 + a θ Q E ) , D = ǫ 0 ( ǫ ∞ E + P L B = µ 0 H , (2) which takes into account the following effects. Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 5
Introduction The model linear instantaneous response ǫ 0 ǫ ∞ E . linear Lorentz response, where ∂ 2 P L ∂ P L + 1 delay delay + ω 2 delay = ω 2 0 P L p E . (3) ∂ t 2 τ ∂ t Here ω 0 , ω p are the resonance and plasma frequencies of the medium. τ − 1 is a damping constant. nonlinear response. P NL = P NL delay = a (1 − θ ) E | E | 2 Kerr + P NL + a θ Q E . � �� � � �� � Raman Kerr Here a , θ are constants. Q describes the natural molecular vibrations within the dielectric material that has frequency many orders of magnitude less than the optical wave frequency, where ∂ 2 Q ∂ t 2 + 1 ∂ Q ∂ t + ω 2 v Q = ω 2 v | E | 2 , (4) τ v where ω v is the resonance frequency of the vibration, and τ − 1 a damping constant. v Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 6
Introduction Previous work for nonlinear Maxwell model with Kerr/Raman effects Relatively fewer papers compared with linear media. FDTD approach: Hile, Kath (96), Sorenson et al (05), Giles et al. (00) Pseudospectral method: Tyrrell et al (05) FVM approach for Kerr media: De La Bourdonnaye (00), Aregba-Driollet (15) DG for Kerr media: Fezoui (15) Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 7
Introduction Simplified 1D model In 1D, by using the ADE approach, we have µ 0 ∂ t H = ∂ x E , (5a) ∂ t D = ∂ x H , (5b) ∂ t P = J , (5c) ∂ t J = − 1 τ J − ω 2 0 P + ω 2 p E , (5d) ∂ t Q = σ, (5e) ∂ t σ = − 1 σ − ω 2 v Q + ω 2 v E 2 , (5f) τ v with the constitutive law D = ǫ 0 ( ǫ ∞ E + P + a (1 − θ ) E 3 + a θ QE ) , (6) Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 8
Introduction Energy relations We consider the model in 1D, and under the assumption of periodic boundary conditions, the energy � J 2 + ǫ 0 ω 2 ( µ 0 2 H 2 + ǫ 0 ǫ ∞ E 2 + ǫ 0 P 2 + ǫ 0 a θ σ 2 0 E = 2 ω 2 2 ω 2 4 ω 2 2 Ω p p v + ǫ 0 a θ 2 QE 2 + 3 ǫ 0 a (1 − θ ) E 4 + ǫ 0 a θ 4 Q 2 ) dx , 4 satisfies the following relation, � � dt E = − ǫ 0 d J 2 dx − ǫ 0 a θ σ 2 dx ≤ 0 . ω 2 p τ 2 ω 2 v τ v Ω Ω Note that E ( t ) is guaranteed non-negative only when θ ∈ [0 , 3 4 ]. Objective of this work : develop nonlinear Maxwell solver with provable energy stability. Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 9
Numerical methods Outline Introduction 1 Numerical methods 2 Temporal discretizations Spatial discretizations Numerical results 3 Conclusion 4 Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 10
Numerical methods Temporal discretizations Outline Introduction 1 Numerical methods 2 Temporal discretizations Spatial discretizations Numerical results 3 Conclusion 4 Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 11
Numerical methods Temporal discretizations Temporal discretizations Scheme1 : Leap-frog staggered in time for the PDE part, and implicit in ODE part H n +1 / 2 − H n D n +1 − D n n n +1 / 2 = ∂ E = ∂ H µ 0 , , (8a) ∆ t / 2 ∂ x ∆ t ∂ x D n +1 = ǫ 0 ( ǫ ∞ E n +1 + P n +1 + a (1 − θ ) Y n +1 + a θ Q n +1 E N +1 ) , (8b) Y n +1 = Y n + 3 2(( E n +1 ) 2 + ( E n ) 2 )( E n +1 − E n ) , (8c) P n +1 − P n Q n +1 − Q n � J n + J n +1 � � σ n + σ n +1 � = 1 = 1 , , (8d) ∆ t 2 ∆ t 2 J n +1 − J n � J n + J n +1 � � P n + P n +1 � � E n + E n +1 � = − 1 2(1 + ω 2 − ω 2 ) , (8e) 0 p ∆ t τ σ n +1 − σ n � σ n + σ n +1 � � Q n + Q n +1 � = − 1 2( 1 + ω 2 − 2 ω 2 v E n E n +1 ) , (8f) v ∆ t τ v H n +1 − H n +1 / 2 n +1 = ∂ E µ 0 . (8g) ∆ t / 2 ∂ x Scheme2 : Implicit trapezoidal, replace (8a), (8g) by H n +1 − H n D n +1 − D n n n +1 n n +1 = 1 2( ∂ E + ∂ E = 1 2( ∂ H + ∂ H µ 0 ) , ) ∆ t ∂ x ∂ x ∆ t ∂ x ∂ x Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 12
Numerical methods Temporal discretizations Discrete energy relation With periodic boundary condition, then we have � � E n +1 − E n = − ǫ 0 ∆ t ( J n +1 + J n ) 2 dx − ǫ 0 a θ ∆ t ( σ n +1 + σ n ) 2 dx ≤ 0 , (9) 4 ω 2 8 ω 2 p τ v τ v Ω Ω where the discrete energy for scheme1 is � ( J n ) 2 + ǫ 0 ω 2 µ 0 2 H n +1 / 2 H n − 1 / 2 + ǫ 0 ǫ ∞ ( E n ) 2 + ǫ 0 E n 0 ( P n ) 2 = (10) 2 ω 2 2 ω 2 2 Ω p p + ǫ 0 a θ ( σ n ) 2 + ǫ 0 a θ 2 Q n ( E n ) 2 + 3 ǫ 0 a (1 − θ ) ( E n ) 4 + ǫ 0 a θ 4 ( Q n ) 2 dx 4 ω 2 4 v the discrete energy for scheme2 is � ( J n ) 2 + ǫ 0 ω 2 µ 0 2 ( H n ) 2 + ǫ 0 ǫ ∞ ( E n ) 2 + ǫ 0 E n 0 ( P n ) 2 = (11) 2 ω 2 2 ω 2 2 Ω p p + ǫ 0 a θ ( σ n ) 2 + ǫ 0 a θ 2 Q n ( E n ) 2 + 3 ǫ 0 a (1 − θ ) ( E n ) 4 + ǫ 0 a θ 4 ( Q n ) 2 dx . 4 ω 2 4 v Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 13
Numerical methods Spatial discretizations Outline Introduction 1 Numerical methods 2 Temporal discretizations Spatial discretizations Numerical results 3 Conclusion 4 Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 14
Numerical methods Spatial discretizations Spatial discretizations We use discontinuous Galerkin (DG) discretizations for the unknowns. We consider central/upwind/alternating type of fluxes. Optimal error estimates are obtained for alternating/upwind fluxes, and suboptimal error estimates are obtained for central flux with assumptions on the smallness of nonlinearity. We have similar type of energy relations as the semi-discrete case (with additional damping from upwind flux). The proof can be done by using same test functions. The trapezoidal schemes are unconditionally stable, while the leap frog scheme has cfl restriction resulted from the positivity requirement of the energy. We have also extend the work to arbitrary even order FDTD methods in a subsequent work. Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 15
Numerical methods Spatial discretizations Spatial discretizations: DG scheme DG methods. Invented by Reed and Hill (73) for neutron transport. First analysis by Lesaint and Raviart (74). Runge-Kutta discontinuous Galerkin (RKDG) method by Cockburn and Shu (89, 90,...) for general conservation laws. Many works on DG methods of various kinds for wave equations, Maxwell’s equations. Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 16
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