Energy Stable Discontinuous Galerkin Methods for Maxwell s Equations in Nonlinear Optical Media

Transcription

1 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

2 Introduction Outline 1 Introduction 2 Numerical methods Temporal discretizations Spatial discretizations 3 Numerical results 4 Conclusion Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 2

3 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

4 Introduction Numerical simulations Common approach: simulate approximate models, such as nonlinear Schrödinger 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

5 Introduction The model under consideration Maxwell s equations in a non-magnetic nonlinear optical medium t B + E = 0, in (0, T ) Ω, t D + J s H = 0, in (0, T ) Ω, B = 0, D = ρ, in (0, T ) Ω, (1a) (1b) (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 D = ɛ 0 (ɛ E + P L delay + a(1 θ)e E 2 + aθqe), 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

6 Introduction The model linear instantaneous response ɛ 0 ɛ E. linear Lorentz response, where 2 P L delay t τ P L delay t + ω 2 0 PL delay = ω2 pe. (3) 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 Kerr + PNL delay = a(1 θ)e E 2 }{{} Kerr + aθqe. }{{} Raman 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 where ω v is the resonance frequency of the vibration, and τ 1 v + 1 τ v Q t + ω2 v Q = ω2 v E 2, (4) a damping constant. Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 6

7 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

8 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 ω2 0P + ω 2 pe, t Q = σ, t σ = 1 τ v σ ω 2 v Q + ω 2 v E 2, (5d) (5e) (5f) with the constitutive law D = ɛ 0 (ɛ E + P + a(1 θ)e 3 + aθqe), (6) Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 8

9 Introduction Energy relations We consider the model in 1D, and under the assumption of periodic boundary conditions, the energy E = + ɛ 0aθ ( µ 0 Ω 2 H2 + ɛ 0ɛ 2 E 2 + ɛ 0 2ωp 2 2 QE 2 + 3ɛ 0a(1 θ) 4 satisfies the following relation, d dt E = ɛ 0 ωpτ 2 Ω J 2 + ɛ 0ω 2 0 2ω 2 p E 4 + ɛ 0aθ 4 Q2 )dx, P 2 + ɛ 0aθ 4ωv 2 σ 2 J 2 dx ɛ 0aθ 2ωv 2 σ 2 dx 0. τ v 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

10 Numerical methods Outline 1 Introduction 2 Numerical methods Temporal discretizations Spatial discretizations 3 Numerical results 4 Conclusion Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 10

11 Numerical methods Temporal discretizations Outline 1 Introduction 2 Numerical methods Temporal discretizations Spatial discretizations 3 Numerical results 4 Conclusion Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 11

12 Numerical methods Temporal discretizations Temporal discretizations Scheme1: Leap-frog staggered in time for the PDE part, and implicit in ODE part µ 0 H n+1/2 H n t/2 = E n, x D n+1 D n t = H n+1/2, (8a) x D n+1 = ɛ 0 (ɛ E n+1 + P n+1 + a(1 θ)y n+1 + aθq n+1 E N+1 ), Y n+1 = Y n ((E n+1 ) 2 + (E n ) 2 )(E n+1 E n ), P n+1 P n = 1 ( J n + J n+1), t 2 J n+1 J n = 1 t 2 ( 1 τ σ n+1 σ n t H n+1 H n+1/2 µ 0 = E n+1. t/2 x (8b) (8c) Q n+1 Q n = 1 ( σ n + σ n+1), t 2 (8d) ( E n + E n+1) ), (8e) ( J n + J n+1) + ω0 2 ( P n + P n+1) ωp 2 = 1 2 ( 1 ( σ n + σ n+1) + ω 2 ( v Q n + Q n+1) 2ωv 2 E n E n+1 ), (8f) τ v Scheme2: Implicit trapezoidal, replace (8a), (8g) by (8g) µ 0 H n+1 H n t = 1 n 2 ( E x + E n+1 ), x D n+1 D n t = 1 n 2 ( H x + H n+1 ) x Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 12

13 Numerical methods Temporal discretizations Discrete energy relation With periodic boundary condition, then we have E n+1 E n = ɛ 0 t 4ωpτ 2 (J n+1 + J n ) 2 dx ɛ 0aθ t 8ωv 2 (σ n+1 + σ n ) 2 dx 0, (9) τ v where the discrete energy for scheme1 is E n = Ω Ω µ 0 2 Hn+1/2 H n 1/2 + ɛ 0ɛ 2 (E n ) 2 + ɛ 0 2ωp 2 2 Qn (E n ) 2 + 3ɛ 0a(1 θ) 4 + ɛ 0aθ 4ωv 2 (σ n ) 2 + ɛ 0aθ the discrete energy for scheme2 is E n = Ω µ 0 2 (Hn ) 2 + ɛ 0ɛ 2 (E n ) 2 + ɛ 0 2ω 2 p + ɛ 0aθ 4ωv 2 (σ n ) 2 + ɛ 0aθ 2 Qn (E n ) 2 + 3ɛ 0a(1 θ) 4 Ω (J n ) 2 + ɛ 0ω0 2 2ωp 2 (P n ) 2 (10) (E n ) 4 + ɛ 0aθ 4 (Qn ) 2 dx (J n ) 2 + ɛ 0ω0 2 2ωp 2 (P n ) 2 (11) (E n ) 4 + ɛ 0aθ 4 (Qn ) 2 dx. Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 13

14 Numerical methods Spatial discretizations Outline 1 Introduction 2 Numerical methods Temporal discretizations Spatial discretizations 3 Numerical results 4 Conclusion Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 14

15 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

16 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

17 Numerical methods Spatial discretizations Semi-discrete DG formulation Let Ω = [x L, x R ] be the computational domain, with mesh x L = x 1/2 < x 3/2 < < x N+1/2 = x R, is introduced. Let I j = [x j 1/2, x j+1/2 ] h j = x j+ 1 x 2 j 1 as its length, and h = max 1 j N h j 2 as the largest meshsize. We now define a finite dimensional discrete space, V k h = {v : v I j P k (I j ), j = 1, 2,, N}. (12) Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 17

18 Numerical methods Spatial discretizations Semi-discrete DG formulation We find H h (t, ), D h (t, ), E h (t, ), P h (t, ), J h (t, ), Q h (t, ), σ h (t, ) Vh k, such that j, µ 0 t H h φdx + E h x φdx (Êhφ I ) j+1/2 + (Êhφ + ) j 1/2 = 0, φ Vh k, j I j t D h φdx + H h x φdx ( H h φ ) j+1/2 + ( H h φ + ) j 1/2 = 0, φ Vh k, I j I j t P h = J h, ( ) 1 t J h = τ J h + ω0p 2 h ωpe 2 h, t Q h = σ h, t σ h φdx = I j I j ( ) 1 σ h + ωv 2 Q h ωv 2 Eh 2 φdx, φ Vh k τ. v The constitutive law is imposed via the L 2 projection, namely, ( D h φdx = ɛ 0 ɛ E h + a(1 θ)eh 3 + P ) h + aθq h E h φdx, I j I j φ V k h. (14) Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 18

19 Numerical methods Spatial discretizations Semi-discrete DG formulation As for numerical fluxes, we take either central fluxes, one of the following alternating flux pair Ê h = {E h }, Hh = {H h }, (15) Ê h = E h, Hh = H + h ; Ê h = E + h, Hh = H h, (16) or the dissipative flux inspired by the upwind flux for the Maxwell system without Kerr, linear Lorentz and Raman effects, Ê h = {E h } + 1 µ0 [H h ], Hh = {H h } + 1 ɛ0 ɛ [E h ]. (17) 2 ɛ 0 ɛ 2 µ 0 Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 19

20 Numerical methods Spatial discretizations Semi-discrete stability Theorem (Semi-discrete stability) Under the assumption of periodic boundary conditions, the semi-discrete DG scheme with central and alternating fluxes satisfies d dt E h = ɛ 0 ωpτ 2 Jh 2 dx ɛ 0aθ Ω 2ωv 2 σh 2 dx 0, τ v Ω and the DG scheme with the upwind flux satisfies d dt E h = ɛ 0 ωpτ 2 Jh 2 dx ɛ 0aθ Ω 2ωv 2 σh 2 τ dx 1 µ0 v Ω 2 ɛ 0 ɛ where E h = Ω ( µ0 2 H2 h + ɛ 0ɛ 2 E 2 h + ɛ 0 2ω 2 p J 2 h + ɛ 0ω 2 0 2ω 2 p N [H h ] 2 j+1/2 1 ɛ0 ɛ N [E h ] 2 j+1/2 2 0, j=1 Ph 2 + ɛ 0aθ 4ωv 2 σh 2 + ɛ 0aθ is the discrete energy. Moreover, E h 0 when θ [0, 3 4 ]. µ 0 j=1 2 Q heh 2 + 3ɛ 0a(1 θ) 4 Eh 4 + ɛ ) 0aθ 4 Q2 h dx (18) Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 20

21 Numerical methods Spatial discretizations Semi-discrete error estimates Theorem (Error estimates of semi-discrete scheme) Let κ err (0, 1) and ρ err (0, 1) be two arbitrary parameters. Assume the periodic boundary condition and E, H, P, Q, J, σ W 1, ([0, T ], H k+1 (Ω)), and Then where under the conditions on θ E W 1, ([0, T ], W 1, (Ω)), Q W 1, ([0, T ], L (Ω)). u u h CC model C(κ err, ρ err )h r, u = E, H, P, Q, J, σ, { k for central flux (15), r = k + 1 for alternating flux (16) and upwind flux (17), and on the strength of nonlinearity, 1 θ [0, 3(1 ρ err ) ], aθc k Q ɛ (1 κ err ), ( 3 θ a Ck 2 ρ te 2 + 3(1 θ)ck 2 te E + θ ) err 2 C k t Q ɛ κ err. 4 Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 21

22 Numerical methods Spatial discretizations Fully discrete stability: leap-frog-dg Assuming the periodic boundary condition, then the fully discrete scheme with central and alternating fluxes, satisfies E n+1 h E n h = ɛ 0 t 4ωp 2τ (J n+1 h + J n h )2 dx ɛ 0aθ t Ω 8ωv 2 (σ n+1 τv h + σ n h )2 dx 0, (19) Ω E n h = µ 0 Ω 2 Hn+1/2 H n 1/2 h h + ɛ 0aθ 4ωv 2 (σ n h )2 + ɛ 0aθ is the discrete energy. In addition, E h 0 if θ [0, 4 3 The fully discrete scheme with the upwind flux satisfies E n+1 h E n h = ɛ 0 t 4ωp 2τ (J n+1 h Ω + ɛ 0ɛ (E n h 2 )2 + ɛ 0 2ωp 2 2 Qn h (E n h )2 + 3ɛ 0a(1 θ) 4 ] and the CFL condition t h + J n h )2 dx ɛ 0aθ t 8ω 2 v τv Ω (J n h )2 + ɛ 0ω0 2 2ωp 2 (P n h )2 (20) (E n h )4 + ɛ 0aθ 4 (Qn h )2 dx C µ 0 ɛ 0 ɛ is satisfied. (σ n+1 h + σ n h )2 dx (21) E n h = t µ0 8 ɛ 0 ɛ N [H n 1/2 h j=1 µ 0 Ω 2 Hn+1/2 H n 1/2 h h + H n+1/2 ] 2 h j+1/2 t ɛ0 ɛ 8 µ 0 + ɛ 0ɛ (E n h 2 )2 + ɛ 0 2ωp 2 (J n h )2 + ɛ 0ω 2 0 2ω 2 p N j=1 [E n h + E n+1 h ] 2 j+1/2 0, (P n h )2 + ɛ 0aθ 4ωv 2 (σ n h )2 + ɛ 0aθ 2 Qn h (E n h )2 + 3ɛ 0a(1 θ) (E n h 4 )4 + ɛ 0aθ 4 (Qn h )2 dx + t µ0 8 ɛ 0 ɛ is the discrete energy. In addition, E h 0 if θ [0, 3 ] and the CFL condition t 4 h N j=1 ([H n 1/2 h ][H n 1/2 h ɛ0 C µ 0 min(1, ɛ ) µ0 ( ɛ0 2+min(1, ɛ )) µ0 + H n+1/2 ]) h j+1/2 (22) is satisfied. Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 22

23 Numerical methods Spatial discretizations Fully discrete stability: trapezoidal-dg The schemes are unconditionally stable. In particular, with central and alternating fluxes satisfies E n+1 h E n h = ɛ 0 t 4ωp 2τ Ω (J n+1 h + J n h )2 dx ɛ 0aθ t 8ω 2 v τv Ω (σ n+1 h + σ n h )2 dx 0, (23) and that with the upwind flux satisfies E n+1 h E n h = ɛ 0 t 4ωp 2τ (J n+1 h + J n h )2 dx ɛ 0aθ t Ω 8ωv 2 (σ n+1 τv h + σ n h )2 dx (24) Ω t µ0 8 ɛ 0 ɛ N [H n h + Hn+1 h ] 2 j+1/2 t ɛ0 ɛ N 8 µ j=1 0 j=1 [E n h + E n+1 h ] 2 j+1/2 0, where E n h = Ω µ 0 2 (Hn h )2 + ɛ 0ɛ 2 + ɛ 0aθ 4ωv 2 (σ n h )2 + ɛ 0aθ (E n h )2 + ɛ 0 2ω 2 p 2 Qn h (E n h )2 + 3ɛ 0a(1 θ) 4 (J n h )2 + ɛ 0ω0 2 2ωp 2 (P n h )2 (25) (E n h )4 + ɛ 0aθ 4 (Qn h )2 dx. It is non-negative when θ [0, 3 4 ]. Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 23

24 Numerical results Outline 1 Introduction 2 Numerical methods Temporal discretizations Spatial discretizations 3 Numerical results 4 Conclusion Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 24

25 Numerical results Kink shaped solution We consider kink shaped solutions (Sorensen et al. 05), where a traveling wave solution was constructed for the instantaneous intensity-dependent Kerr response neglecting the influence of damping, i.e., θ = 0, τ =. ɛ = 2.25, ɛ s = 5.25, β 1 = ɛ s ɛ, ω 0 = , ω p = ω 0 β1, a = ɛ /3, v = / ɛ, E(0) = 0, Φ(0) = Solutions are of the form u(x, t) = u 0(x vt). We compute until T = 6/v, when the solutions recover its initial state. (a) Initial condition E(x, 0). (b) Reference solution E(x, t). Figure: A traveling kink and antikink wave: the electric field. Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 25

26 Numerical results Energy conservation (a) Leap-frog scheme. k = 1. (b) Leap-frog scheme. k = 1. Figure: A traveling kink and antikink wave: the time evolution of the relative deviation in energy. N = 400 grid points. Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 26

27 upwind flux central flux alternating flux I alternating flux II Table: A traveling kink and antikink wave: errors and orders of accuracy of E. k = 1. N Leap-frog scheme Fully implicit scheme L 2 errors order L error order L 2 errors order L error order E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E

28 upwind flux central flux alternating flux I alternating flux II Table: A traveling kink and antikink wave: errors and orders of accuracy of E. k = 2. N Leap-frog scheme Fully implicit scheme L 2 errors order L error order L 2 errors order L error order E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E

29 Numerical results Soliton propagation (Giles et al, 00) - Third harmonic generation Initially, all fields are zero. ɛ = 2.25, ɛ s = 5.25, β 1 = ɛ s ɛ, 1/τ = , 1/τ v = 29.2/32, a = 0.07, θ = 0.3, Ω 0 = 12.57, ω 0 = 5.84, ω v = 1.28, ω p = ω 0 β1. The left boundary is injected with an incoming solitary wave, for which the electric field is prescribed as E(x = 0, t) = f (t) cos(ω 0 t), (26) where f (t) = M sech(t 20). The boundary condition of H can be approximated from the linearized dispersion relation. Right boundary: absorbing wall. We simulate the transient fundamental (M = 1) and second-order (M = 2) temporal soliton evolutions. Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 29

30 Numerical results Boundary treatment: left boundary The boundary condition of H can be approximated from the linearized dispersion relation. Assuming a space-time harmonic variation e i(ωt kx) of all fields, the exact dispersion relation associated with the linear parts of the system is ɛ ω 4 i ɛ τ ω3 (ɛ ω ω 2 p + k 2 )ω 2 + i 1 τ k2 ω + k 2 ω 2 0 = 0. (27) The solution corresponding to the wave propagating to the right is Then we take the approximate value of H as H(x = 0, t) = k = ω ωp/ɛ ɛ 1 2 ω 2 iω/τ ω0 2. (28) 1 [ 8 2 m=0 Ĥ(ω)e iωt dω ( i) m ( 1 m! Z )(m) ω=ω0 f (m) (t) ] e iω0t + c.c., (29) where c.c. denotes the complex conjugate of the first term, f (m) (t) is the m-th derivative of f (t), and ( 1 Z )(m) is the m-th derivative of Z = ω/k with respect to ω. Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 30

31 Numerical results Boundary treatment: right boundary We treat the right boundary as an absorbing wall corresponding to the linearized system, similar to the procedure performed in Hile et al. (96). Neglecting the nonlinear effects and the delayed response, we have t (H + ɛ E) = 1 ɛ x (H + ɛ E) t (H ɛ E) = 1 ɛ x (H ɛ E). Because only waves that propagate to the right are allowed, the left going characteristic variable H + ɛ E is set to be zero at the right boundary x R = x N+1/2. Therefore, for semi-discrete scheme, we require (H h + ɛ E h ) + N+1/2 = 0, (H h ɛ E h ) + N+1/2 = (H h ɛ E h ) N+1/2. This corresponds to rewriting the central flux (and also the alternating fluxes) as Ê h N+1/2 = 3 4 E h N+1/2 1 4 H h ɛ N+1/2, Hh N+1/2 = 3 4 H h N+1/2 ɛ 4 E h N+1/2, (30) and rewriting the upwind flux as Ê h N+1/2 = 1 2 E h N+1/2 1 2 ɛ H h N+1/2, Hh N+1/2 = 1 2 H h N+1/2 Energy analysis with boundary effects has been conducted. ɛ 2 E h N+1/2. (31) Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 31

32 Numerical results Simulation results Figure: leap-frog scheme. N = 6400 grid points. k = 3, alternating flux I. Left:M = 1, right: M = 2. Results agree with literature. (For upwind flux with k = 1, the daughter pulse is not evident due to numerical dissipation.) Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 32

33 Figure: leap-frog scheme. N = 6400 grid points. k = 3, alternating flux I. M = 1. Energy relation.

34 Conclusion Outline 1 Introduction 2 Numerical methods Temporal discretizations Spatial discretizations 3 Numerical results 4 Conclusion Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 34

35 Conclusion Conclusion We developed energy-stable DG methods for nonlinear Maxwell equations with Lorentz, Kerr and Raman effects in 1D. Main ingredients: second order time discretizations with special treatment of nonlinear terms, DG spatial discretizations. Overall, the alternating fluxes show the best performance. The scheme has been extended to arbitrary order FDTD method on staggered mesh. Study of numerical dispersion for the linearized Lorentz model is on-going. Future work : higher order, higher dimensions other nonlinear models. Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 35

36 Conclusion Reference V. A. Bokil, Y. Cheng, Y. Jiang and F. Li, Energy stable discontinuous Galerkin methods for Maxwells equations in nonlinear optical media, Journal of Computational Physics, v350 (2017), pp V. A. Bokil, Y. Cheng, Y. Jiang, F. Li and P. Sakkaplangkul, High spatial order energy stable FDTD methods for Maxwells equations in nonlinear optical media, Journal of Scientific Computing, to appear. V. A. Bokil, Y. Cheng, Y. Jiang, F. Li and P. Sakkaplangkul, Dispersion Analysis of Finite Difference and Discontinuous Galerkin Schemes for Maxwell s Equations in Linear Lorentz Media, preprint. Yingda Cheng (MSU) Energy stable DG for nonlinear Maxwell ICERM workshop, June 2018 Page 36

37 The END! Thank You!