Energy Stable Discontinuous Galerkin Methods for Maxwell s Equations in Nonlinear Optical Media
|
|
- Randall Francis
- 5 years ago
- Views:
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!
ENERGY STABLE DISCONTINUOUS GALERKIN METHODS FOR MAXWELL S EQUATIONS IN NONLINEAR OPTICAL MEDIA
ENERGY STABLE DISCONTINUOUS GALERKIN METHODS FOR MAXWELL S EQUATIONS IN NONLINEAR OPTICAL MEDIA VRUSHALI A. BOKIL, YINGDA CHENG, YAN JIANG, AND FENGYAN LI Abstract. Te propagation of electromagnetic waves
More informationarxiv: v1 [math.na] 3 Oct 2018
DISPERSION ANALYSIS OF FINITE DIFFERENCE AND DISCONTINUOUS GALERKIN SCHEMES FOR MAXELL S EQUATIONS IN LINEAR LORENTZ MEDIA YAN JIANG, PUTTHA SAKKAPLANGKUL, VRUSHALI A. BOKIL, YINGDA CHENG, AND FENGYAN
More informationStability and dispersion analysis of high order FDTD methods for Maxwell s equations in dispersive media
Contemporary Mathematics Volume 586 013 http://dx.doi.org/.90/conm/586/11666 Stability and dispersion analysis of high order FDTD methods for Maxwell s equations in dispersive media V. A. Bokil and N.
More information13.1 Ion Acoustic Soliton and Shock Wave
13 Nonlinear Waves In linear theory, the wave amplitude is assumed to be sufficiently small to ignore contributions of terms of second order and higher (ie, nonlinear terms) in wave amplitude In such a
More informationElectromagnetically Induced Flows in Water
Electromagnetically Induced Flows in Water Michiel de Reus 8 maart 213 () Electromagnetically Induced Flows 1 / 56 Outline 1 Introduction 2 Maxwell equations Complex Maxwell equations 3 Gaussian sources
More informationOptical Solitons. Lisa Larrimore Physics 116
Lisa Larrimore Physics 116 Optical Solitons An optical soliton is a pulse that travels without distortion due to dispersion or other effects. They are a nonlinear phenomenon caused by self-phase modulation
More informationApplying Asymptotic Approximations to the Full Two-Fluid Plasma System to Study Reduced Fluid Models
0-0 Applying Asymptotic Approximations to the Full Two-Fluid Plasma System to Study Reduced Fluid Models B. Srinivasan, U. Shumlak Aerospace and Energetics Research Program, University of Washington, Seattle,
More informationCONSTRUCTION AND ANALYSIS OF WEIGHTED SEQUENTIAL SPLITTING FDTD METHODS FOR THE 3D MAXWELL S EQUATIONS
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 5 Number 6 Pages 747 784 c 08 Institute for Scientific Computing and Information CONSTRUCTION AND ANALYSIS OF WEIGHTED SEQUENTIAL SPLITTING
More informationNumerical resolution of discontinuous Galerkin methods for time dependent. wave equations 1. Abstract
Numerical resolution of discontinuous Galerkin methods for time dependent wave equations Xinghui Zhong 2 and Chi-Wang Shu Abstract The discontinuous Galerkin DG method is known to provide good wave resolution
More informationDerivation of the General Propagation Equation
Derivation of the General Propagation Equation Phys 477/577: Ultrafast and Nonlinear Optics, F. Ö. Ilday, Bilkent University February 25, 26 1 1 Derivation of the Wave Equation from Maxwell s Equations
More information3 Constitutive Relations: Macroscopic Properties of Matter
EECS 53 Lecture 3 c Kamal Sarabandi Fall 21 All rights reserved 3 Constitutive Relations: Macroscopic Properties of Matter As shown previously, out of the four Maxwell s equations only the Faraday s and
More informationMacroscopic dielectric theory
Macroscopic dielectric theory Maxwellʼs equations E = 1 c E =4πρ B t B = 4π c J + 1 c B = E t In a medium it is convenient to explicitly introduce induced charges and currents E = 1 B c t D =4πρ H = 4π
More informationNonlinear wave-wave interactions involving gravitational waves
Nonlinear wave-wave interactions involving gravitational waves ANDREAS KÄLLBERG Department of Physics, Umeå University, Umeå, Sweden Thessaloniki, 30/8-5/9 2004 p. 1/38 Outline Orthonormal frames. Thessaloniki,
More informationHigh Order Accurate Runge Kutta Nodal Discontinuous Galerkin Method for Numerical Solution of Linear Convection Equation
High Order Accurate Runge Kutta Nodal Discontinuous Galerkin Method for Numerical Solution of Linear Convection Equation Faheem Ahmed, Fareed Ahmed, Yongheng Guo, Yong Yang Abstract This paper deals with
More informationNONLINEAR OPTICS. Ch. 1 INTRODUCTION TO NONLINEAR OPTICS
NONLINEAR OPTICS Ch. 1 INTRODUCTION TO NONLINEAR OPTICS Nonlinear regime - Order of magnitude Origin of the nonlinearities - Induced Dipole and Polarization - Description of the classical anharmonic oscillator
More informationFull-Wave Maxwell Simulations for ECRH
Full-Wave Maxwell Simulations for ECRH H. Hojo Plasma Research Center, University of Tsukuba in collaboration with A. Fukuchi, N. Uchida, A. Shimamura, T. Saito and Y. Tatematsu JIFT Workshop in Kyoto,
More informationPolynomial Chaos Approach for Maxwell s Equations in Dispersive Media
Polynomial Chaos Approach for Maxwell s Equations in Dispersive Media Prof. Nathan L. Gibson Department of Mathematics Applied Mathematics and Computation Seminar March 15, 2013 Prof. Gibson (OSU) PC-FDTD
More informationPart VIII. Interaction with Solids
I with Part VIII I with Solids 214 / 273 vs. long pulse is I with Traditional i physics (ICF ns lasers): heating and creation of long scale-length plasmas Laser reflected at critical density surface Fast
More informationEnergy-Conserving Numerical Simulations of Electron Holes in Two-Species Plasmas
Energy-Conserving Numerical Simulations of Electron Holes in Two-Species Plasmas Yingda Cheng Andrew J. Christlieb Xinghui Zhong March 18, 2014 Abstract In this paper, we apply our recently developed energy-conserving
More information1 Fundamentals of laser energy absorption
1 Fundamentals of laser energy absorption 1.1 Classical electromagnetic-theory concepts 1.1.1 Electric and magnetic properties of materials Electric and magnetic fields can exert forces directly on atoms
More informationPseudospectral and High-Order Time-Domain Forward Solvers
Pseudospectral and High-Order Time-Domain Forward Solvers Qing H. Liu G. Zhao, T. Xiao, Y. Zeng Department of Electrical and Computer Engineering Duke University DARPA/ARO MURI Review, August 15, 2003
More informationConservation Laws & Applications
Rocky Mountain Mathematics Consortium Summer School Conservation Laws & Applications Lecture V: Discontinuous Galerkin Methods James A. Rossmanith Department of Mathematics University of Wisconsin Madison
More informationWeighted Essentially Non-Oscillatory limiters for Runge-Kutta Discontinuous Galerkin Methods
Weighted Essentially Non-Oscillatory limiters for Runge-Kutta Discontinuous Galerkin Methods Jianxian Qiu School of Mathematical Science Xiamen University jxqiu@xmu.edu.cn http://ccam.xmu.edu.cn/teacher/jxqiu
More informationSemi-Lagrangian Formulations for Linear Advection Equations and Applications to Kinetic Equations
Semi-Lagrangian Formulations for Linear Advection and Applications to Kinetic Department of Mathematical and Computer Science Colorado School of Mines joint work w/ Chi-Wang Shu Supported by NSF and AFOSR.
More informationApplication of Nodal Discontinuous Glaerkin Methods in Acoustic Wave Modeling
1 Application of Nodal Discontinuous Glaerkin Methods in Acoustic Wave Modeling Xin Wang ABSTRACT This work will explore the discontinuous Galerkin finite element method (DG-FEM) for solving acoustic wave
More informationFourier analysis for discontinuous Galerkin and related methods. Abstract
Fourier analysis for discontinuous Galerkin and related methods Mengping Zhang and Chi-Wang Shu Abstract In this paper we review a series of recent work on using a Fourier analysis technique to study the
More informationStability and instability of solitons in inhomogeneous media
Stability and instability of solitons in inhomogeneous media Yonatan Sivan, Tel Aviv University, Israel now at Purdue University, USA G. Fibich, Tel Aviv University, Israel M. Weinstein, Columbia University,
More informationComparison between Staggered and Unstaggered Finite-Difference Time-Domain Grids for Few-Cycle Temporal Optical Soliton Propagation
Journal of Computational Physics 161, 379 400 (000) doi:10.1006/cph.000.6460, available online at http://www.idealibrary.com on Comparison between Staggered and Unstaggered Finite-Difference Time-Domain
More informationElectromagnetic Waves Across Interfaces
Lecture 1: Foundations of Optics Outline 1 Electromagnetic Waves 2 Material Properties 3 Electromagnetic Waves Across Interfaces 4 Fresnel Equations 5 Brewster Angle 6 Total Internal Reflection Christoph
More informationComparison of a Finite Difference and a Mixed Finite Element Formulation of the Uniaxial Perfectly Matched Layer
Comparison of a Finite Difference and a Mixed Finite Element Formulation of the Uniaxial Perfectly Matched Layer V. A. Bokil a and M. W. Buksas b Center for Research in Scientific Computation a North Carolina
More informationA Space-Time Expansion Discontinuous Galerkin Scheme with Local Time-Stepping for the Ideal and Viscous MHD Equations
A Space-Time Expansion Discontinuous Galerkin Scheme with Local Time-Stepping for the Ideal and Viscous MHD Equations Ch. Altmann, G. Gassner, F. Lörcher, C.-D. Munz Numerical Flow Models for Controlled
More informationCONVERGENCE ANALYSIS OF YEE SCHEMES FOR MAXWELL S EQUATIONS IN DEBYE AND LORENTZ DISPERSIVE MEDIA
INTRNATIONAL JOURNAL OF NUMRICAL ANALYSIS AND MODLING Volume Number 4 ages 657 687 c 04 Institute for Scientific Computing and Information CONVRGNC ANALYSIS OF Y SCHMS FOR MAXWLL S QUATIONS IN DBY AND
More informationA Survey of Computational High Frequency Wave Propagation II. Olof Runborg NADA, KTH
A Survey of Computational High Frequency Wave Propagation II Olof Runborg NADA, KTH High Frequency Wave Propagation CSCAMM, September 19-22, 2005 Numerical methods Direct methods Wave equation (time domain)
More informationNonlinear Drude Model
Nonlinear Drude Model Jeremiah Birrell July 8, 009 1 Perturbative Study of Nonlinear Drude Model In this section we compare the 3rd order susceptibility of the Nonlinear Drude Model to that of the Kerr
More information(a) Show that the amplitudes of the reflected and transmitted waves, corrrect to first order
Problem 1. A conducting slab A plane polarized electromagnetic wave E = E I e ikz ωt is incident normally on a flat uniform sheet of an excellent conductor (σ ω) having thickness D. Assume that in space
More informationCONVERGENCE ANALYSIS OF YEE SCHEMES FOR MAXWELL S EQUATIONS IN DEBYE AND LORENTZ DISPERSIVE MEDIA
INTRNATIONAL JOURNAL OF NUMRICAL ANALYSIS AND MODLING Volume XX Number 0 ages 45 c 03 Institute for Scientific Computing and Information CONVRGNC ANALYSIS OF Y SCHMS FOR MAXWLL S QUATIONS IN DBY AND LORNTZ
More informationSummary of Beam Optics
Summary of Beam Optics Gaussian beams, waves with limited spatial extension perpendicular to propagation direction, Gaussian beam is solution of paraxial Helmholtz equation, Gaussian beam has parabolic
More informationThe WKB local discontinuous Galerkin method for the simulation of Schrödinger equation in a resonant tunneling diode
The WKB local discontinuous Galerkin method for the simulation of Schrödinger equation in a resonant tunneling diode Wei Wang and Chi-Wang Shu Division of Applied Mathematics Brown University Providence,
More informationAn efficient implementation of the divergence free constraint in a discontinuous Galerkin method for magnetohydrodynamics on unstructured meshes
An efficient implementation of the divergence free constraint in a discontinuous Galerkin method for magnetohydrodynamics on unstructured meshes Christian Klingenberg, Frank Pörner, Yinhua Xia Abstract
More informationFDTD for 1D wave equation. Equation: 2 H Notations: o o. discretization. ( t) ( x) i i i i i
FDTD for 1D wave equation Equation: 2 H = t 2 c2 2 H x 2 Notations: o t = nδδ, x = iδx o n H nδδ, iδx = H i o n E nδδ, iδx = E i discretization H 2H + H H 2H + H n+ 1 n n 1 n n n i i i 2 i+ 1 i i 1 = c
More informationA Discontinuous Galerkin Method for Vlasov Systems
A Discontinuous Galerkin Method for Vlasov Systems P. J. Morrison Department of Physics and Institute for Fusion Studies The University of Texas at Austin morrison@physics.utexas.edu http://www.ph.utexas.edu/
More informationElectromagnetic Waves in Materials
Electromagnetic Waves in Materials Outline Review of the Lorentz Oscillator Model Complex index of refraction what does it mean? TART Microscopic model for plasmas and metals 1 True / False 1. In the Lorentz
More informationFDM for wave equations
FDM for wave equations Consider the second order wave equation Some properties Existence & Uniqueness Wave speed finite!!! Dependence region Analytical solution in 1D Finite difference discretization Finite
More informationTime stepping methods
Time stepping methods ATHENS course: Introduction into Finite Elements Delft Institute of Applied Mathematics, TU Delft Matthias Möller (m.moller@tudelft.nl) 19 November 2014 M. Möller (DIAM@TUDelft) Time
More information1 Macroscopic Maxwell s equations
This lecture purports to the macroscopic Maxwell s equations in the differential forms and their revealation about the propagation of light in vacuum and in matter. Microscopic Maxwell s equations and
More informationGeneralized Nonlinear Wave Equation in Frequency Domain
Downloaded from orbit.dtu.dk on: Dec 03, 208 Generalized Nonlinear Wave Equation in Frequency Domain Guo, Hairun; Zeng, Xianglong; Bache, Morten Publication date: 203 Document Version Early version, also
More informationPHYS 408, Optics. Problem Set 1 - Spring Posted: Fri, January 8, 2015 Due: Thu, January 21, 2015.
PHYS 408, Optics Problem Set 1 - Spring 2016 Posted: Fri, January 8, 2015 Due: Thu, January 21, 2015. 1. An electric field in vacuum has the wave equation, Let us consider the solution, 2 E 1 c 2 2 E =
More informationDISCONTINUOUS GALERKIN METHOD FOR TIME DEPENDENT PROBLEMS: SURVEY AND RECENT DEVELOPMENTS
DISCONTINUOUS GALERKIN METHOD FOR TIME DEPENDENT PROBLEMS: SURVEY AND RECENT DEVELOPMENTS CHI-WANG SHU Abstract. In these lectures we give a general survey on discontinuous Galerkin methods for solving
More informationDevelopment of a stable coupling of the Yee scheme with linear current
Development of a stable coupling of the Yee scheme with linear current Martin Campos Pinto (LJLL), Bruno Després (LJLL) Stéphane Heuraux (IJL), Filipe Da Silva (IPFN+IST) 15 octobre 2013 Munich: Pereverzev
More informationTypical anisotropies introduced by geometry (not everything is spherically symmetric) temperature gradients magnetic fields electrical fields
Lecture 6: Polarimetry 1 Outline 1 Polarized Light in the Universe 2 Fundamentals of Polarized Light 3 Descriptions of Polarized Light Polarized Light in the Universe Polarization indicates anisotropy
More informationNumerical Oscillations and how to avoid them
Numerical Oscillations and how to avoid them Willem Hundsdorfer Talk for CWI Scientific Meeting, based on work with Anna Mozartova (CWI, RBS) & Marc Spijker (Leiden Univ.) For details: see thesis of A.
More informationChemistry 24b Lecture 23 Spring Quarter 2004 Instructor: Richard Roberts. (1) It induces a dipole moment in the atom or molecule.
Chemistry 24b Lecture 23 Spring Quarter 2004 Instructor: Richard Roberts Absorption and Dispersion v E * of light waves has two effects on a molecule or atom. (1) It induces a dipole moment in the atom
More informationGuangye Chen, Luis Chacón,
JIFT workshop! Oct. 31, 2014 New Orleans, LA.! Guangye Chen, Luis Chacón, CoCoMANs team Los Alamos National Laboratory, Los Alamos, NM 87545, USA gchen@lanl.gov 1 Los Alamos National Laboratory Motivation
More informationFull Wave Analysis of RF Signal Attenuation in a Lossy Rough Surface Cave Using a High Order Time Domain Vector Finite Element Method
Progress In Electromagnetics Research Symposium 2006, Cambridge, USA, March 26-29 425 Full Wave Analysis of RF Signal Attenuation in a Lossy Rough Surface Cave Using a High Order Time Domain Vector Finite
More informationElectrodynamics I Final Exam - Part A - Closed Book KSU 2005/12/12 Electro Dynamic
Electrodynamics I Final Exam - Part A - Closed Book KSU 2005/12/12 Name Electro Dynamic Instructions: Use SI units. Short answers! No derivations here, just state your responses clearly. 1. (2) Write an
More informationElectromagnetic fields and waves
Electromagnetic fields and waves Maxwell s rainbow Outline Maxwell s equations Plane waves Pulses and group velocity Polarization of light Transmission and reflection at an interface Macroscopic Maxwell
More informationBielefeld Course on Nonlinear Waves - June 29, Department of Mathematics University of North Carolina, Chapel Hill. Solitons on Manifolds
Joint work (on various projects) with Pierre Albin (UIUC), Hans Christianson (UNC), Jason Metcalfe (UNC), Michael Taylor (UNC), Laurent Thomann (Nantes) Department of Mathematics University of North Carolina,
More informationExact analytical Helmholtz bright and dark solitons
Exact analytical Helmholtz bright and dark solitons P. CHAMORRO POSADA Dpto. Teoría de la Señal y Comunicaciones e Ingeniería Telemática ETSI Telecomunicación Universidad de Valladolid, Spain G. S. McDONALD
More informationBasics of electromagnetic response of materials
Basics of electromagnetic response of materials Microscopic electric and magnetic field Let s point charge q moving with velocity v in fields e and b Force on q: F e F qeqvb F m Lorenz force Microscopic
More informationIntegration of Vlasov-type equations
Alexander Ostermann University of Innsbruck, Austria Joint work with Lukas Einkemmer Verona, April/May 2017 Plasma the fourth state of matter 99% of the visible matter in the universe is made of plasma
More informationPlasma waves in the fluid picture I
Plasma waves in the fluid picture I Langmuir oscillations and waves Ion-acoustic waves Debye length Ordinary electromagnetic waves General wave equation General dispersion equation Dielectric response
More informationTwo-scale numerical solution of the electromagnetic two-fluid plasma-maxwell equations: Shock and soliton simulation
Mathematics and Computers in Simulation 76 (2007) 3 7 Two-scale numerical solution of the electromagnetic two-fluid plasma-maxwell equations: Shock and soliton simulation S. Baboolal a,, R. Bharuthram
More informationIntroduction to Nonlinear Optics
Introduction to Nonlinear Optics Prof. Cleber R. Mendonca http://www.fotonica.ifsc.usp.br Outline Linear optics Introduction to nonlinear optics Second order nonlinearities Third order nonlinearities Two-photon
More informationA High Order Conservative Semi-Lagrangian Discontinuous Galerkin Method for Two-Dimensional Transport Simulations
Motivation Numerical methods Numerical tests Conclusions A High Order Conservative Semi-Lagrangian Discontinuous Galerkin Method for Two-Dimensional Transport Simulations Xiaofeng Cai Department of Mathematics
More informationRelativistic laser beam propagation and critical density increase in a plasma
Relativistic laser beam propagation and critical density increase in a plasma Su-Ming Weng Theoretical Quantum Electronics (TQE), Technische Universität Darmstadt, Germany Joint work with Prof. Peter Mulser
More informationDesign of optimal Runge-Kutta methods
Design of optimal Runge-Kutta methods David I. Ketcheson King Abdullah University of Science & Technology (KAUST) D. Ketcheson (KAUST) 1 / 36 Acknowledgments Some parts of this are joint work with: Aron
More information9 Atomic Coherence in Three-Level Atoms
9 Atomic Coherence in Three-Level Atoms 9.1 Coherent trapping - dark states In multi-level systems coherent superpositions between different states (atomic coherence) may lead to dramatic changes of light
More informationA DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR TIME DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS WITH HIGHER ORDER DERIVATIVES
MATHEMATCS OF COMPUTATON Volume 77, Number 6, April 008, Pages 699 730 S 005-5718(07)0045-5 Article electronically published on September 6, 007 A DSCONTNUOUS GALERKN FNTE ELEMENT METHOD FOR TME DEPENDENT
More informationSingularity Formation in Nonlinear Schrödinger Equations with Fourth-Order Dispersion
Singularity Formation in Nonlinear Schrödinger Equations with Fourth-Order Dispersion Boaz Ilan, University of Colorado at Boulder Gadi Fibich (Tel Aviv) George Papanicolaou (Stanford) Steve Schochet (Tel
More informationBeam propagation method for waveguide device simulation
1/29 Beam propagation method for waveguide device simulation Chrisada Sookdhis Photonics Research Centre, Nanyang Technological University This is for III-V Group Internal Tutorial Overview EM theory,
More informationElectromagnetic wave propagation in complex dispersive media
Electromagnetic wave propagation in complex dispersive media Associate Professor Department of Mathematics Workshop on Quantification of Uncertainties in Material Science January 15, 2016 (N.L. Gibson,
More informationChapter 3. Finite Difference Methods for Hyperbolic Equations Introduction Linear convection 1-D wave equation
Chapter 3. Finite Difference Methods for Hyperbolic Equations 3.1. Introduction Most hyperbolic problems involve the transport of fluid properties. In the equations of motion, the term describing the transport
More informationHierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws
Hierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws Dedicated to Todd F. Dupont on the occasion of his 65th birthday Yingjie Liu, Chi-Wang Shu and Zhiliang
More informationMEMORANDUM-4. n id (n 2 + n2 S) 0 n n 0. Det[ɛ(ω, k)]=0 gives the Dispersion relation for waves in a cold magnetized plasma: ω 2 pα ω 2 cα ω2, ω 2
Fundamental dispersion relation MEMORANDUM-4 n S) id n n id n + n S) 0 } n n 0 {{ n P } ɛω,k) E x E y E z = 0 Det[ɛω, k)]=0 gives the Dispersion relation for waves in a cold magnetized plasma: n P ) [
More informationThe Finite-Difference Time-Domain (FDTD) Algorithm
The Finite-Difference Time-Domain (FDTD Algorithm James R. Nagel 1. OVERVIEW It is difficult to overstate the importance of simulation to the world of engineering. Simulation is useful because it allows
More informationElectromagnetic Modeling and Simulation
Electromagnetic Modeling and Simulation Erin Bela and Erik Hortsch Department of Mathematics Research Experiences for Undergraduates April 7, 2011 Bela and Hortsch (OSU) EM REU 2010 1 / 45 Maxwell s Equations
More informationAn electric field wave packet propagating in a laser beam along the z axis can be described as
Electromagnetic pulses: propagation & properties Propagation equation, group velocity, group velocity dispersion An electric field wave packet propagating in a laser beam along the z axis can be described
More informationOverview in Images. S. Lin et al, Nature, vol. 394, p , (1998) T.Thio et al., Optics Letters 26, (2001).
Overview in Images 5 nm K.S. Min et al. PhD Thesis K.V. Vahala et al, Phys. Rev. Lett, 85, p.74 (000) J. D. Joannopoulos, et al, Nature, vol.386, p.143-9 (1997) T.Thio et al., Optics Letters 6, 197-1974
More informationThe Finite-Difference Time-Domain (FDTD) Algorithm
The Finite-Difference Time-Domain (FDTD) Algorithm James R. Nagel Overview: It is difficult to overstate the importance of simulation to the world of engineering. Simulation is useful because it allows
More informationPlasma Processes. m v = ee. (2)
Plasma Processes In the preceding few lectures, we ve focused on specific microphysical processes. In doing so, we have ignored the effect of other matter. In fact, we ve implicitly or explicitly assumed
More informationProgramming of the Generalized Nonlinear Paraxial Equation for the Formation of Solitons with Mathematica
American Journal of Applied Sciences (): -6, 4 ISSN 546-99 Science Publications, 4 Programming of the Generalized Nonlinear Paraxial Equation for the Formation of Solitons with Mathematica Frederick Osman
More informationChapter 1. Introduction to Nonlinear Space Plasma Physics
Chapter 1. Introduction to Nonlinear Space Plasma Physics The goal of this course, Nonlinear Space Plasma Physics, is to explore the formation, evolution, propagation, and characteristics of the large
More informationHydraulic Modelling for Drilling Automation
Hydraulic Modelling for Drilling Automation CASA Day Harshit Bansal April 19, 2017 Where innovation starts Team: Supervisors at TU/e : W.H.A. Schilders, N. van de Wouw, B. Koren, L. Iapichino Collaborators:
More informationElectromagnetic Wave Propagation Lecture 13: Oblique incidence II
Electromagnetic Wave Propagation Lecture 13: Oblique incidence II Daniel Sjöberg Department of Electrical and Information Technology October 2016 Outline 1 Surface plasmons 2 Snel s law in negative-index
More informationImplicit-Explicit Time Integration of a High-Order Particle-in-Cell Method with Hyperbolic Divergence Cleaning.
Implicit-Explicit Time Integration of a High-Order Particle-in-Cell Method with Hyperbolic Divergence Cleaning. G.B. Jacobs a J.S. Hesthaven b a Department of Aerospace Engineering & Engineering Mechanics,
More information9 The conservation theorems: Lecture 23
9 The conservation theorems: Lecture 23 9.1 Energy Conservation (a) For energy to be conserved we expect that the total energy density (energy per volume ) u tot to obey a conservation law t u tot + i
More informationA local-structure-preserving local discontinuous Galerkin method for the Laplace equation
A local-structure-preserving local discontinuous Galerkin method for the Laplace equation Fengyan Li 1 and Chi-Wang Shu 2 Abstract In this paper, we present a local-structure-preserving local discontinuous
More informationA Comparison between the Two-fluid Plasma Model and Hall-MHD for Captured Physics and Computational Effort 1
A Comparison between the Two-fluid Plasma Model and Hall-MHD for Captured Physics and Computational Effort 1 B. Srinivasan 2, U. Shumlak Aerospace and Energetics Research Program University of Washington,
More informationFDM for parabolic equations
FDM for parabolic equations Consider the heat equation where Well-posed problem Existence & Uniqueness Mass & Energy decreasing FDM for parabolic equations CNFD Crank-Nicolson + 2 nd order finite difference
More informationFull-wave Electromagnetic Field Simulations in the Lower Hybrid Range of Frequencies
Full-wave Electromagnetic Field Simulations in the Lower Hybrid Range of Frequencies P.T. Bonoli, J.C. Wright, M. Porkolab, PSFC, MIT M. Brambilla, IPP, Garching, Germany E. D Azevedo, ORNL, Oak Ridge,
More informationAn iterative solver enhanced with extrapolation for steady-state high-frequency Maxwell problems
An iterative solver enhanced with extrapolation for steady-state high-frequency Maxwell problems K. Hertel 1,2, S. Yan 1,2, C. Pflaum 1,2, R. Mittra 3 kai.hertel@fau.de 1 Lehrstuhl für Systemsimulation
More informationMODELING OF AN ECR SOURCE FOR MATERIALS PROCESSING USING A TWO DIMENSIONAL HYBRID PLASMA EQUIPMENT MODEL. Ron L. Kinder and Mark J.
TECHCON 98 Las Vegas, Nevada September 9-11, 1998 MODELING OF AN ECR SOURCE FOR MATERIALS PROCESSING USING A TWO DIMENSIONAL HYBRID PLASMA EQUIPMENT MODEL Ron L. Kinder and Mark J. Kushner Department of
More informationSpatial and Modal Superconvergence of the Discontinuous Galerkin Method for Linear Equations
Spatial and Modal Superconvergence of the Discontinuous Galerkin Method for Linear Equations N. Chalmers and L. Krivodonova March 5, 014 Abstract We apply the discontinuous Galerkin finite element method
More informationDiscontinuous Galerkin method for hyperbolic equations with singularities
Discontinuous Galerkin method for hyperbolic equations with singularities Chi-Wang Shu Division of Applied Mathematics Brown University Joint work with Qiang Zhang, Yang Yang and Dongming Wei Outline Introduction
More informationWave Turbulence and Condensation in an Optical Experiment
Wave Turbulence and Condensation in an Optical Experiment S. Residori, U. Bortolozzo Institut Non Linéaire de Nice, CNRS, France S. Nazarenko, J. Laurie Mathematics Institute, University of Warwick, UK
More informationTheory of optical pulse propagation, dispersive and nonlinear effects, pulse compression, solitons in optical fibers
Theory of optical pulse propagation, dispersive and nonlinear effects, pulse compression, solitons in optical fibers General pulse propagation equation Optical pulse propagation just as any other optical
More informationHigh Order Semi-Lagrangian WENO scheme for Vlasov Equations
High Order WENO scheme for Equations Department of Mathematical and Computer Science Colorado School of Mines joint work w/ Andrew Christlieb Supported by AFOSR. Computational Mathematics Seminar, UC Boulder
More informationElectromagnetic Wave Propagation Lecture 13: Oblique incidence II
Electromagnetic Wave Propagation Lecture 13: Oblique incidence II Daniel Sjöberg Department of Electrical and Information Technology October 15, 2013 Outline 1 Surface plasmons 2 Snel s law in negative-index
More informationPositivity-preserving high order schemes for convection dominated equations
Positivity-preserving high order schemes for convection dominated equations Chi-Wang Shu Division of Applied Mathematics Brown University Joint work with Xiangxiong Zhang; Yinhua Xia; Yulong Xing; Cheng
More informationLet s consider nonrelativistic electrons. A given electron follows Newton s law. m v = ee. (2)
Plasma Processes Initial questions: We see all objects through a medium, which could be interplanetary, interstellar, or intergalactic. How does this medium affect photons? What information can we obtain?
More information