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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Exponential Recursion for Multi-Scale Problems in Electromagnetics

Matthew F. Causley

Department of Mathematics Kettering University

Computational Aspects of Time Dependent Electromagnetic Wave Problems in Complex Materials ICERM, Brown University June 28, 2018

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Outline

1 Introduction and Motivation 2 Havriliak-Negami Model 3 Wave Solver 4 Conclusions

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Table of Contents

1 Introduction and Motivation 2 Havriliak-Negami Model 3 Wave Solver 4 Conclusions

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Exponential recursion is a simple concept, but a powerful tool. We focus on EM wave propagation in complex media. (1) Anomalous dielectric relaxation (fractional relaxation models)

1 Complex hetergoeneous materials (soil, biological tissues) 2 Empirical dispersion models involve (iω)α. 3 Power-law decay, requires time history of fields. 4 Useful for similar problems in acoustics, solid mechanics, etc.

(2) Plasmas

1 Plasma phenomena occur at vastly disparate time scales. 2 Geometry, fine spatial scales must be resolved. 3 Experiments often have non-local effects. 4 Pros and Cons of kinetic vs. fluid models of plasma.

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

In time, exponential recursion truncates time history. Consider ˙ φ + yφ = f (t), 0 < t < T Multiply by integrating factor, and integrate over [t − δ, t] t

t−δ

• eytφ

′ dt = t

t−δ

eytf (t) dt eytφ(t) − ey(t−δ)φ(t − δ) = t

t−δ

eyτf (τ) dτ Rearranging, we find the exponential recursion φ(t) = e−yδφ(t − δ) + δ e−yuf (t − u) du. Discretize using any exponential integrator.

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

In space, exponential recursion localizes the solution. Consider w − 1 α2 w′′ = u = ⇒ wp(x) = α 2 b

a

e−α|x−y|u(y) dy Split the integral at y = x, and let wp = wL + wR. Then wL(x) = α 2 x

a

e−α(y−x)u(y) dy, wR(x) = α 2 b

x

e−α(x−y)u(y) dy. A few steps of algebra produce the exponential recursion wL(x) = e−αδLwL(x − δL) + δL e−αzu(x − z) dz wR(x) = e−αδRwR(x + δR) + δR e−αzu(x + z) dz. Discretize using any collocation method.

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Table of Contents

1 Introduction and Motivation 2 Havriliak-Negami Model 3 Wave Solver 4 Conclusions

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Consider EM propagation through a region Ω ∂D ∂t = ∇ × H − J, ∇ · D = ρ ∂B ∂t = −∇ × E, ∇ · B = 0 In the absence of external charges and currents, ρ = 0, J = 0. Assume the material is non-magnetic, so B = µ0H. Anomalous dielectric relaxation, ˆ D = ǫ0ˆ ǫ ˆ E, where ˆ ǫ(s) = ǫ∞ + ∆ǫ (1 + (sτ)α)β with ǫ∞ ≥ 1, ∆ǫ > 0, 0 < α, β ≤ 1.

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Decompose D = ǫ0 (ǫ∞E + P), where (I + τ α

cDα t )β P = ∆ǫE

fractional PDO! Solve for the polarization using Laplace transforms. P(x, t) = t χ(u)E(x, t − u) du, ˆ χ(s) = ∆ǫ (1 + (sτ)α)β χ(t) ∼

• tαβ−1

t ≪ 1 t−α−1 t ≫ 1

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

We first re-cast the susceptibility as an integral over R χ(t) = ∆ǫ 2πi ζ+i∞

ζ−i∞

est (1 + (sτ)α)β ds = ∞ f (y)e−yt/τ dy = ∞

−∞

f (ez)ez−ezt/τ dz, where f (y) = ∆ǫ πτ sin

• β cos−1
• yα cos (πα)+1

√

y2α+2 cos (πα)yα+1

• (y2α + 2 cos (πα)yα + 1)β/2

.

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

In general, consider K(t) = ∞ f (y)e−yt dy for non-negative f ∈ L1(R+). We set y = ez. Then, for h > 0 the Poisson summation formula yields h

∞

• n=−∞

f (enh)enh−enht =

∞

• k=−∞

ˆ f 2πk h

• where

ˆ f (k) = ∞

−∞

• f (ez)ez−ezt

eikz dz, ˆ f (0) = K(t) Under mild assumptions, the integrand is analytic for Im{z} ≤ θ. |ˆ f (k)| ≤ Cke−|k|θ.

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

K(t) = h

∞

• n=−∞

f (enh)enh−enht −

• k=0

ˆ f 2πk h

• ≈ h
• nh<zr

f (enh)enh−enht + O

• e−π2/h
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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Given ǫ > 0, for t ∈ [∆t, T]

1 Discretization: Choose h so that e−π2/h ≈ ǫ. 2 Truncation: Choose zr so that hf (ezr )ezr−∆tezr ≈ ǫ. 3 Compression: Choose zℓ so that a minimal number J

(typically 2 or 3) compressed nodes satisfies h

• z≤zℓ

f (enh)enh−enhT =

J

• j=1

wje−yjT + O(ǫ) Next, merge the weights and nodes. Then, we have a uniform relative error bound max

∆t<t<T

• K(t) −

M

• m=1

wme−ymt

• ≤ K(t)ǫ
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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

For t ∈ [0, T], P(x, t) = ∆t χ(u)E(x, t − u) du +

M

• m=1

wmφm(t) + E(t) where we have the diagonalized linear system ˙ φm + ymφm = E. Discretize using any exponential integrator. For E ∈ L2([0, T]), ||E(t)||L2 ≤

• χ(t) −

M

• m=1

wme−ymt

• L1

||E(x, t)||L2 ≤ T

• χ(t) −

M

• m=1

wme−ymt

• L∞

||E(x, t)||L2 ≤ ǫT||E(x, t)||L2

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Replace the electric field with a polynomial interpolant, and perform product integration to arrive at1 P(t) =

L

• ℓ=0

AℓE(x, t − ℓ∆t) +

M

• m=1

wmφm(t) φm(x, t) = e−ym∆tφm(x, t − ∆t) +

L

• ℓ=0

Bℓ,mE(x, t − ℓ∆t). The polarization law can now be evaluated with L levels of time history, and M = O (log Nt) terms in memory. The operation count is O(LM). Here, Nt = ⌈ T

∆t ⌉.

1JCP 2013, with S. Jiang and P. Petropoulos

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

• Theorem. The numerical scheme, based on the second order

accurate FDTD method, and is stable under the standard CFL stability condition. We solve a signaling problem (1d TEM wave) using the finite difference time domain (FDTD) technique, with square pulse E(0, t) = 1 td (H(t) − H(t − td)) .

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Similar asymptotics for Cole-Cole, Cole-Davidson models. 2

2IEEE Trans. Ant. 2011 with P. Petropoulos

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Table of Contents

1 Introduction and Motivation 2 Havriliak-Negami Model 3 Wave Solver 4 Conclusions

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Consider the Vlasov-Maxwell system for a plasma ǫ0µ0 ∂E ∂t = ∇ × B − µ0J, ∇ · E = ρ ǫ0 ∂B ∂t = −∇ × E, ∇ · B = 0 ρ = q

• v

f (x, v, t) dv, J = q

• v

vf (x, v, t) dv ∂f ∂t + v · ∇xf + F m · ∇vf = 0 F = q (E + v × B)

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Define EM fields using scalar potential φ and vector potential A by E = −∂A ∂t − ∇φ, B = ∇ × A. Impose the Lorenz gauge

1 c2 ∂φ ∂t + ∇ · A = 0, so that

1 c2 ∂2A ∂t2 − ∇2A = µ0J 1 c2 ∂2φ ∂t2 − ∇2φ = 1 ǫ0 ρ

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

1 Apply the method of lines transpose (MOLT)

Discretize in time first. Re-formulate as a semi-discrete boundary integral.

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

1 Apply the method of lines transpose (MOLT)

Discretize in time first. Re-formulate as a semi-discrete boundary integral.

2 Build a (1d) spatial solver

Spatial discretization over a (perhaps nonuniform) mesh. Fast O(N) matrix-free convolution, via exponential recursion.

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

1 Apply the method of lines transpose (MOLT)

Discretize in time first. Re-formulate as a semi-discrete boundary integral.

2 Build a (1d) spatial solver

Spatial discretization over a (perhaps nonuniform) mesh. Fast O(N) matrix-free convolution, via exponential recursion.

3 Multi-dimensional solver

Spatial solver works in a ”line-by-line” fashion. Embedded boundaries. Local interpolation for normal derivatives.

• M. Causley

Kettering University Exponential Recursion 22 / 49

Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

1 Apply the method of lines transpose (MOLT)

Discretize in time first. Re-formulate as a semi-discrete boundary integral.

2 Build a (1d) spatial solver

Spatial discretization over a (perhaps nonuniform) mesh. Fast O(N) matrix-free convolution, via exponential recursion.

3 Multi-dimensional solver

Spatial solver works in a ”line-by-line” fashion. Embedded boundaries. Local interpolation for normal derivatives.

4 Raise order of accuracy in time

Taylor expansion reformulated with convolution operators. Coefficients determined with resolvent expansion. Stability guaranteed by introducing a free parameter β > 0.

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

1 c2 utt − ∇2u = S

2Math Comp. 2014 with A. Christlieb, B. Ong, and L. Van Groningen

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

1 c2 utt − ∇2u = S Discretize utt in time utt(x, tn) ≈ un+1 − 2un + un−1 ∆t2 .

2Math Comp. 2014 with A. Christlieb, B. Ong, and L. Van Groningen

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

1 c2 utt − ∇2u = S Discretize utt in time utt(x, tn) ≈ un+1 − 2un + un−1 ∆t2 . The Laplacian is treated semi-implicitly, so that un+1 − 2un + un−1 (c∆t2) − ∇2

• un + un+1 − 2un + un−1

β2

• ≈ Sn,

with 0 < β ≤ 2.

2Math Comp. 2014 with A. Christlieb, B. Ong, and L. Van Groningen

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

L

• un + un+1 − 2un + un−1

β2

• = un + Sn

α2 , where the modified Helmholtz operator is L[u](x) :=

• 1 − 1

α2 ∇2

• u(x),

α = β c∆t . Update equation un+1 = 2un − un−1 − β2D[un](x) + β2L−1 Sn α2

• (x),

where D[u](x) := u(x) − L−1[u](x) ≈ − 1 α2 ∇2u.

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Rather than invert the 3d Helmholtz operator, we utilize dimensional splitting L ≈ LxLyLz In one spatial dimension, un+1 = 2un − un−1 − β2D[un](x) + β2L−1 Sn α2

• (x),

where L−1[u](x) := α 2 b

a

u(y)e−α|x−y|dy

• Particular Solution

+ Ae−α(x−a) + Be−α(b−x)

• Homogeneous Solution

. Theorem: This semi-discrete scheme is second order accurate in time, and A-stable for β ∈ (0, 2].

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

wp(x) = I[u](x) = α 2 b

a

e−α|x−y|u(y)dy. Split the particular solution at y = x, I[u](x) = α 2 x

a

e−α(x−y)u(y)dy

• I L(x)

+ α 2 b

x

e−α(y−x)u(y)dy

• I R(x)

. Each ”characteristic” is updated using exponential recursion I L(x + δ) =e−αδI L(x) + α 2 x+δ

x

e−α(xj−y)u(y)dy I R(x − δ) =e−αδI R(x) + α 2 x

x−δ

e−α(y−xj)u(y)dy.

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Partition [a, b] into N subintervals [xj−1, xj], hj = xj − xj−1. Replace u with a local Lagrange interpolant, of order 2M. Then I L(xj) = e−αhjI L(xj−1) +

M

• k=−M

wL

k u(xj+k),

j = 1, . . . N, I R(xj) = e−αhj+1I R(xj+1) +

M

• k=−M

wR

k u(xj+k),

j = N − 1, . . . 0. Convolution computed in O(MN) operations.

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Transmission conditions can be formulated, for domain decomposition and (1d) outflow BCs. L−1[u](x) := α 2 b

a

u(y)e−α|x−y|dy

• Particular Solution

+ Ae−α(x−a) + Be−α(b−x)

• Homogeneous Solution

. Compare with the free space solution. Then, for x ∈ [a, b], we see A(t) = α ∞ u(a − y, t)e−αydy. Using exponential recursion, storing time history at the boundary An = α a

a−ctn

e−α(a−y)u(y, tn)dy = e−βAn−1 + α c∆t e−αyu

• a, tn − y

c

• dy.
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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

The exponential recursion can also be employed hierarchically.3

Ω Ωm Ω0 ΩM−1

JR

m

JL

m+1

(a) Fine-Coarse Pass

Ω Ωm Ω0 ΩM−1

Am Bm

(b) Coarse-Fine Pass

I L(Xj) = e−α(Xj−Xj−1)I L(Xj−1) + JL(Xj), JL(Xj) = α Xj

Xj−1

e−α(Xj−y)u(y)dy

1 Compute each local particular solution on Ωj. 2 Find particular solution using global exponential recursion. 3 Pass transmission conditions, boundary conditions to each Ωj.

3Arxiv, with A. Christlieb, Y. Guclu and E. Wolf

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Undergraduate thesis with C. Seipp

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

In order to approximate higher order powers of the Laplacian

• perator using dimensional splitting4, we first define

Lγ := 1 − ∂2

γ

α2 , Dγ := 1 − L−1

γ ,

γ = x, y, z, and Cxyz := L−1

y L−1 z Dx + L−1 z L−1 x Dy + L−1 x L−1 y Dz.

so that

• −∇2

α2 m = Cm

xyz ∞

• p=m

p − 1 m − 1

• Dp−m

xyz .

4SINUM 2013, with A. Christlieb

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

un+1 − 2un + un−1 =

∞

• m=1

2β2m (2m)! ∇2 α2 m un =

∞

• m=1

(−1)m 2β2m (2m)!Cm

∞

• p=m

p − 1 m − 1

• Dp−m[un]

=

∞

• p=1

p

• m=1

(−1)m 2β2m (2m)! p − 1 m − 1

• CmDp−m[un]

=

P

• p=1

p

• m=1

Ap,m(β)CmDp−m[un] + O(∆t2P+2).

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

P = 1 P = 2 P = 3 ∆t Error Rate Time (s) Error Rate Time (s) Error Rate Time (s) 0.4 7.81E-1 * 1.5 7.19E-1 * 4.1 8.16E-1 * 8.2 0.2 2.46E-1 1.67 3.9 1.07E-1 2.74 11.2 7.80E-2 3.39 19.7 0.1 7.15E-2 1.78 7.1 1.03E-2 3.38 23.3 2.83E-3 4.78 44.1 0.05 1.89E-2 1.92 15.1 7.36E-4 3.81 48.3 5.74E-5 5.63 90.0 0.025 4.84E-3 1.96 30.0 4.97E-5 3.89 94.2 2.29E-6 4.64 186.2

Table: Refinement and computational efficiency for a 2d rectangular mode u(x, y, 0) = sin(πx) sin(πy). The mesh is held fixed at ∆x = ∆y = 0.003125.

The algorithm scales linearly with the number of spatial points.

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

The characteristic polynomial satisfied by the amplification factor is ρ2 − 2ρ + 1 = ρ  

P

• p=1

Ap(β) ˆ Dp   , β > 0, 0 ≤ ˆ D ≤ 1.

• Def. The scheme will be A-stable provided that |ρ| ≤ 1.
• Theorem. For each finite P, there exists βmax such that the

semi-discrete scheme will be A-stable for 0 < β ≤ βmax, and where

P

• p=1

Ap(βmax) = 4. P 1 2 3 4 5 Order 2 4 6 8 10 βmax 2 1.4840 1.2345 1.0795 0.9715

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

(a) 2nd order (b) 4th order

Figure: Propagation due to a point source in 2d, on a 80 × 80 mesh,with CFL number 2.

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

1 Add a ghost point, exterior to the domain.5 2 Interpolate uI and uII from local interior grid points (bilinear). 3 Interpolant through uI and uII, such that ∂u ∂n(ξB) = 0. 4 Extrapolate to find uG, and update only local grid points.

Iterate to convergence.

• 5J. Sci. Comp. 2017, with A. Christlieb and E. Wolf
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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

C =

• (x, y) :

x + y 4 2 + (x − y)2 = 1

• .

−2 −1 1 2 −2 −1 1 2 x y

(a) x-sweep

−2 −1 1 2 −2 −1 1 2 x y

(b) y-sweep

Figure: The boundary points (red) close each line of the x and y sweeps.

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Solution is 6th order in time and space. No stability restriction.

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

(a) t = 0 (b) t = 0.1 (c) t = 0.2 (d) t = 0.25 (e) t = 0.45 (f) t = 0.7 (g) t = 0.8 (h) t = 1.0

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Implicit PIC, quasi-electrostatics.6 Initial Prediction

1 Charge:

{ξn

i } → ρn. 2 Potential (MOLT):

φn−1, φn, ρn → φ∗.

3 Positions:

ξ∗

i = ξn i + ∆tvn i . 4 Fields:

φ∗ → E ∗ → E ∗

i =

E ∗(ξ∗

i ).

Correction Iteration

5 Velocities:

v∗

i = vn i + ∆t(α

E ∗

i + (1 − α)

E n

i ). 6 Positions:

ξ∗

i = ξn i + ∆t(αv∗ i + (1 − α)vn i ). 7 Charge:

{ξ∗

i } → ρ∗. 8 Potential/Fields:

φn−1, φn, ¯ ρ → φ∗ → E ∗

i . 9 Repeat steps 5-8 to convergence.

6JCP 2016, with M. Bettencourt A. Christlieb and E. Wolf

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

1 2 3 4 5 6 7 x 10

−12

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10

−18

Total Energy time (s) energy (J)

α = 0.5 α = 0.6 α = 1

50 periods of an electron-ion oscillatory system. The CFL is 10, with 100 cells in the domain. Standard spatial smoothing operators are applied to grid quantities. The relaxation parameter is varied, no grid heating is observed when α = 0.5.

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

1 2 3 4 5 6 7 x 10

−12

1 2 3 4 5 6 x 10

−18

Kinetic, Field and Total Energies, α = 0.5 time (s) energy (J)

Field Energy Kinetic Energy Total Energy

Total energy is conserved, as the underlying field solver is non-dissipative.

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Ph.D. thesis of M. Thavappiragasam

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Ph.D. thesis of M. Thavappiragasam

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Table of Contents

1 Introduction and Motivation 2 Havriliak-Negami Model 3 Wave Solver 4 Conclusions

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Future Work:

1 Analysis of outflow boundary conditions 2 Domain decomposition in multiple dimensions 3 Consistent treatment of particles 4 Fully parallel 3d Maxwell solver

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Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions

Thank you!

References The Christlieb Group

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