Forum for Electromagnetic Research Methods and Application Technologies (FERMAT)

The Precise Integration Time Domain (PITD) Method - A Supplement to the Computational Electromagnetics

Xikui Ma1 and Tianyu Dong 1,2

1 State Key Laboratory of Electrical Insulation and Power Equipment, School of Electrical

Engineering, Xi'an Jiaotong University, Xi'an 710049, China

2 State Key Lab for Strength and Vibration of Mechanical Structures, School of Aerospace,

Xi'an Jiaotong University, Xi'an 710049, China

Abstract: The finite-difference time-domain (FDTD) method [1] is one of the most widely used full-wave

electromagnetic simulation tools for solving Maxwell’s curl equations due to its simplicity,

straightforwardness and easy implementation. However, the Courant-Friedrich-Levy (CFL) stability

condition, i.e., Δ𝑡FDTD ≤ 1/ (𝑣√1

Δ𝑥2+

1

Δ𝑦2+

1

Δ𝑧2), and the increasing numerical dispersion error have

limited the intense utilization of the FDTD. Thus, the development of accurate and efficient numerical

algorithms for solving Maxwell’s equations is still an attractive topic in our computational electromagnetics

(CEM) society.

The precise integration time domain (PITD) method [2] is proposed for solving Maxwell’s equations,

which breaks through the CFL limit and improves the computational efficiency. The fundamental idea of

the PITD method consists of reducing Maxwell’s curl equations to a set of ordinary differential equations

(ODEs) by approximating the spatial derivative with difference only, and solving the ODEs by using the

precise integration (PI) technique with an accuracy of machinery precision. Compared to the

convectional FDTD method, the PITD method posseses some striking features, which are summerized

hereinafter.

First, different with that of the conventional FDTD method, the stability condition of the PITD

method can be expressed as Δ𝑡PITD ≤ √2𝑙/ (𝑣√1

Δ𝑥2+

1

Δ𝑦2+

1

Δ𝑧2) , where 2Nl and N is a preselected

integer. It can be clearly seen that the upper limit of the time-step size of the PITD method is determined by

the preselected integer N and the spatial mesh size. Hence, the requirement of the time-step size can be

satisfied by the alternative choices of the integer N . Meanwhile, it should be emphasized that the time-step

size of the PITD method is much larger than that of the FDTD method.

Second, although the numerical dispersion error of the PITD method is slightly larger than that of the

FDTD method, the numerical dispersion performance of the PITD method is much better than other

unconditionally stable algorithms, such as alternating direction implicit FDTD (ADI-FDTD) method.

Furthermore, the numerical dispersion errors can be made nearly independent of the time-step size [3, 4].

Finally, the PITD method has advantages over the conventional Yee’s FDTD method in using a large

time step, and over the ADI-FDTD method in having high computational accuracy, respectively, and

numerical dispersion errors of the PITD scheme can be made nearly independent of the time-step size.

Therefore, we can use large time-step size to solve Maxwell’s equations without increasing the

computational error.

Due to its excellent performance on computational electromagnetics, the PITD method attracts more

and more attentions [2-7]. Recently, various low-dispersion and low-memory-requirement PITD methods

have been proposed, such as the fourth-order accurate PITD [PITD (4)] method [3], the wavelet Galerkin

scheme-based PITD (WG-PITD) method [4], the leapfrog scheme-based PITD (L-PITD) method [5], the

unified split-step PITD (SS-PITD) method [6], the compact PITD (CPITD) method [7] and so on. With the

rapid development of the computer hardware technology, the PITD method still has

considerable development potential in the future.

Keywords: computational electromagnetics, matrix-exponential methods, time-domain analysis, large time-steps, and precise integration technique.

References

[1] K. S. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE

Trans. Antennas Propag., 3 (1966) 302–307.

[2] X. K. Ma, X. T. Zhao, and Y. Z. Zhao, A 3-D precise integration time-domain method without the restrains of the Courant-

Friedrich-Levy stability condition for the numerical solution of Maxwell’s equations, IEEE Trans. Microw. Theory Tech.

7 (2006) 3026-3037.

[3] Z. M. Bai, X. K. Ma, and G. Sun, A low-dispersion realization of precise integration time-domain method using a fourth-

order accurate finite difference scheme, IEEE Trans. Antennas Propag., 4 (2011) 1311-1320.

[4] G. Sun, X. K. Ma, and Z. M. Bai, A low dispersion precise integration time domain method based on wavelet Galerkin

scheme, IEEE Microw. Wireless Compon. Lett., 12 (2010) 651-653.

[5] G. Sun, X. K. Ma, and Z. M. Bai, A low-memory-requirement realization of precise integration time domain method using

a leapfrog scheme, IEEE Microw. Wireless Compon. Lett., 6 (2012) 294-296.

[6] Q. Liu, X. K. Ma, and F. Chen, Unified split-step precise integration time-domain method for dispersive media, Electron.

lett. 18 (2013) 1135-1136.

[7] Z. Kang, X. K. Ma, and X. Zhuansun, An efficient 2-D compact precise-integration time-domain method for longitudinally

invariant waveguiding structures, IEEE Trans. Microw. Theory Tech. 7 (2013) 2535-2544.

Xikui Ma was born in Shaanxi, China, in 1958. He received the B.Sc. and M.Sc. degrees

in electrical engineering from Xi'an Jiaotong University, Shaanxi, China, in 1982 and

1985, respectively. In 1985, he joined the School of Electrical Engineering, Xi'an

Jiaotong University, as a Lecturer, and became a Professor in 1992. From 1994 to 1995,

he was a Visiting Scientist with the Department of Electrical Engineering and Computer,

University of Toronto. He has authored or coauthored over 140 scientific and technical

papers. He has authored five books in electromagnetic fields. His main areas of research

interests include electromagnetic field theory and its applications, analytical and

numerical methods in solving electromagnetic problems, chaotic dynamics and their

applications in power electronics, and the applications of digital control to power

electronics.

Tianyu Dong is an Assistant Professor in the School of Electrical Engineering at Xi’an

Jiaotong University, China. He is affiliated with the Group for Advanced Electrical

Technologies (GAET), an interdisciplinary center for research and education with an

emphasis on electromagnetics, circuits, power electronics, nonlinear phenomenon and

renewable energies. He is also affiliated with the State Key Laboratory of Electrical

Insulation and Power Equipment (SKLEI), and the State Key Laboratory for Strength and

Vibration of Mechanical Structures (MSSV) as a postdoctoral researcher. Prior to joining

XJTU as a tenure-tracked faculty at the beginning of the year of 2015, he received the BS

and PhD degrees from Xi'an Jiaotong University, China, respectively in 2008 and 2014.

During the PhD, he spent two years from 2011 to 2013 in the Electromagnetic

Communication (EMC) Lab of Electrical Engineering Department at The Pennsylvania

State University (Penn State) in the United States, where he was hosted by Prof. Dr. Raj Mittra. His current

research interests include graphene plasmonics, metasurfaces, scattering of nanoparticles, and wireless power

transfer.

*This use of this work is restricted solely for academic purposes. The author of this work owns the copyright and no reproduction in any form is permitted without written permission by the author. *

The Precise Integration Time Domain

(PITD) Method— A Supplement to the Computational Electromagnetics

Xikui Ma1 and Tianyu Dong1,2

maxikui@mail.xjtu.edu.cn tydong@mail.xjtu.edu.cn

September 19, 20161 SKLEI, School of Electrical Engineering, XJTU

2 MSSV, School of Aerospace, XJTU

Overview Methods Examples Outlooks

Outline

1 Overview

2 MethodsIntroduction to PITDNumerical dispersion and stabilitySource and boundary conditionsImproved methods

3 ExamplesResonant cavityMicrostrip LPFSmall antenna

4 Outlooks

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Overview Methods Examples Outlooks

CEM map ˚

˚source: http://feko.info/product-detail/numerical methods

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Overview Methods Examples Outlooks

FDTD timeline

birth of F

DTD (Yee)

Engquist-M

ajda ABC

1965 1970 1975 1980

1985199019952000

20052010 2015

Coining of FDTD

Mur ABC

NTFF/RCS

Liao ABCPML (Berenger)

UPML (Sack

s)

CPML (Gedney)

ADI(Namiki; Z

heng et al)

CFDTD (Mitt

ra);

PSTD (Liu)

PI tech

nique to CEM (M

a)

Nonlinear m

edia/circu

its

Recursi

ve convolutio

n

ADE Dispersi

ve dielectric

Unconstr

uctured grid

Contour p

ath

subce

ll model

3D-PITD (Ma)

High-ord

er FDTD

FDTD “Bible” (T

a�ove)

LOD-FDTDMetamateria

l modelin

g

(Hao & M

ittra)

Photonics

& Nanotech

.

(Ta�ove et. a

l.)

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Overview Methods Examples Outlooks

FDTD major technical paths

4 Absorbing boundary conditions• Mur; Engquist-Majda; Berenger PML, UPML, CPML

4 Numerical dispersion• High-order space differences; MRTD; PSTD

4 Numerical stability• ADI techniques; PITD; One-step Chebyshev method

4 Conforming grids• Locally/globally conforming; Stable hybrid FETD/FDTD

4 Digital signal processing• Near-to-far-field transformation

4 Dispersive and nonlinear materials• Isotropic/anisotropic dispersions; Nonlinear dispersions

4 Multiphysics

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Overview Methods Examples Outlooks

PITD timeline

PI techniqe forLTI ODEs (Zhong)

1990

1995

2000

2005

2010

2015

PI techniqe for CEM (Ma)

3D-PITD (Ma) Engquist-Madja & PML ABCs for PITD (Ma)

PITD in cylin

drical c

oord. (M

a);

Numerical d

ispersi

on analysis

& stabilit

y conditio

n (Chan)

Closed-su

rface

crite

rion (M

a) Sub-domain PITD (Ma);Wavelet Galerkin PITD (Ma)

High order PITD (Ma)

Leap-frog scheme (Ma);Lossy media (Ma)

Compact PITD (Ma);Split-step-scheme-based PITD (Ma)

Uni�ed split-step PITD (Ma)

Hybrid PITD-FDTDKrylov subspace-based PITD

PITD monograph (Sci. Press, Ma)

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Overview Methods Examples Outlooks

Maxwell’s equations

And God said:

BE

Bt“ ε´1 ¨∇ˆH,

BH

Bt“ ´µ´1 ¨∇ˆ E,

and then there was light.

J. C. Maxwell

“From a long view of the history of mankind the most significant event of

the nineteenth century will be judged as Maxwell’s discovery of the laws

of electrodynamics.” — Richard P. Feynman

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Overview Methods Examples Outlooks

Discretization and Yee cells

FDTD• Finite difference in space

• Finite difference in time

PITD

• Finite difference in space

• ODEs in time

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Updating equations / ODEs

FDTD

pR` FqXn`1 “ pR´ FqXn ` fn`1

where

Xn “

„

En

Hn`1{2

• Dε|µ|σe |σm — diagonal matricescontaining ε, µ, σe , σm for each cell

• K — arises from the discretization ofthe curl operators

• fn`1 — sources

PITD

dXptq

dt“MXptq ` fptq

where

Xptq “

„

EptqHptq

• M — matrix containing materialproperties and the discretization of thecurl operators

• f — sources

R “1

2

«

2∆t

Dε ´K

´KT 2∆t

Dµ

ff

F “1

2

„

Dσe K

´KT Dσm

E and H are staggered in time. E and H are non-staggered in time.

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Overview Methods Examples Outlooks

Formal solution to dXdt “MX` fptq

• Analytical form

Xptq “ exppMtqX p0q `

ż t

0exprMpt ´ sqsfpsqds

• Recursive form

Xn`1 “ TXn ` Tn`1

ż tn`1

tn

expp´sMqfpsqds

T “ eM∆t1

2

Key points!

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Overview Methods Examples Outlooks

Ways to compute eM∆t

• Series methods

• ODE methods

• Polynomial methods

• Matrix decomposition methods

• Splitting methods

• Krylov space methods

• ...

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Overview Methods Examples Outlooks

PI technique to compute T “ eM∆t

1 scaling and squaring

T “ eM∆t “

˜

eM ∆t{`

¸`

“

´

eMτ¯`

2 series expansion of eMτ

eMτ “ I` Ta « I` pMτq `pMτq2

2!`pMτq3

3!`pMτq4

4!

3 compute T “`

eMτ˘`“ pI` Taq

` — to be contd.

τ “ ∆t{`

` “ 2N

N – preselected integer

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Overview Methods Examples Outlooks

PI technique to compute T “ eM∆t (contd.)

3 compute T “`

eMτ˘`“ pI` Taq

`

T “ pI` Taq2N“ pI` Taq

2N´1

ˆ pI` Taq2N´1

“ ...

with pI` Taq2“ I` 2Ta ` Ta ˆ Ta.

Pseudo code

do i “ 1, ...,NTa Ð 2Ta ` Ta ˆ Ta

end doTÐ I` Ta

The algorithm holds thecomputational precision.

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Overview Methods Examples Outlooks

Evaluating T

ż tn`1

tn

e´sMfpsqds & the recursive formula

(1) Analytical solution under the linear representation of fptq

• 1st order Taylor approximation of fptq, i.e.,

f “ f0 ` pt ´ tnqf1, t P ptn, tn`1q

• integrate T

ż tn`1

tn

e´sMfpsqds analytically

• recursive form for Xn`1:

Xn`1 “T“

Xn `M´1pf0 `M´1f1q‰

´M´1“

f0 `M´1f1 ` f1∆t‰

compute M´1? It is noninvertible generally!

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Evaluating T

ż tn`1

tn

e´sMfpsqds & the recursive formula (contd.)

(2) Recursive formula by using the Gaussian quadrature rule

• three-point Gaussian quadrature

• recursive form for Xn`1:

Xn`1 “ TXn `5

18e

´

1`?

3{5¯

M∆t{2f”

tn `´

1´a

3{5¯

∆t{2ı

`5

18e

´

1´?

3{5¯

M∆t{2f”

tn `´

1`a

3{5¯

∆t{2ı

`8

18eM∆t{2f ptn `∆t{2q

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Overview Methods Examples Outlooks

Remarks

dX

dt“MX` f, X “ rE,HsT

1 finite difference in space, but differential in time

2 scaling and squaring for exppM∆tq

3 Ta Ð 2Ta ` Ta ˆ Ta to guarantee the computationalprecision

4 Gaussian quadrature for the excitation term

5 non-staggered E and H in time

Precise Integration

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Numerical stability condition

FDTD CFL criteria: ∆tFDTDupperbound

“1

cb

1p∆xq2

` 1p∆yq2

` 1p∆zq2

PITD

Stability conditions vary for different orders of Taylor approximation:

• 1st/2nd-order: unstable;

• 3rd-order: ∆t ă?

3{2 ` ∆tFDTDupperbound

“?

3{2 2N ∆tFDTDupperbound

• 4th-order: ∆t ă?

2`∆tFDTDupperbound

“ 2N`1{2∆tFDTDupperbound

• 5th-order:?

2p15´?

65q

4?

15´?

65`∆tFDTD

upperbound

ă ∆t ă?

2p15`?

65q

4?

15`?

65`∆tFDTD

upperbound

Almost unconditionally

stable for large N.

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Overview Methods Examples Outlooks

Numerical dispersion analysis

FDTD numerical dispersion relation:

W 2t {c

2 “W 2x `W 2

y `W 2z ,

where Wx |y |z “sinpk̃x|y |z∆x |y |z{2q

∆x |y |z{2 , Wt “sinpω∆t{2q

∆t{2 .

PITD

tan2

ˆ

ω∆t

`

˙

“

`

ΛPITD ´ Λ3PITD{3!

˘2

1` Λ2PITD{2!´ Λ4

PITD{4!

where ΛPITD “c∆t`

b

W 2x `W 2

y `W 2z .

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Overview Methods Examples Outlooks

Numerical dispersion analysis (contd.)

• numerical dispersion is slightlyworse than that of FDTD

• independent of the time step

• dense griding improves theaccuracy

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Overview Methods Examples Outlooks

Source and boundary conditions

Source conditions

• Hard sources

• Plane waves & TS/SF technique

• ...

Boundary conditions

• Engquist-Majda ABC

• PMLs

• ...

“它山之石，可以攻玉。” —《诗经·小雅·鹤鸣》

“Stones from other hills may serve topolish the jade of this one.”

— Classic of Poetry ‚ Lesser Court Hymns‚ Singing of Cranes

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Overview Methods Examples Outlooks

Remarks

Characteristics of the PITD method:

3 preselected N determines the upper bound of ∆tPITD

3 ∆tPITDupperbound

ąą ∆tFDTDupperbound

3 slight worse numerical dispersion compared with that of theFDTD method

3 numerical dispersion can be independent of ∆t

3 technique paths of the FDTD method can be learned

“Stones from other hills may serve to polish the jade of this one.”— Classic of Poetry ‚ Lesser Court Hymns ‚ Singing of Cranes

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Overview Methods Examples Outlooks

Improved methods

• Fourth-order PITD [PITD(4)] method

• Wavelet Galerkin PITD (WG-PITD) method

• Leapfrog PITD (L-PITD) method

• Compact PITD (CPITD) method

• Hybrid PITD-FDTD method

• Krylov subspace method

• ...

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Overview Methods Examples Outlooks

Improved methods — PITD(4) 1

4th-order spatial difference scheme is used as:

BuiBx

“1

∆x

„

1

24

`

ui´3{2 ´ ui`3{2

˘

´27

24

`

ui´1{2 ´ ui`1{2

˘

`O”

p∆xq4ı

Numerical dispersion

1IEEE T-AP, 59(4), 2011: 1311-1320.

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Overview Methods Examples Outlooks

Improved methods — WG-PITD method 2

Discretization form of Maxwell equation(s) in space:

d Ex |l`1{2,m,n

dt“ ´

σ|l`1{2,m,n

ε|l`1{2,m,n

Ex |l`1{2,m,n `ÿ

i

ai

«

Hz |l`1{2,m`i`1{2,n

ε|l`1{2,m,n ∆y´

Hyˇ

ˇ

l`1{2,m,n`i`1{2

ε|l`1{2,m,n ∆z

ff

where ai “ş8

´8

dφ1{2pxq

dx φ´i pxqdx .

Numerical dispersion S11-parameter

2IEEE MWCL, 20(12), 2010: 651-653

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Overview Methods Examples Outlooks

Improved methods — Hybrid PITD-FDTD method 3

How to handle multiscale problems with fine geometrical features?

• Subgriding in FDTD — ∆t depends on ∆xmin

• PITD — need to compute eM∆t , but ∆t can be relaxed

Hybriding

3Please refer to PA-11 (Mon. 14:00-15:30, Function Room 2, 2F)

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Overview Methods Examples Outlooks

Improved methods — Krylov space method 4

Recursive form of the PITD method:

Xn`1 “ eM∆tXn `ÿ

i

αieMβi∆tfptn ` γi∆tq

number of nonzeroelements

Denseness Memory cost (MB)

M 1558 0.0039 0.003eM∆t 370482 0.9334 0.76

Evaluate eM∆t explicitly? 7 ÝÑ Estimate eM∆tv directly. 3

4Please refer to OC2-6 (Tue. 08:30-10:00, Function Room 3, 3F)

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Overview Methods Examples Outlooks

Improved methods — Krylov space method 4 (contd.)

Direct estimation of eM∆tv

1 mth-order Krylov subspace: KmpM, vq “ spanpv,Mv, . . . ,Mm´1vq

2 Arnoldi process

• Vm “ rv1, v2, . . . , vmsT – orthogonal basis of Km

• Hm « VTmMVm – matrix generated during the Arnoldi process

3 eM∆tv « VmeHm∆tVT

mv “ VmeHm∆te1, e1 “ r1, 0, 0, . . . , 0s

TP Rmˆ1

CPU time (s)Memory

cost (MB)Krylov-PITD 23.78 0.30

FDTD 137.28 1.34

4Please refer to OC2-6 (Tue. 08:30-10:00, Function Room 3, 3F)

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Overview Methods Examples Outlooks

Rectangular cavity

fana. (GHz)FDTD scheme ADI-FDTD scheme PITD scheme

∆t = 1 ps ∆t = 60 ps ∆t = 60 psf (GHz) rel. err. f (GHz) rel. err. f (GHz) rel. err.

3.125 2.983 4.54% 2.900 7.20% 2.983 4.54%4.881 4.750 2.68% 4.650 4.73% 4.750 2.68%5.340 5.450 2.06% 5.580 4.49% 5.450 2.06%7.289 7.333 0.60% 6.817 6.92% 7.333 0.60%7.529 7.567 0.51% 7.000 7.03% 7.567 0.51%

• relative error increases as the time-stepincreases for the ADI-FDTD method

• relative error is independent of the timestep for the PITD method

8 cm

4 cm

6 cm

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Overview Methods Examples Outlooks

Microstrip low pass filter 5

Comparison between the FDTD method and the L-PITD (Leapfrog PITD) method :

∆x ∆y ∆z ∆t memory CPU time

FDTD 0.41 mm 0.26 mm 0.42 mm 0.441 ps 24.44 MB 1024 sL-PITD 0.41 mm 0.26 mm 0.42 mm 0.884 ps 248.4 MB 851 s: Simulations were performed on Intel® CoreTM Duo CPU T8100 2.10 GHz PC.

5IEEE MWCL, 22(6), 2012: 294 – 296.

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Overview Methods Examples Outlooks

Small antenna

Comparison between the FDTD method and the Krylov-PITD method

CPU time (s) Memory cost (MB)

Krylov-PITD 23.78 0.30FDTD 137.28 1.34

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Overview Methods Examples Outlooks

Remarks

4 almost unconditionally stable

4 relative large time step size can be used

4 PI technique maintains the computational precision

4 relative error independent of the time-step

4 hybrid PITD-FDTD technique suitable for multiscale problems

4 memory cost can be relaxed by using the Kyrlov space method

“瑕不掩瑜。” —《礼记·聘义》

“One flaw cannot obscure the splendor of the jade.”— Book of Rites ‚ Meaning of Interchange of Missions twixt Different Courts

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Overview Methods Examples Outlooks

Outlook

Future work :

• Sub-domain technique

• Parallel computing technique

• Extend to complex materials

Future prospects :

• Nanophotonics and nanoplasmonics. Ultimately, combinationof quantum and classical electrodynamics

• Multiphysics

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Overview Methods Examples Outlooks

Acknowledgements & References

Acknowledgements

Dr. Jinquan Zhao, Mr. Min Tang, Ms. Mei Yang, Dr. Xintai Zhao, Dr. Gang Sun,

Dr. Zhongming Bai, Dr. Qi Liu and Dr. Zhen Kang are kindly acknowledged.

References• IEEE T-MTT, 2006, 54(7): 3026

• IEEE MWCL, 2007, 17(7): 471

• PIER, 2007, 69: 201

• IEEE T-MTT, 2008, 56(12): 2859

• IEEE MWCL, 2010, 20(12): 651

• IEEE T-AP, 2011, 59(4): 1311

• IEEE T-MTT, 2012, 60(9): 2723

• IEEE MWCL, 2012, 22(6): 294

• Electron. Lett., 2013, 49(18): 1135

• COMPEL, 2013, 33(1/2): 85

• IEEE T-MTT, 2013, 61(7): 2535

• Electron. Lett., 2014, 50(18): 1297

• IEEE MWCL, 2016, 26(2): 83

• Ma, Xikui. “The precise integrationtime domain method.”(in Chinese)Beijing: Science Press (2015).

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Thank you!Q & A.

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