Multiscale Methods and Analysis for the Dirac Equation in

Multiscale Methods and Analysis for the Dirac Equation in the Nonrelativistic Limit Regime

Weizhu Bao

Department of MathematicsNational University of SingaporeEmail: matbaowz@nus.edu.sg

URL: http://www.math.nus.edu.sg/~bao

Collaborators: Yongyong Cai (CSRC, Beijing), Xiaowei Jia (China);Qinglin Tang (postdoc, Hong Kong), Jia Yin (PhD, NUS)

Outline

The Dirac equation Numerical methods and error estimates– Finite difference time domain (FDTD) methods– Exponential wave integrator Fourier spectral (EWI-FP) method– Time-splitting Fourier pseudospectral (TSFP) method

A uniformly accurate (UA) method Extension to nonlinear Dirac equationConclusion & future challenges

The Dirac equation

– : spatial coordinates– : complex-valued vector wave function ``spinorfield’’– : real-valued electrical potential– : real-valued magnetic potential – : electric field– : magnetic field

Refs: [1] P.A.M. Dirac, Proc. R. Soc. London A, 127 (1928) & 126 (1930). [2]. Principles of Quantum Mechanics, Oxford Univ Press, 1958.[3] http://en.wikipedia.org/wiki/Dirac_equation

3 32

41 1

( ) ( )t j j j jj j

i ic mc e V x I A xα β α= =

∂ Ψ = − ∂ + Ψ + − Ψ

∑ ∑

41 2 3 4( , ) ( , , , )Tt x ψ ψ ψ ψΨ = Ψ = ∈

0V A= −

1 2 3( , , )TA A A A=

0t tE V A A A= −∇ − ∂ = ∇ − ∂B A= ∇×

31 2 2( , , ) (or ( , , ) )T Tx x x x x y z= ∈

The Dirac equation

• : 4-by-4 matrices

• : 2-by-2 Pauli matrices

• : 4-by-4 matrices

• The Dirac equation

1 2 3, , ,α α α β

31 2 21 2 3

31 2 2

00 0 0, , ,

00 0 0I

Iσσ σ

α α α βσσ σ

= = = = −

0 1 2 3, , ,γ γ γ γ

1 2 3, ,σ σ σ

1 2 3

0 1 0 1 0, ,

1 0 0 0 1i

iσ σ σ

− = = = −

0 0, , 1,2,3kk kγ β γ γ α= = =

( ) 0i mc e Aµ µµ µγ γ∂ − + Ψ =

2 24 ,

0, 0j

j l l j j j

Iα β

α α α α α β βα

= =

+ = + =

The Dirac equation

Derived by British physicist Paul Dirac in 1928 to describe all spin-1/2 massive particles such as electrons and quarksIt is consistent with both the principles of quantum mechanics and the theory of the special relativityThe first theory to account fully for special relativity in quantum mechanics

Accounted for fine details of the hydrogen spectrum in a completely rigorous way, implied the existence of a new form of matter, antimatter & predated experimental discovery of positron

In special limits, it implies the Pauli, Schrodinger and Weyl equations!

Par Dirac with Newton, Maxwell & Einstein!! Be awarded the Nobel Prize in 1933 (with Schrodinger). Host Lucasian Professor of Mathematics at the University of Cambridge at the age of 30!!

Quantum Mechanics with Relativistic

Making the Schroedinger equation relativistic

Klein-Gordon equation for spinless –pion (Oskar Klein&Walter Gordon, 1926)

2 2;2( ) ( ) :

2 2tE i p i

tpE V x i V x Hm m

ψ ψ ψ ψ→ ∂ →− ∇

= + ⇒ ∂ = − ∇ + =

( )2

22 2 4 2 2 22

2 2 2;2

2 2 21tE i p i

EE m c cp p m cc

m cc t

φ φ→ ∂ →− ∇

= + ⇔ − =

∂⇒ ∇ − = ∂

::

0

t t

t

JJ

ρ φ φ φ φ

φ φ φ φρ

= ∂ − ∂

= ∇ − ∇∂ +∇ =

Quantum Mechanics with Relativistic

Dirac’s coup– Square-root of an operator

– Dirac equation

22 2 2 22 2 2 2

2 2 2 21

x y z tE i m cp m c A B C Dc c t c

∂ − = ⇒∇ − = ∂ + ∂ + ∂ + ∂ = ∂

22

2 0 0x y z tE i mcp mc A B C Dc c

ψ − − = ⇒ ∂ + ∂ + ∂ + ∂ − =

2 22 2 2 2 2

2 2( ) ( ) 0E EE pc mc p mc p mcc c

= + ⇔ − = ⇔ − − =

3;2 2 2 2

1( ) ( )

tE i p i

t j jj

E pc mc i ic mcα β→ ∂ →− ∇

=

= + ⇒ ∂ Ψ = − ∂ + Ψ

∑

2 2

1 2 3

0, & 1, , , Clifford Algebra over 4d!!

AB BA A BA i B i C i Dβα βα βα β+ = = = =

⇒ = = = = ⇔

Typical Applications

Graphene/graphites and/or 2D materials – K. Novoselov, A. Geim, etc., Science, 2004; K. Novoselov, A. Geim , etc., Nature, 2005; K. Novoselov, …, A. Geim, Science, 2007; D.A. Abanin, etc., Science, 2011; A.H.C. Neto, etc., Rev. Mod. Phys., 2009, … (Nobel Prize in 2010!!!!)

– Same dispersion relation at Dirac cone

Typical Applications

Chiral confinement of quasirelativistic BEC – M. Merkl et al., PRL,2010

Atto-second laser on molecule -F. Fillion-Gourdeau, E. Lorin and A.D. Bandrauk, PRL, 13’&JCP14’

Neutron interaction in nuclear physics--H. Liang, J. Meng &Z.G. Zhou, Phys. Rep., 15’

The Dirac equation

Dimensionless Dirac equation in d-dimension (d=3,2,1)

Different parameter regimes– Standard scaling: – Semi-classical limit regime: – Nonrelativistic limit regime:– Massless regime:

02

0

0 : 1; 0 : 1; 0 : 1s

s

x vv mt c c e m

κε η λ< = = ≤ < = ≤ < = ≤

421 1

( ) ( ) ,d d

dt j j j j

j j

ii V x I A x xη λη α β αε ε= =

∂ Ψ = − ∂ + Ψ + − Ψ ∈

∑ ∑

1ε η λ= = =1& 0 1ε λ η= = <

1& 0 1η λ ε= = <

1& 0 1ε η λ= = <

3 32

41 1

( ) ( )t j j j jj j

i ic mc e V x I A xα β α= =

∂ Ψ = − ∂ + Ψ + − Ψ

∑ ∑

Different limits of the Dirac equation

Dirac Equation

WeylEquation

massless

1, 10( 0)m

µ ελ

= =→ → Schrodinger

orPauli Equationnonrelativistic

1, 10( )c

η λε= =→ →∞

relativistic Euler Equation

Euler equation

semiclasical1, 10( 0)

λ εη= =→ →

10( )c

λε=→ →∞

10

( 0)

λη=→→

421 1

( ) ( ) ,d d

t j j j jj j

ii V x I A xη λη α β αε ε= =

∂ Ψ = − ∂ + Ψ + − Ψ

∑ ∑

The Dirac equation

Dimensionless Dirac equation in d-dimension (d=3,2,1)

– Initial data

– Dispersive PDE & time symmetric – Mass & energy conservation

0 : 1s

s

x vt c c

ε< = = ≤

421 1

1 ( ) ( ) ,d d

dt j j j j

j j

ii V x I A x xα β αε ε= =

∂ Ψ = − ∂ + Ψ + − Ψ ∈

∑ ∑

0(0, ) ( ), dx x xΨ = Ψ ∈

41 2 3 4( , , , )Tψ ψ ψ ψΨ = ∈

1η λ= =

Conservations laws

Position and current densities

Conservation lawMass conservation

Energy (or Hamiltonian) conservation

4 2* *1 2 3

1

1: , ( , , ) with :Tj l l

jJ J J J Jρ ψ α

ε=

= Ψ Ψ = = = Ψ Ψ∑

0, dt J xρ∂ +∇ = ∈

2 2 20: ( , ) ( ) 1

d d

t x dx x dxΨ = Ψ ≡ Ψ =∫ ∫

2* * *2

1 1

1( ) : ( ) ( ) (0)d

d d

j j j jj j

iE t V x A x dx Eα β αε ε= =

= − Ψ ∂ Ψ + Ψ Ψ + Ψ − Ψ Ψ ≡

∑ ∑∫

421 1

1 ( ) ( )d d

t j j j jj j

ii V x I A xα β αε ε= =

∂ Ψ = − ∂ + Ψ + − Ψ

∑ ∑

The Dirac equation

In 2D/1D

– Initial data

– Dispersive PDE & time symmetric– Mass & energy conservation

1 2 1 4 2 3( , ) with =( , ) or ( , )T T Tφ φ ψ ψ ψ ψΦ = Φ

3 221 1

1 ( ) ( ) ,d d

dt j j j j

j j

ii V x I A x xσ σ σε ε= =

∂ Φ = − ∂ + Φ + − Φ ∈

∑ ∑

0(0, ) ( ), dx x xΦ = Φ ∈

Conservations laws

Position and current densities

Conservation lawMass conservation

Energy (or Hamiltonian) conservation

2 2* *1 2

1

1: , ( , ) with :Tj l l

jJ J J Jρ φ σ

ε=

= Φ Φ = = = Φ Φ∑

0, dt J xρ∂ +∇ = ∈

2 2 20: ( , ) ( ) 1

d d

t x dx x dxΦ = Φ ≡ Φ =∫ ∫

2* * *32

1 1

1( ) : ( ) ( )d

d d

j j j jj j

iE t V x A x dxσ σ σε ε= =

= − Φ ∂ Φ + Φ Φ + Φ − Φ Φ

∑ ∑∫

3 221 1

1 ( ) ( )d d

t j j j jj j

ii V x I A xσ σ σε ε= =

∂ Φ = − ∂ + Φ + − Φ

∑ ∑

Two typical regimes & results

Standard regime– Analytical study on existence & multiplicity of solutions: Gross, 66’;

Gesztesy, Grosse & Thaller, 84’; Das & Kay, 89’; Das, 93’; Esteban & Sere, 97’; Dolbeault, Esteban & Sere, 00’; Esteban & Sere, 02’; Booth, Legg & Jarvis, 01’; Fefferman & Weistein, J. Amer. Math. Soc., 12’; CMP, 14; Ablowitz & Zhu, 12’; ……

– Numerical methods• Leap-frog finite difference (LFFD) method: Shebalin, 97’; Nraun, Su & Grobe, 99’; Xu,

Shao & Tang, 13’; Brinkman, Heitzinger & Markowich, 14’; Hammer, Potz & Arnold, 15’; Antoine, Lorin, Sater, Fillion-Gourdeau & Bandrauk, 15’, …..

• Time-splitting Fourier pseudospectral (TSFP) method: Bao & Li, 04’; Huang, Jin, Markowich, Sparber & Zheng, 05’; Xu, Shao & Tang, 13’; …..

• Gaussian beam method: Wu, Huang, Jin & Yin, 12’, ….

Nonrelativistic limit regime

( ) 1v O c ε= ⇔ =

( )20 1v c Oε ω ε −⇔ < ⇒ =

Existing results in nonrelativistic limit regime

Nonrelativistic limits: Gross, 66’; Hunziker, 75’; Foldy & Wouthuysen, 78’; Schoene, 79’; Cirincione & Chernoff, 81’; Grigore, Nenciu & Purice, 89’; Najman, 92’; Gerad, Markowich, Mauser & Poupaud, 97’; Bechouche, Mauser & Poupaud, 98’; Bolte & Keppeler, 99’; Spohn, 00’; Kammerer, 04’; Bechouche, Mauser & Selberg, 05; ……

– Main difficulty: Solution propagates waves with wavelength in time & in space– Plane wave solutions

: (or : ) ??? when 0ε ε εΨ = Ψ Φ = Φ → →( ) is indefinite & unbounded when 0!!!E t ε →

2( )O ε

( )0 0 2 23 22

1

1 ,d

jj j

j

kB A V I B B Oω σ σ ω ε

ε ε−

=

= − + + ∈ ⇒ =

∑

(1)O( )( , ) i k x tt x B e ω−Φ =

Numerical results

Numerical results

Existing results in nonrelativistic limit regime

– Asymptotic and rigorous results: Gross, 66’; Hunziker, 75’; Foldy & Wouthuysen, 78’; Schoene, 79’; Cirincione & Chernoff, 81’; Grigore, Nenciu & Purice, 89’; Najman, 92’; Gerad, Markowich, Mauser & Poupaud, 97’; Bechouche, Mauser & Poupaud, 98’; Bolte & Keppeler, 99’; Spohn, 00’; Kammerer, 04’; Bechouche, Mauser & Selberg, 05; ……

• The Schrodinger equation

• Semi-nonrelativistic limit: Bechouche, Mauser & Poupaud, 98

Highly oscillatory dispersive PDEs:

2 2/ / 0: ( ), 0

0it ite e Oε ε εφ

ε εφ

+ −

−

Φ = Φ = + + →

1 ( ) ,2

dti V x xφ φ φ± ± ±∂ = ∆ + ∈

Numerical methods for Dirac equation

Finite difference time domain (FDTD) methods

– Mesh size– Time step– Numerical approximation

( )1 3 2 1 12

0

1 ( , ) ( , ) , , 0

( , ) ( , ), ( , ) ( , ), 0;( ,0) ( ), [ , ]

t x

x x

ii V t x I A t x x t

a t b t a t b t tx x x a b

σ σ σε ε

∂ Φ = − ∂ + Φ + − Φ ∈Ω >

Φ = Φ ∂ Φ = ∂ Φ ≥

Φ = Φ ∈Ω =

: , , 0,1, ,jb ah x x a jh j MM−

= ∆ = = + =

: 0, , 0,1,nt t n nτ τ= ∆ > = =

( , ) , 0,1, , , 0,1,nj n jx t j M nΦ ≈ Φ = =

Numerical methods for Dirac equation

Finite difference discretization operators

Leap-frog finite difference (LFFD) method

Semi-implicit finite difference (SIFD1) method

Numerical methods for Dirac equation

Semi-implicit finite difference (SIFD2) method

Energy conservative finite difference (CNFD) method

– Initial and boundary data

– First step for LFFD, SIFD1 &SIFD2

Properties of FDTD methods

Time symmetric, unchanged if

Stability – CNFD is unconditionally stable– LFFD, SIFD1 & SIFD2 are conditionally stable

Energy conservation: CNFD conserves mass & energy vs others not

Computational cost– CNFD needs solve a linear coupled system per time step!! LFFD is explicit!– SIFD1 & SIFD2 can be solved very almost explicit !!

Resolution in nonrelativistic limit regime

1 1 &n n τ τ+ ↔ − ↔ −

? ?( ) & ( ), 0 1h O Oε τ ε ε= = <

Error estimates for FDTD methods

Define `error’ function

Assumptions– For the solution of the Dirac equation --- (A)

– For the electronic & magnetic potentials – (B)

Error estimates for FDTD methods

Theorem Under some stability conditions, we have error estimates for LFFD, SIFD1, SIFD2 & CNFD as (Bao, Cai , Jia & Tang, JSC, 16’)

– Resolution ---- (under resolution)

Spatial Errors of CNFD

Temporal Errors of CNFD

Spatial Errors of CNFD

EWI-FP method

Apply Fourier spectral method for spatial derivatives

– with

Take Fourier transform, we get ODEs for l=-M/2,…,M/2-1

Exponential wave integrator (EWI) for 1st ODEs

Take and approximate the integral --W. Gautschi(61’); P. Deuflhard (79’); E. Hairer, Ch. Lubich, G. Wanner, A. Iserles, V. Grimm, M. Hochbruck, D. Cohen, …..

EWI-FP method

s τ=

Symmetric Exponential wave integrator (sEWI)

Take and approximate the integral

sEWI-FP method

s τ= ±

Error estimates for EWI-FP method

Theorem Under some stability conditions, we have error estimates for EWI-FP and sEWI-FP as (Bao, Cai, Jia & Tang, JSC,16’)

– Resolution --- (optimal resolution)

01/2 2( ) ( ), ( ) (1), 0 1mO O h O Oτ ε δ ε δ ε= = = = <

Spatial Errors of EWI-FP

Temporal Errors of EWI-FP

Time-splitting Fourier spectral (TSFP) method

From , apply time splitting technique – Step 1

– Step 2

Thm. Under proper assumptions, we have

– Resolution --- (optimal resolution)

1[ , ]n nt t +

01/2 2( ) ( ), ( ) (1), 0 1mO O h O Oτ ε δ ε δ ε= = = = <

Spatial Errors of TSFP

Temporal Errors of TSFP

Thm. If time step satisfies (Bao, Cai, Jia&Yin, SCM, 16’)

– P. Chartier,F. Mehats,M. Thalhammer &Y. Zhang, superconvergnnce for TSSP for NLSE, 14’

22 / Nτ πε=

2Cτ ε≤ ⇒

Comparison

Comparison

A uniformly accurate (UA) method

T can be diagonalizable (Bechouche, Mauser & Poupaud, 98’)

– With

– Satisfying

2

1 3 2 1 1

0

1 ( , ) , , 0

, ( , ) ( , ) ( , )(0, ) ( ),

t

x

i T W t x x t

T i W t x V t x I A t xx x x

εεσ σ σ

∂ Φ = Φ + Φ ∈ >

= − ∂ + = −Φ = Φ ∈

2 21 1T ε ε+ −= − ∆ Π − − ∆ Π

( ) ( )1/2 1/22 22 2

1 11 , 12 2

I T I Tε ε− −

+ − Π = + − ∆ Π = − − ∆

22 , 0,I+ − + − − + ± ±Π + Π = Π Π = Π Π = Π = Π

A uniformly accurate (UA) method

Given initial data at :Multiscale decomposition in frequency (MDF) :(Bechouche, Mauser & Poupaud, 98’)

Two sub-problems

nt t=( , ) ( ),n nt x x xΦ = Φ ∈

2 2/ 1, 1, / 2, 2,( , ) ( , ) ( , ) , 0is n n is n nnt s x e s x e s x sε ε τ−

+ − + − Φ + = Ψ +Ψ + Ψ +Ψ ≤ ≤

( ) ( )

( ) ( )

1, 2 1, 1, 1,2

1, 2 1, 1, 1,2

1, 1,

1( , ) 1 1 ( , ) ( , ) ( , )

1( , ) 1 1 ( , ) ( , ) ( , )

(0, ) ( ), (0, ) 0, with : ( , )

n n n ns

n n n ns

n nn n

i s x s x W s x W s x

i s x s x W s x W s x

x x x W W t s x

εε

εε

+ + + + −

− − − + −

+ + −

∂ Ψ = − ∆ − Ψ + Π Ψ + Ψ

∂ Ψ = − − ∆ − Ψ + Π Ψ + Ψ

Ψ = Π Φ Ψ = = +

1, 1, 2(1), ( )n nO O ε+ −Ψ = Ψ =

A uniformly accurate (UA) method

Two sub-problems

Solve the two sub-problems via EW-FP

Reconstruct the solution at

( ) ( )

( ) ( )

2, 2 2, 2, 2,2

2, 2 2, 2, 2,2

2, 2,

1( , ) 1 1 ( , ) ( , ) ( , )

1( , ) 1 1 ( , ) ( , ) ( , )

(0, ) 0, (0, ) ( ), with : ( , )

n n n ns

n n n ns

n nn n

i s x s x W s x W s x

i s x s x W s x W s x

x x x W W t s x

εε

εε

+ + + + −

− − − + −

+ − −

∂ Ψ = − ∆ + Ψ +Π Ψ + Ψ

−∂ Ψ = − ∆ − Ψ +Π Ψ + Ψ

Ψ = Ψ = Π Φ = +

2, 2 2,( ), (1)n nO Oε+ −Ψ = Ψ =

1, 2,( , ) & ( , )n nx xτ τ± ±Ψ Ψ1nt t +=

2 2/ 1, 1, / 2, 2,1( , ) ( , ) ( , )i n n i n n

nt x e x e xτ ε τ ετ τ−+ + − + − Φ = Ψ +Ψ + Ψ +Ψ

Multiscale Decomposition 0.5ε =

Multiscale Decomposition 0.25ε =

Error estimates for MTI-FP method

Theorem Under proper assumptions on the solution, we have error estimates for the MTI-FP method (Bao, Cai, Jia & Tang, SINUM 15’)

– Which yields a uniform error bound

– Resolution ---- (super resolution)01/( ) (1), ( ) (1), 0 1mO O h O Oτ δ δ ε= = = = <

Key Steps in the Proof

Step 1. Some properties of micro variables

Step 2. Local error bounds for micro variables0

2 2

02 2

0 02 2

0 02 2

1, 1, 1 2, 1

2, 2, 1 2, 1

1, 1, 2 1, 1, 2 2, ,

2, 2, 2 2, 2, 2, ,

( ,.) ( ) ( ),

( ,.) ( ) ( ),

( ), ( / ),

( ), ( /

mn n nh n IL L

mn n nh n IL L

m mn n n nh hL L

m mn n n nh hL L

t h

t h

h h

h h

τ τ

τ τ

τ ε τ τ ε

τ ε τ τ

−+ + −

−− − −

− − − −

+ + + +

Ψ −Ψ ≤ Φ −Φ + +

Ψ −Ψ ≤ Φ −Φ + +

Ψ −Ψ ≤ + Ψ −Ψ ≤ +

Ψ −Ψ ≤ + Ψ −Ψ ≤ +

2 )ε

1, 2, 1, 2, 1, 2,

1, 2, 2 1, 2, 1, 2,2

(1)

1( ), (1), ( )

n n n n n ns s ss ss

n n n n n ns s ss ss

O

O O Oεε

+ − + − + −

− + − + − +

Ψ + Ψ + ∂ Ψ + ∂ Ψ + ∂ Ψ + ∂ Ψ ≤

Ψ + Ψ ≤ ∂ Ψ + ∂ Ψ ≤ ∂ Ψ + ∂ Ψ ≤

Key Steps in the Proof

Step 3. Local error bounds for macro variables

Step 4. The energy method via discrete Gronwall’s inequality

Step 5. Uniform error bound

2 2 2 2 2

02

02 2

1, 1, 2, 2, 1, 1, 2, 2,, , , ,

21 2

1 2

1 2 21

( ,.) ( )

( ,.) ( ) ( )

( ,.) ( ) ( ,.) ( ) ( )

n n n n n n n n nn I h h h hL L L L L

mnn I L

mn nn I n IL L

t

t h

t t h

ττ τε

τ τ ε

+ + − − − − + +

−−

−−

Φ −Φ ≤ Ψ −Ψ + Ψ −Ψ + Ψ −Ψ + Ψ −Ψ

≤ Φ −Φ + + +

Φ −Φ ≤ Φ −Φ + + +

0 02

22 2

2( ,.) ( ) min , , 0 1m mnn I L

t h hττ ε τ εε

Φ −Φ ≤ + + ≤ + < ≤

Spatial Errors of MTI-FP

Temporal Errors of MTI-FP

The nonlinear Dirac equation (NLDE)

The nonlinear Dirac equation in d-dimension (d=3,2,1)

– Initial data– Dispersive PDEs & time symmetric – Soliton solution in 1D when – Mass & energy conservation

0 : / 1v cε< = ≤

( )*42

1 1

1 ( ) ( ) ,d d

t j j j jj j

ii V x I A xα β α δ β βε ε= =

∂ Ψ = − ∂ + Ψ + − Ψ + Ψ Ψ Ψ

∑ ∑

0(0, ) ( ), dx x xΨ = Ψ ∈

1ε =

( )

2 2 20

22* * * *2

1 1

: ( , ) ( ) 1

1: ( ) ( )2

d d

d

d d

j j j jj j

t x dx x dx

iE V x A x dxδα β α βε ε= =

Ψ = Ψ ≡ Ψ =

= − Ψ ∂ Ψ + Ψ Ψ + Ψ − Ψ Ψ + Ψ Ψ

∫ ∫

∑ ∑∫

The nonlinear Dirac equation

In 2D/1D (d=2,1):

– Initial data – Dispersive PDEs & time symmetric – Soliton solution in 1D when – Mass & energy conservation

( )*3 2 3 32

1 1

1 ( ) ( ) ,d d

t j j j jj j

ii V x I A xσ σ σ δ σ σε ε= =

∂ Φ = − ∂ + Φ + − Φ + Φ Φ Φ

∑ ∑

1 2 1 4 2 3( , ) with =( , ) or ( , )T T Tφ φ ψ ψ ψ ψΦ = Φ

0(0, ) ( ), dx x xΦ = Φ ∈

1ε =

( )

2 2 20

22* * * *3 32

1 1

: ( , ) ( ) 1

1: ( ) ( )2

d d

d

d d

j j j jj j

t x dx x dx

iE V x A x dxδσ σ σ σε ε= =

Φ = Φ ≡ Φ =

= − Φ ∂ Φ + Φ Φ + Φ − Φ Φ + Φ Φ

∫ ∫

∑ ∑∫

Extension to nonlinear Dirac equation

FDTD methods– LFFD

– SIFD1

– SIFD2

– CNFD ---- mass & energy conservation!!

( ) ( )*1 3 2 1 1 3 32

1 ( , ) ( , )t xii V t x I A t xσ σ σ δ σ σε ε

∂ Φ = − ∂ + Φ + − Φ + Φ Φ Φ

Error estimates of FDTD for NLDE

Thm. Under some stability conditions and we have error estimates for LFFD, SIFD1, SIFD2 & CNFD as (Bao, Cai, Jia & Yin, SCM,16’)

– Resolution

3 1/ 4 2 /3&C h h Cτ ε ε≤ ≤

Extension to nonlinear Dirac equation

EWI-FP methodTSFP method– Step 1– Step 2

– Time symmetric, dispersive relation, mass conservation!

321

1d

t j jj

ii σ σε ε=

∂ Φ = − ∂ + Φ

∑

( )

( ) ( )

( )

*2 3 3

1

* *3 3

*2 3 3

1

( , ) ( ) ( ) ( , ) ( , ) ( , )

( , ) ( , ),

( , ) ( ) ( ) ( , ) ( , ) ( , )

d

t j jj

n n

d

t j j nj

i t x V x I A x t x t x t x

t x t x t t

i t x V x I A x t x t x t x

σ δ σ σ

σ σ

σ δ σ σ

=

=

∂ Φ = − Φ + Φ Φ Φ

⇒ Φ Φ = Φ Φ ≥

∂ Φ = − Φ + Φ Φ Φ

∑

∑

( ) ( )*1 3 2 1 1 3 32

1 ( , ) ( , )t xii V t x I A t xσ σ σ δ σ σε ε

∂ Φ = − ∂ + Φ + − Φ + Φ Φ Φ

Error estimates of EWI-FP for NLDE

Theorem Under some stability conditions and , we have error estimates for EWI-FP as (Bao, Cai, Jia & Yin, SCM, 16’)

– Resolution

01/2 2( ) ( ), ( ) (1), 0 1mO O h O Oτ ε δ ε δ ε= = = = <

2 1/ 4C hτ ε≤

Error estimates of TSFP for NLDE

Thm. Under some stability conditions and , we have error estimates for TSFP as (Bao, Cai, Jia & Yin, SCM, 16’)

– Resolution

Thm. If time step satisfies

01/2 2( ) ( ), ( ) (1), 0 1mO O h O Oτ ε δ ε δ ε= = = = <

22 / Nτ πε=

2Cτ ε≤

Spatial Errors of TSFP for NLDE

Temporal Errors of TSFP for NLDE

2Cτ ε≤ ⇒

TSFP for Dirac without magnetic potential

0 02 2

22 2( , ) ( ) ; ( , ) ( ) uniform accuratem mn n

n M n ML Lt I h t I hτ τ ε

εΦ − Φ ≤ + Φ − Φ ≤ + + ⇒

TSFP for NLDE without magnetic potential

0 02 2

22 2( , ) ( ) ; ( , ) ( ) uniform accuratem mn n

n M n ML Lt I h t I hτ τ ε

εΦ − Φ ≤ + Φ − Φ ≤ + + ⇒

Conclusion & future challenges

Conclusion– For Dirac equation in nonrelativistic limit regime

• FDTD methods:• An EWI-FP method:• A TSFP method :

– A uniformly accurate (UA) method

Future challenges– Extension of the UA method to nonlinear Dirac equation (Y. Cai & Y.Wang, 17’)

– For coupled systems – Klein-Gordon-Dirac, Maxwell-Dirac, …

– For other parameter limit regimes

( )2 2 6 3/ / ( ) & ( )O h h O Oε τ ε ε τ ε+ ⇒ = =

( )2 4 2/ (1) & ( )mO h h O Oτ ε τ ε+ ⇒ = =

( )2 2 2 2/ (1) & ( )mO h h O Oτ ε τ ε τ ε≤ ⇒ + ⇒ = =

( )2

220 1

max min , (1) & (1)m mh O h h O Oε

τ ε τ τε< ≤

+ = + ⇒ = =