Multiscale Methods and Analysis for the Dirac Equation in ...
Transcript of 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: [email protected]
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< ≤
+ = + ⇒ = =