2020 Ver.3
1 TDGL Affine Integrator (AFI) AFITetsuya Matsuno, Edmund Soji Otabe, and Yasunori Mawatari “Explicit Integrators Based on a Bipartite Lattice and a Pair of Affine Transforma-
tions to Solve Quantum Equations with Gauge Fields,” J. Phys. Soc. Jpn. 89, 054006
(2020)
https://journals.jps.jp/doi/abs/10.7566/JPSJ.89.054006
AFI SI SI SI TDGL AFI TDGLTDGPAFI Lie-Trotter-SuzukiLTS LTS AFI 4Runge-Kutta4 AFI
1
dq
∂p (4)
exp(τ(Q + P )) Lie-Trotter-Suzuki LTS [1]exp(τQ) exp(τ P )LTS τ
exp(τ(Q+ P )) = exp(τQ) exp(τ P ) +O(τ 2) (5)
exp(τ(Q+ P )) = exp((τ/2)Q) exp(τ P ) exp((τ/2)Q) +O(τ 3) (6)
Symplectic Integrator(SI)
exp
( τp
(4)∂/∂q p∂/∂p q separable
2
exp
( τp
E = p2
2 + K
3 SI
dq
3
exp
( τp
)( q
p
) (21)
exp(τ(Q+ P )) = exp(τQ) exp(τ P ) +O(τ 2) (23)
exp(τ(Q+ P )) = exp((τ/2)Q) exp(τ P ) exp((τ/2)Q) +O(τ 3) (24)
SI exponential integrator[2] EI ζ −→ 0 SISI
4 Affine Integrator(AFI)
AFITime-Dependent

dp
(26)
4
dq
dp
(27)

−β|q|2q −→ −β|p|2q, −β|p|2p −→ −β|q|2p (28)
(27)
∂p
)( q
p
) (29)
ζq = Kq + α + β|p|2, ζp = Kp + α + β|q|2 (30)
(28)
exp
)( q
p
) (31)
exp
)( q
p
) (32)

2. q p q p
SI AFI
5
du1 dt
du3 dt
du2 dt
du4 dt
(33)

u1 = (w12u2 + w14u4)/2,
u3 = (w32u2 + w34u4)/2,
u2 = (w21u1 + w23u3)/2,
u4 = (w41u1 + w43u3)/2
(33)wij −2 2 1 7.4
ui 7.5 ui ui
ζi = ni + αi + βi|ui|2, i = 1, · · · , N (34)
ni = 2, i = 1, · · · , N ; N = 4 (33)
dui dt
6
d
dt
(36)
q p ζQ ζP
q =
)( q
p
) (39)
W † W T (39)
d
dt
( q
p
∂
exp(τQ)
( q
p
) (43)
7
ai = exp(−ζiτ), bi = (1− ai)/ζi, i = 1, · · · , N (44)
(41) (42) Q P (q,p)T 4 AFI
1 2 2 z 3 3 3YouTube
https://www.youtube.com/user/pftetsuyaGPU/videos
8
exp(τ(Q+ P )) (45)
Lie-Trotter-Suzuki(LTS) Q P Q P
exp(τ(Q+ P )) = exp(τQ) exp(τ P ) +O(τ 2) (46)
τ nAFI 2
exp(τ(Q+ P )) = exp((1/2)τQ) exp(τ P ) exp((1/2)τQ) +O(τ 3) (47)
2 LTS [3] 1
γ ∂ψ
γ ∂ψ
α = β = 0AFI LTS LTS α = 0 β = 0AFI
LTS
exp(τ(A+B)) = exp((1/2)τA) exp(τB) exp((1/2)τA) +O(τ 3) (50)
[1]2 [3]
exp(τ(A+B)) = exp(a1τA) exp(b1τB) exp(a2τA) exp(b1τB) exp(a1τA) +O(τ 3),
a1 = 1
6 (3−
1
n
n
2 + (α + β|ψmax(t)|2)h2 ≥ 0, for all t (53)
|ψmax| t (52) (53)AFI (52) (53) 2LTSLTS

exp(τ(A+B)) = exp(a1τA) exp(b1τB) exp(a2τA) exp(b2τB) exp(a3τA) exp(b3τB)× exp(a3τA) exp(b2τB) exp(a2τA) exp(b1τB) exp(a1τA) +O(τ 5),
a1 = 0.095848502741203681182,
a2 = 0.078111158921637922695,
a3 = 1
b2 = 0.12039526945509726545, b3 = 1− 2(b1 + b2)
42 1 AFI-4 CFL(Courant-Friedrichs-Lewy)τ = h2 5 4Runge-Kutta 2 4AFIAFI44Runge-KuttaRK4 AFI AFIRK 2 AFI4 1RK4 2
10
ER R
RK stable AFI-4 stable
2: L = 10, N = 100, h = 0.1, γ = 1, T = 10 τERRh,τ
Affine Integrator (AFI): τ ( h = 0.1)
τ 10-4 10-3 10-2 10-1

3: LTS [3] LTS
6.3 LTSLTS LTS [3] LTS 3 LTS AFIRunge-Kutta(RK) LTSLTS RKDuFort-Frankel(DF)DF 2AFI-4-opt DFCFLAFI4-opt

11
Total Energy: AFI
Total Prob: AFI
5AFI LTS 4 LTS LTS 3 6 SI
LTSLTS
5: AFI
6: Symplectic integrator (SI) SILTS
7.3
7: 2 E. Babaev
13

ψi+1 = ψ(x+ h), ψi−1 = ψ(x− h) (55)

wi x = exp
(58)
dx (x) +
x ψi−1 − 2ψi)/h 2 +O(h) (62)
(62)
=
ψi = 1
2 (wi
ψi (59) (63)
exp
( −ihAx
( x+
h
2
2 (wi
ψi ψi h2
15
7.6 AFI d q q′ 1
q′ = Aq + b (68)
A 1 d× db d (68) d 1 d + 1 d d+ 1 1 8(a) 2d1 8(a) AFI qp 8(b) d 2d 1 8(c) qp
bq = Bqp, bp = Bpq (69)
Bq Bp
q' A q b= +
(c)
8: (a): 1(b): AFI (c): AFI
16
P
Q
WPQ
Q
R
WPR
WQP
WRP
WQR
WRQ
[1] M. Suzuki, “Fractal decomposition of exponential operators with applications to
many-body theories and Monte Carlo simulations,” Phys. Lett. 146 (1990) 319.
[2] S. M. Cox and P. C. Matthews, “Exponential Time Differencing for Stiff Systems,”
J. Compt. Phys. 176 (2002) 430.
[3] T. Barthel and Y. Zhang, “Optimized Lie-Trotter-Suzuki decompositions for two
and three non-commuting terms, ” Annals of Physics 418 (2020) 168165.
17