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Evaluation Warning : The document was created with Spire.PDF for Python.
I
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I
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t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
14
,
No
.
6
,
Dec
em
b
e
r
20
24
:
6
5
1
9
-
6
5
2
5
6520
im
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3
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r
war
d
to
g
r
asp
as its
s
y
n
tax
clo
s
ely
r
esem
b
les th
at
o
f
th
e
C
p
r
o
g
r
a
m
m
in
g
lan
g
u
ag
e.
W
ith
GPUs
em
er
g
in
g
as
a
v
iab
le
s
u
b
s
titu
te
f
o
r
p
r
o
ce
s
s
o
r
s
in
p
ar
allel
c
o
m
p
u
tin
g
,
v
ar
io
u
s
p
ar
allel
tr
ian
g
u
latio
n
s
o
lv
er
s
an
d
h
y
b
r
i
d
ap
p
r
o
ac
h
es h
av
e
b
ee
n
d
e
p
lo
y
ed
o
n
GPUs
[
5
]
–
[
1
3
]
.
Fo
r
in
s
tan
ce
,
Z
h
an
g
et
a
l
.
[
5
]
in
itially
in
tr
o
d
u
ce
d
th
e
p
ar
allel
cy
clic
r
ed
u
ctio
n
(
PC
R
)
m
eth
o
d
an
d
s
u
b
s
eq
u
e
n
tly
s
u
g
g
ested
a
h
y
b
r
id
cy
clic
r
ed
u
ctio
n
-
p
ar
allel
cy
cli
c
r
ed
u
ctio
n
(
CR
-
PC
R
)
alg
o
r
it
h
m
.
T
h
e
h
y
b
r
id
PC
R
-
T
h
o
m
as
m
eth
o
d
was
also
p
r
o
p
o
s
ed
an
d
s
tu
d
ie
d
b
y
So
u
r
i
[
1
3
]
.
T
h
e
liter
atu
r
e
p
r
o
v
id
es m
an
y
e
x
am
p
les o
f
t
h
e
s
u
cc
ess
f
u
l
u
s
e
o
f
GPUs
f
o
r
wav
e
p
r
o
p
ag
atio
n
m
o
d
elin
g
[
1
4
]
–
[
2
4
]
.
I
n
th
is
wo
r
k
,
we
aim
to
ad
d
r
ess
th
e
n
u
m
er
ical
im
p
lem
en
t
atio
n
o
f
two
-
d
im
e
n
s
io
n
al
ac
o
u
s
tic
wav
e
p
r
o
p
a
g
atio
n
o
n
a
GPU,
p
r
o
v
id
in
g
in
s
ig
h
ts
in
to
t
h
e
p
o
ten
t
ial
ad
v
an
tag
es
an
d
ef
f
icien
c
y
o
f
th
is
ap
p
r
o
ac
h
.
L
ev
er
ag
in
g
th
e
p
ar
allel
co
m
p
u
tin
g
ca
p
ab
ilit
ies
o
f
GPUs
ca
n
s
ig
n
if
ican
tly
ac
ce
ler
ate
th
e
co
m
p
u
tatio
n
tim
e
,
m
ak
in
g
it
f
ea
s
ib
le
to
s
im
u
late
lar
g
er
an
d
m
o
r
e
co
m
p
lex
wav
e
f
ield
s
in
a
r
ea
s
o
n
ab
le
tim
ef
r
am
e.
Ad
d
itio
n
ally
,
th
is
im
p
lem
en
tatio
n
ca
n
o
f
f
er
a
s
ca
lab
le
s
o
lu
tio
n
f
o
r
r
ea
l
-
ti
m
e
ap
p
licatio
n
s
a
n
d
e
x
ten
s
iv
e
p
ar
am
eter
s
tu
d
ies,
h
ig
h
lig
h
tin
g
th
e
GPU'
s
r
o
le
as
a
p
o
wer
f
u
l to
o
l in
co
m
p
u
tatio
n
al
ac
o
u
s
tics
.
2.
G
O
VE
RNING
E
Q
UAT
I
O
N
AND
NUM
E
R
I
CA
L
SI
M
U
L
AT
I
O
N
W
e
in
v
esti
g
ate
th
e
ac
o
u
s
tic
wav
e
eq
u
atio
n
in
two
d
im
en
s
io
n
s
.
2
2
−
2
(
,
)
(
2
2
+
2
2
)
=
(
,
,
)
,
(
,
,
)
[
0
,
]
×
[
0
,
]
×
[
0
,
]
,
(
1
)
T
h
e
in
itial c
o
n
d
itio
n
s
ar
e
g
iv
e
n
as:
(
0
,
,
)
=
0
(
,
)
,
,
[
0
,
]
×
[
0
,
]
(
2
)
an
d
(
0
,
,
)
=
0
,
,
[
0
,
]
×
[
0
,
]
(
3
)
T
h
e
b
o
u
n
d
a
r
y
co
n
d
itio
n
s
ar
e:
(
,
,
0
)
=
0
,
(
,
,
)
=
0
,
[
0
,
]
,
[
0
,
]
×
[
0
,
]
(
4
)
an
d
(
,
0
,
)
=
0
,
(
,
,
)
=
0
,
[
0
,
]
,
[
0
,
]
×
[
0
,
]
(
5
)
I
n
th
i
s
c
o
n
t
ex
t
,
H
r
ep
r
e
s
en
t
s
th
e
w
av
e
f
u
n
c
t
i
o
n
,
q
d
e
n
o
t
e
s
th
e
s
o
u
r
c
e
t
e
r
m
,
a
n
d
c
in
d
i
ca
te
s
t
h
e
w
av
e
s
p
e
e
d
.
F
o
r
t
h
e
h
o
m
o
g
e
n
e
o
u
s
s
c
e
n
a
r
i
o
,
c
r
e
m
a
in
s
u
n
if
o
r
m
,
a
n
d
a
G
a
u
s
s
i
a
n
f
u
n
c
t
io
n
i
s
em
p
lo
y
e
d
a
s
th
e
s
o
u
r
c
e
t
e
r
m
.
I
n
o
u
r
n
u
m
er
ical
m
o
d
elin
g
a
p
p
r
o
ac
h
,
we
estab
lis
h
a
s
p
ac
e
-
tim
e
g
r
id
with
in
cr
em
en
ts
ℎ
1
,
ℎ
2
,
an
d
τ
,
r
esp
ec
tiv
ely
,
f
o
r
th
e
v
a
r
iab
les
,
,
an
d
:
ℎ
1
ℎ
2
=
{
=
ⅈ
ℎ
1
,
ⅈ
=
0
,
̅
̅
̅
̅
̅
;
=
ℎ
2
,
=
0
,
̅
̅
̅
̅
̅
;
=
,
=
0
,
1
,
2
…
}
(
6
)
Her
e,
we
s
et
ℎ
1
=
/
1
,
ℎ
2
=
/
2
,
an
d
=
/
.
Usi
n
g
th
is
g
r
id
,
we
ap
p
ly
t
h
e
f
in
ite
d
if
f
er
en
ce
tech
n
i
q
u
e
to
ap
p
r
o
x
im
ate
th
e
d
if
f
er
en
tial
eq
u
atio
n
p
r
esen
ted
in
p
r
o
b
le
m
(
1
)
alo
n
g
with
th
e
ac
co
m
p
a
n
y
in
g
co
n
d
itio
n
s
(
2
)
to
(
5
)
.
Fo
r
s
im
p
licity
,
let'
s
d
ef
in
e
as
1
=
2
an
d
d
en
o
te
ℎ
as
ℎ
1
=
ℎ
2
.
No
w,
let's
co
n
s
id
er
th
e
im
p
licit
f
in
ite
d
if
f
er
e
n
ce
s
ch
em
e
f
o
r
ad
d
r
ess
in
g
th
e
p
r
o
b
lem
o
u
tlin
ed
in
(
1
)
-
(
5
)
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
P
a
r
a
llel n
u
merica
l simu
la
tio
n
o
f th
e
2
D
a
co
u
s
tic
w
a
ve
eq
u
a
tio
n
(
A
r
s
h
yn
A
ltyb
a
y)
6521
ⅈ
,
+
1
−
2
ⅈ
,
+
ⅈ
,
−
1
2
−
ⅈ
,
ℎ
2
(
+
1
,
+
1
−
2
,
+
1
+
−
1
,
+
1
+
,
+
1
+
1
−
2
,
+
1
+
,
−
1
+
1
)
=
,
,
(
7
)
F
or
(
;
ⅈ
;
)
ℎ
1
ℎ
2
with
in
itial c
o
n
d
itio
n
s
,
0
=
,
,
,
1
−
,
0
=
,
,
(
8
)
F
or
(
ⅈ
,
)
∈
0
,
̅
̅
̅
̅
̅
×
0
,
̅
̅
̅
̅
̅
,
an
d
with
b
o
u
n
d
a
r
y
co
n
d
itio
n
s
0
,
=
0
,
,
=
0
,
,
0
=
0
,
,
=
0
,
(
9
)
F
or
(
,
)
∈
0
,
̅
̅
̅
̅
̅
×
0
,
̅
̅
̅
̅
̅
̅
an
d
(
ⅈ
,
)
∈
0
,
̅
̅
̅
̅
̅
×
0
,
̅
̅
̅
̅
̅
̅
,
r
esp
ec
t
iv
el
y
.
T
h
e
i
m
p
li
cit
s
ch
em
e
is
i
n
h
e
r
e
n
tl
y
s
t
a
b
le
r
e
g
a
r
d
less
o
f
t
h
e
c
h
o
s
e
n
s
te
p
s
izes
an
d
ac
h
i
e
v
es
a
n
ac
c
u
r
ac
y
o
f
(
+
|
ℎ
2
|
)
as
d
is
c
u
s
s
e
d
i
n
[
2
5
]
.
T
h
e
d
i
f
f
er
e
n
ce
(
7
)
is
ad
d
r
ess
ed
u
s
i
n
g
th
e
im
p
l
ici
t
alt
er
n
ati
n
g
d
ir
ec
ti
o
n
m
et
h
o
d
(
AD
I
)
,
w
h
i
ch
i
n
v
o
lv
es
s
p
l
itt
in
g
it
in
t
o
tw
o
s
e
p
a
r
at
e
s
u
b
t
ask
s
.
H
ⅈ
,
j
n
+
1
−
2
H
ⅈ
,
j
n
+
H
ⅈ
,
j
n
−
1
/
2
τ
2
−
c
ⅈ
,
j
h
2
(
H
i
+
1
,
j
n
+
1
2
−
2
H
i
,
j
n
+
1
2
+
H
i
−
1
,
j
n
+
1
2
)
=
q
i
,
j
n
,
(
1
0
)
H
ⅈ
,
j
n
+
1
−
2
H
ⅈ
,
j
n
+
1
/
2
+
H
ⅈ
,
j
n
−
1
/
2
τ
2
−
c
ⅈ
,
j
h
2
(
H
i
,
j
+
1
n
+
1
−
2
H
i
,
j
n
+
1
+
H
i
,
j
−
1
n
+
1
)
=
q
i
,
j
n
+
1
/
2
,
(
1
1
)
3.
NUM
E
RIC
AL
M
E
T
H
O
D
T
h
e
alter
n
atin
g
d
ir
ec
tio
n
im
p
licit
(
ADI
)
m
eth
o
d
,
wh
ich
u
t
ilizes
f
in
ite
d
if
f
er
e
n
ce
tec
h
n
i
q
u
es,
h
as
h
is
to
r
ically
b
ee
n
a
well
-
estab
lis
h
ed
m
eth
o
d
f
o
r
ad
d
r
ess
in
g
d
if
f
e
r
en
tial
eq
u
atio
n
s
i
n
co
m
p
lex
,
h
ig
h
-
d
im
en
s
io
n
al
s
ettin
g
s
.
I
n
itiall
y
p
r
o
p
o
s
ed
b
y
Peac
em
an
an
d
R
atch
f
o
r
d
[
2
6
]
,
t
h
e
m
eth
o
d
h
as
s
ee
n
s
ev
er
al
r
ef
in
em
en
ts
o
v
er
tim
e
[
2
7
]
–
[
3
1
]
.
C
h
ar
ac
ter
ized
b
y
im
p
lic
it
f
in
ite
d
if
f
er
en
ce
o
p
e
r
atio
n
s
,
th
e
ADI
m
eth
o
d
en
s
u
r
es
f
u
ll
s
tab
ilit
y
in
p
r
o
b
le
m
s
d
ev
o
id
o
f
m
ix
e
d
d
er
i
v
ativ
es
an
d
m
ain
tain
s
a
co
n
s
id
er
ab
le
s
tab
ilit
y
m
ar
g
in
in
ca
s
es
wh
er
e
m
ix
ed
d
er
i
v
ativ
es
ar
e
p
r
esen
t
[
1
0
]
.
B
y
em
p
lo
y
in
g
th
e
im
p
licit
s
u
b
-
cir
cu
it,
we
ap
p
ly
th
e
cy
cli
c
r
ed
u
ctio
n
(
C
R
)
m
eth
o
d
alo
n
g
th
e
x
-
d
ir
ec
tio
n
to
co
m
p
u
te
th
e
g
r
id
f
u
n
ctio
n
H
i
,
j
k
+
1
/
2
.
Su
b
s
eq
u
en
tly
,
in
th
e
s
ec
o
n
d
f
r
ac
tio
n
al
tim
e
s
tep
u
tili
zin
g
s
u
b
-
cir
cu
it,
th
e
C
R
m
eth
o
d
is
u
tili
ze
d
alo
n
g
th
e
y
-
ax
i
s
d
ir
ec
tio
n
,
y
ield
in
g
th
e
g
r
id
f
u
n
ctio
n
H
i
,
j
k
+
1
.
T
h
e
ADI
m
eth
o
d
d
em
o
n
s
tr
ates
s
ec
o
n
d
-
o
r
d
er
ac
cu
r
ac
y
o
f
(
τ
2
+
ℎ
2
)
.
T
h
e
en
s
u
in
g
n
u
m
er
ical
s
im
u
latio
n
s
ar
e
d
et
ailed
b
elo
w.
All
ca
lcu
latio
n
s
wer
e
p
er
f
o
r
m
ed
with
th
e
Py
th
o
n
p
r
o
g
r
am
m
in
g
lan
g
u
ag
e
an
d
em
p
lo
y
ed
th
e
c
y
clic
r
ed
u
ctio
n
tech
n
iq
u
e.
Fo
r
th
e
s
im
u
latio
n
s
,
a
tim
e
s
tep
o
f
=
0
.
0
0
1
an
d
a
s
p
atial
s
tep
o
f
ℎ
=
1
wer
e
u
s
ed
.
R
e
s
u
lts
wer
e
v
is
u
alize
d
with
th
e
Ma
tp
lo
tlib
lib
r
a
r
y
,
an
d
th
e
f
i
n
d
in
g
s
f
r
o
m
th
e
n
u
m
er
ical
e
x
p
er
im
e
n
ts
ar
e
illu
s
tr
ated
in
Fig
u
r
es 1
to
3.
T
h
ese
f
ig
u
r
es
d
ep
ict
b
o
th
th
e
tim
e
f
u
n
ctio
n
o
f
th
e
s
o
u
r
ce
an
d
th
e
p
r
o
p
ag
atio
n
o
f
th
e
wav
e
th
r
o
u
g
h
a
m
ed
iu
m
with
v
ar
y
i
n
g
p
r
o
p
er
ti
es
at
m
u
ltip
le
tim
e
p
o
in
ts
.
I
n
s
u
ch
a
m
ed
iu
m
,
th
e
s
p
ee
d
o
f
t
h
e
wav
e
f
lu
ctu
ates
ac
co
r
d
in
g
to
th
e
m
ed
iu
m
'
s
s
tr
u
ctu
r
al
v
ar
iatio
n
s
.
Ou
r
an
al
y
s
is
f
o
cu
s
es
o
n
h
o
w
th
e
wav
e
p
r
o
p
ag
ates
th
r
o
u
g
h
a
m
ed
iu
m
with
d
is
tin
ct
wav
e
v
e
lo
cities in
th
e
wh
ite
an
d
b
l
u
e
r
eg
io
n
s
.
Fig
u
r
e
1
.
So
u
r
ce
f
u
n
ctio
n
a
n
d
two
-
lay
er
h
eter
o
g
en
e
o
u
s
m
ed
i
u
m
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
14
,
No
.
6
,
Dec
em
b
e
r
20
24
:
6
5
1
9
-
6
5
2
5
6522
Fig
u
r
e
2
.
W
av
e
p
r
o
p
a
g
atio
n
at
d
if
f
er
en
t tim
e
s
tep
s
Fig
u
r
e
3
.
W
av
e
p
r
o
p
a
g
atio
n
th
r
o
u
g
h
a
h
eter
o
g
en
e
o
u
s
m
ed
iu
m
at
d
if
f
er
e
n
t tim
e
in
ter
v
als
4.
P
ARAL
L
E
L
E
X
E
CU
T
I
O
N
USI
NG
T
H
E
G
RAP
H
I
C
S P
RO
CE
SS
I
NG
UN
I
T
A
g
r
ap
h
ics
p
r
o
ce
s
s
in
g
u
n
it
(
GPU)
r
ep
r
esen
ts
a
m
ass
iv
ely
p
ar
allel,
m
u
lti
-
co
r
e
p
r
o
ce
s
s
o
r
r
en
o
wn
e
d
f
o
r
its
s
u
b
s
tan
tial
c
o
m
p
u
tatio
n
al
ca
p
ac
ity
.
I
ts
af
f
o
r
d
ab
ilit
y
,
i
m
p
r
ess
iv
e
f
lo
atin
g
p
o
in
t
th
r
o
u
g
h
p
u
t,
an
d
ef
f
icien
t
m
em
o
r
y
ac
ce
s
s
h
av
e
g
ar
n
er
e
d
g
r
o
win
g
in
te
r
est
am
o
n
g
e
x
p
er
ts
in
ad
v
an
ce
d
co
m
p
u
tin
g
p
er
f
o
r
m
an
ce
[
2
4
]
.
T
h
e
C
UDA
im
p
lem
en
tatio
n
en
co
m
p
ass
es
th
r
ee
p
r
im
ar
y
s
tag
e
s
o
f
co
m
p
u
tatio
n
:
d
ata
tr
an
s
f
er
to
GPU
g
lo
b
al
m
em
o
r
y
,
e
x
ec
u
tio
n
o
f
th
e
C
U
DA
k
er
n
el
[
2
5
]
,
an
d
r
etr
iev
al
o
f
r
esu
lts
f
r
o
m
GPU
m
em
o
r
y
to
C
PU
m
em
o
r
y
.
Alth
o
u
g
h
v
ar
io
u
s
p
ar
allel
ap
p
r
o
ac
h
es
wer
e
i
n
v
esti
g
ated
i
n
[
3
2
]
,
th
is
s
tu
d
y
o
p
ts
f
o
r
th
e
C
UDA
ap
p
r
o
ac
h
,
u
tili
zin
g
th
e
cy
clic
r
e
d
u
ctio
n
t
ec
h
n
iq
u
e.
T
h
e
p
r
o
ce
d
u
r
e
f
o
r
s
o
lv
in
g
th
e
p
r
o
b
lem
is
d
etailed
in
Alg
o
r
ith
m
1
.
Alg
o
r
ith
m
1
.
E
x
ec
u
tio
n
o
f
2
D
wav
e
eq
u
atio
n
1.
Compute initial condition matrix H
0
Initialize matrix H with H
0
based on the initial condition (2).
2. Time
-
stepping loop
:
while
t < t
end
do
:
3.Solve in the x
-
direction
:
For each row j=0,.
..
,
m
For each column i=0,..
.
,
m
calculate
the
tridiagonal system elemen
ts a
i
,b
i
,c
i
,f
i
call
the
function
Cyclic_Re
(a
i
,b
i
,c
i
,f
i
,y
i
,
m
)
to solve the tridiagonal system.
calculate matrix H
x
4.Solve in the y
-
direction
:
For each column
i=0,...
,
m
For each row j=0,...
,
m
calculate
the
tridiagonal system elements a
j
,b
j
,c
j
,f
j
call
the
function
Cyclic_Re
(a
j
,b
j
,c
j
,f
j
,y
j
,
m
)
calculate matrix H
y
5.Update the matrices
:
Swap
H
with
H
x
Swap
H
0
with
H
y
.
Update the time step
t
by incrementing
t
by
Δt
.
6
. End of time
-
stepping loop
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
P
a
r
a
llel n
u
merica
l simu
la
tio
n
o
f th
e
2
D
a
co
u
s
tic
w
a
ve
eq
u
a
tio
n
(
A
r
s
h
yn
A
ltyb
a
y)
6523
I
n
th
is
co
n
tex
t,
,
0
,
,
r
ep
r
esen
t
H
i
,
j
k
−
1
/
2
,
H
i
,
j
k
,
H
i
,
j
k
+
1
/
2
,
H
i
,
j
k
+
1
r
esp
ec
tiv
ely
.
W
ith
in
th
e
ⅈ
_
(
)
f
u
n
ctio
n
,
t
h
er
e
a
r
e
th
r
ee
d
ev
i
ce
f
u
n
ctio
n
s
:
(
)
,
_
(
)
,
an
d
(
)
,
alo
n
g
s
id
e
o
n
e
h
o
s
t
f
u
n
ctio
n
,
ⅈ
(
)
.
I
n
itially
,
th
e
b
lo
ck
s
ize
n
ee
d
s
to
b
e
ca
lcu
l
ated
b
ased
o
n
th
e
m
atr
ix
s
ize
an
d
th
e
f
o
r
war
d
a
n
d
b
ac
k
war
d
s
u
b
-
s
tep
s
.
T
h
is
in
v
o
lv
es a
cy
cle
wh
er
e
we
iter
ate
o
v
er
th
e
co
m
p
u
tatio
n
for (i=0;
i<log2(
m
+1)
-
1;
i++) {
ste
_N
=(
m
-
pow(2.0, i+1))/pow(2.0, i+1)+1;
calDim
(
ste_N
,&dimBlock,&dimGrid);
CRForwa
<<>>(d_a,
d_b,
d_c,
d_f,
n,
ste_N
,
i);
}
wh
er
e
2
(
+
1
)
−
1
is
th
e
s
tep
n
u
m
b
er
,
an
d
th
e
_
v
ar
iab
le
is
u
s
ed
to
d
ete
r
m
in
e
th
e
r
e
q
u
ir
ed
b
l
o
ck
s
ize.
Af
ter
th
e
ⅈ
(
)
f
u
n
ctio
n
h
as
d
eter
m
in
ed
t
h
e
b
lo
ck
s
ize,
th
e
(
)
f
u
n
ctio
n
is
ex
ec
u
ted
2
(
+
1
)
−
1
tim
es.
T
h
u
s
,
th
e
s
y
s
tem
ca
n
r
ed
u
ce
th
e
e
q
u
atio
n
b
y
o
n
e.
T
h
e
b
lo
ck
s
ar
e
s
y
n
ch
r
o
n
ize
d
,
f
o
llo
wed
b
y
th
e
in
v
o
ca
tio
n
o
f
th
e
_
(
)
f
u
n
ctio
n
,
wh
ic
h
co
m
p
u
tes
two
u
n
k
n
o
w
n
s
.
Af
ter
th
at,
a
lo
o
p
is
u
s
ed
with
th
e
f
o
llo
win
g
s
tr
u
ct
u
r
e:
for (i=log2(
m
+1)
-
2;
i>=0;
i) {
ste_N
=(
m
-
pow(2.0,
i+1))/pow(2.0, i+1)+1;
calDim
(
ste_N
, &dimBlock, &dimGrid);
CRBackw
<<>>(d_a,
d_b,
d_c,
d_f,
d_x,
n,
ste_N
,
i);
}
I
n
th
is
co
n
tex
t,
t
h
e
b
ac
k
war
d
s
u
b
s
titu
tio
n
p
r
o
ce
s
s
is
ex
ec
u
ted
2
(
+
1
)
–
2
tim
es,
as
th
e
in
itial
b
ac
k
wa
r
d
s
u
b
s
titu
tio
n
s
u
b
-
s
tep
is
h
an
d
led
b
y
th
e
ⅈ
f
u
n
ctio
n
.
Su
b
s
eq
u
en
tly
,
th
e
d_x
ar
r
ay
is
co
m
p
u
te
d
,
an
d
th
en
th
is
co
m
p
u
ted
d
ata
is
tr
an
s
f
er
r
ed
f
r
o
m
th
e
d
ev
ice
t
o
th
e
h
o
s
t
u
s
in
g
th
e
cu
d
a
Memcp
y
f
u
n
ctio
n
with
th
e
p
ar
am
eter
s
,
_
,
ⅈ
(
)
∗
,
an
d
ⅈ
.
5.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
I
n
th
e
f
o
llo
win
g
s
ec
tio
n
,
we
p
r
esen
t
th
e
f
in
d
in
g
s
d
er
i
v
ed
f
r
o
m
ex
p
e
r
im
en
ts
co
n
d
u
cte
d
o
n
a
d
esk
to
p
s
y
s
tem
eq
u
ip
p
ed
with
a
GeFo
r
ce
R
T
X
2
0
8
0
s
etu
p
f
ea
tu
r
i
n
g
4
3
5
2
c
o
r
es.
T
h
e
s
y
s
tem
also
co
m
p
r
is
es
an
NVI
DI
A
GPU,
I
n
tel
C
o
r
e
(
T
M)
i7
-
9
8
0
0
X
Pro
ce
s
s
o
r
clo
ck
ed
at
3
.
8
0
GHz
,
an
d
6
4
GB
R
AM
.
T
h
e
m
o
d
elin
g
p
ar
am
eter
s
ar
e
co
n
f
ig
u
r
ed
as
f
o
llo
ws:
th
e
g
r
id
s
ize
r
em
ai
n
s
co
n
s
is
ten
t
in
b
o
th
d
ir
ec
tio
n
s
,
with
∆
=
∆
,
wh
ile
th
e
tim
e
s
tep
∆
is
s
et
to
0
.
0
5
.
T
h
e
s
im
u
latio
n
d
u
r
atio
n
is
T
=
1
0
.
0
,
r
esu
ltin
g
i
n
a
t
o
tal
o
f
2
0
0
-
tim
e
s
tep
s
.
Fo
r
a
m
o
r
e
co
m
p
r
eh
en
s
iv
e
ev
alu
atio
n
,
we
c
o
n
d
u
cted
test
s
u
s
in
g
f
o
u
r
d
if
f
er
en
t
c
o
m
p
u
tatio
n
al
d
o
m
ain
s
izes: 5
1
2
×5
1
2
,
1
0
2
4
×1
0
2
4
,
2
0
4
8
×2
0
4
8
,
an
d
4
0
9
6
×4
0
9
6
.
T
h
e
ef
f
ec
tiv
en
ess
o
f
a
p
ar
alle
l
alg
o
r
ith
m
is
ass
es
s
ed
b
y
m
ea
s
u
r
in
g
its
ac
ce
ler
atio
n
.
Acc
eler
atio
n
is
d
eter
m
in
ed
b
y
c
o
m
p
ar
in
g
th
e
b
est
r
u
n
tim
e
o
f
a
s
eq
u
e
n
tial
alg
o
r
ith
m
with
th
e
lo
n
g
est
r
u
n
t
im
e
o
f
th
e
p
ar
allel
alg
o
r
ith
m
f
o
r
a
g
iv
en
p
r
o
b
lem
.
T
ab
le
1
p
r
o
v
id
es
in
s
ig
h
ts
in
to
th
e
ex
ec
u
tio
n
d
u
r
atio
n
s
(
in
s
ec
o
n
d
s
)
f
o
r
b
o
t
h
th
e
s
eq
u
en
tial
im
p
lem
en
tatio
n
(
C
PU
tim
e)
an
d
th
e
C
UDA
im
p
lem
en
tatio
n
(
GPU
tim
e)
.
Ad
d
itio
n
ally
,
th
e
tab
le
p
r
esen
ts
th
e
ex
ec
u
tio
n
tim
es
o
f
cy
clic
r
ed
u
ctio
n
m
eth
o
d
s
f
o
r
task
s
,
alo
n
g
with
th
e
co
r
r
esp
o
n
d
in
g
ac
ce
l
er
atio
n
c
o
ef
f
icien
ts
ac
h
iev
ed
o
n
d
iv
er
s
e
d
e
v
ices.
T
h
is
co
m
p
ar
ativ
e
an
al
y
s
is
h
ig
h
lig
h
ts
th
e
ef
f
icien
cy
g
ain
s
o
f
f
er
ed
b
y
th
e
GPU
im
p
lem
en
tatio
n
,
d
em
o
n
s
tr
atin
g
th
e
p
o
ten
tial
f
o
r
s
u
b
s
tan
tial
p
er
f
o
r
m
an
ce
im
p
r
o
v
em
e
n
ts
wh
en
u
s
in
g
p
ar
allel
co
m
p
u
tin
g
tech
n
iq
u
es o
v
er
tr
a
d
itio
n
al
s
eq
u
en
tial m
eth
o
d
s
.
T
ab
le
1
.
Per
f
o
r
m
an
ce
m
ea
s
u
r
e
m
en
ts
an
d
ac
ce
ler
atio
n
an
aly
s
i
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ru
.
Evaluation Warning : The document was created with Spire.PDF for Python.