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An Optimal Nearly-Analytic Discrete Method for
2D Acoustic and Elastic Wave Equations
Dinghui Yang
Depart. of Math., Tsinghua University
Joint with Dr. Peng, McMaster University
Supported by the MCME of China and the MITACS
Outline
• Introduction• Basic Nearly-Analytic Discrete Method(NADM)• Optimal Nearly-Analytic Discrete Method
(ONADM)• Numerical Errors and Comparisons• Wave-Field Modeling• Conclusions
Introduction
Computational GeophysicsGeophysics: a subject of studying the earth problems such as inner
structure and substance, earthquake, motional and changing law, and evolution process of the earth.
Computational Geophysics: a branch of Geophysics, using computational mathematics to study Geophysical problems.
Example: Wave propagation.
Model
fz
u
x
u
t
u
2
2
2
2
2
2
2
1
Problems to be solved: acoustic and elastic wave equations derived from Geophysics.
Computational Issues:• numerical dispersion, computational efficiency, computational costs and storages, accuracy.
Mathematical Model
For the 2D case, the wave equation can be written as
3,1;3,2,1,2
2
jit
uf
xi
i
j
ij
(1)
:ij :ifStress, Force source.:iu displacement component,
Let ,,, zyx uuuU P
t
U
2
2
,,,T
z
U
x
UUU
,,,T
z
P
x
PPP
Ut
W
Computational Methods
1. High-order finite-difference (FD) schemes (Kelly et al.,1976; Wang et al., 2002)
2. Lax-Wendroff methods (Dablain, 86)
3. Others like optimally accurate schemes (Geller et al., 1998, 2000),
pseudo-spectral methods (Kosloff et al., 1982)
Basic Nearly-Analytic Discrete Method (NADM)
Using the Taylor expansion, we have
(2)
(3)
Where denotes the time increment.
We converted these high-order time derivatives to the spatial
derivatives and included in Eqs. (2) and (3).
n
ji
nji
n
ji
n
ji
n
jit
PtP
tWtUU
,
3
,
2
,,
1
,6
)(
2
)(
,24
)(
,
2
24n
jit
Pt
nji
nji
n
ji
n
jit
PtPtWW ,
2
,,
1
, )(2
)(
n
jit
Pt,2
23
)(6
)(
t
n
ji
lk
lk
zx
W
,
n
ji
lk
lk
zx
U
,
Actually, equation (1) can be rewritten as follows
with the operators
Where and are known elastic constant matrices.
So we have
,1
)( 21 FULL
+Pt
U
2
2
),(1
211 zC
xC
xL
),(
1432 zC
xC
zL
321 ,, CCC 4C
,1
)( 21 Ft
WLLPt
etc.
To determine the high-order spatial derivatives, the NADM introduced the following interpolation function
,)(!
1),(
4
0
Uz
Zx
Xr
ZXG r
r
,1
)(221 x
FFx
ULLx
Px
,1
))(()(2
2
212
212
2
t
FFLLULLP
t
Interpolation connections
At the grid point (i-1, j):
,)0,( ,1,n
ji
n
ji UxG
,)()0,( ,1
,
nji
n
ji
Ux
xGX
nji
n
ji
Uz
xGZ ,1
,
)()0,(
Spatial derivatives expressed in term of the wave displacement and its gradients.
nji
nji
nji
nji UUU
xU
x ,1,,12,2
2
2()(
2)(
],)()[(2
1,1,1
nji
nji U
xUxx
)2()(
2)( 1,,1,2,2
2nji
nji
nji
nji UUU
zU
z
],)()[(2
11,1,
nji
nji U
zUzz
])()[(2
1])()[(
2
1)( 1,1,,1,1,
2nji
nji
nji
nji
nji U
xUxz
Uz
Uzx
Uzx
),(4
11,11,11,11,1
nji
nji
nji
nji UUUU
zx
etc.
Ideas: use the forward FD to approximate the derivatives of the so-called “velocity” , i.e.,
Computational Cost and Accuracy: 1. Needs to compute the so-called velocity and it’s
derivatives. 2. In total, 57 arrays are needed for storing the
displacement U, the velocity, and their derivatives.3. 2-order accuracy in time (Yang, et al 03)
)4(,
1
,,,
tzx
U
zx
U
zx
Wn
ji
lk
lkn
ji
lk
lkn
ji
lk
lk
W
Optimal Nearly-Analytic Discrete Method
Improving NADM:
• Reduce additional computational cost • Save storage in computation• Increase time accuracy
Observation
We have
)5(,24
)(
6
)(
2
)(
,
2
24
,
3
,
2
,,
1
,
n
ji
n
ji
nji
n
ji
n
ji
n
jit
Pt
t
PtP
tWtUU
nji
nji
n
ji
n
ji PtUUU ,21
,,
1
, )(2 )6(,
12
)(
,
2
24n
jit
Pt
Merits of ONADM
1. No needs to compute the velocity and it’s derivatives in (4);
2. Save storage (53%): in total only 27 arrays are used based on the formula (6);
3. Higher time accuracy: ONADM (4-order) VS NADM(2-order);
• ONADM enjoys the same space accuracy as NADM.
Numerical Errors and comparisons
The relative errors are defined by for the 1D case
and for the 2D case
(%)rE 100),()],([
1
2
1
1
2
1
2
N
iin
niN
iin
xtuuxtu
(%)rE 100),,(
),,(
1
2
1
1 1
2
,
1 1
2
N
i
N
jjin
njiN
i
N
jjin
zxtuuzxtu
1D caseInitial problem
,1
2
2
22
2
t
u
x
u
),2
cos(),0( xf
xu
).2
sin(2),0(
xf
ft
xu
and
Its exact solution
)(2cos),(
xtfxtu
Fig. 1. The relative errors of the Lax-Wendroff correction (line 1), the NADM (line 2), and the ONADM (line 3).
,10mx st 0001.0
Fig. 2. The relative errors of the Lax-Wendroff correction (line 1), the NADM (line 2), and the ONADM (line 3).
,30mx st 001.0
2D caseInitial problem
,1
2
2
22
2
2
2
t
u
z
u
x
u
),sin2
cos2
cos(),,0( 00 zf
xf
zxu
xf
ft
zxu 0cos
2sin(2
),,0(
)sin2
0 zf
Its exact solution
)sincos(2cos),,( 00
zxtfzxtu
Fig. 3. The relative errors of the second-order FD (line 1), the NADM (line 2), and the ONADM (line 3) for case 1.
Fig. 4. The relative errors of the second-order FD (line 1), the NADM (line 2), and the ONADM (line 3) for case 2.
Fig. 5. The relative errors of the second-order FD (line 1), the NADM (line 2), and the ONADM (line 3) for case 3.
Wave field modeling
Wave propagation equations
32
2
2
22
2
2
22
2
2
2
2
2
1
2
2
2
2
2
2
2
)2()(
)(
)()2(
fz
u
x
u
zx
u
t
u
fz
u
x
u
t
u
fzx
u
z
u
x
u
t
u
zzxz
yyy
zxxx
The time variation of the source function fi is
)2sin( 0tf )4/exp( 220
2 tf with f0=15 Hz.
Fig. 6. Three-component snapshots at time 1.4s, computed by the NADM.
Fig. 7. Three-component snapshots at time 1.4s, computed by the ONADM. It took about 3.4 minutes.
Conclusions
• The new ONADM is proposed.• The ONADM is more accurate than the NADM, L
ax-Wendroff, and second-order methods.
• Significant improvement over NADM in storage (53%) and computational cost (32%).
• Much less numerical dispersion confirmed by numerical simulation.
Future works
• Theoretical analyses in numerical dispersion, stability, etc.
• Applications in heterogeneous and porous media cases.
• 3-D ONADM.