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

TIM case

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.

Figure 8. Synthetic seismograms.

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.

Thanks

Absorbing boundary conditions

0)cos

(01

xi

n

i

ux

v

t

Where 2/0 i