GEOPHYSICS, VOL. 61, NO. 4 (JULY-AUGUST 1996); P. 1160-1174, 13 FIGS. I TABLE.

Spectral-domain decomposition methods for the solutionof acoustic and elastic wave equations

Ezio Faccioli *, Fabio Maggio, Alfio Quarteroni **, and Aldo Tagliani§

ABSTRACT

A new spectral-domain decomposition method is pre-sented for acoustic and elastodynamic wave propagationin 2-D heterogeneous media. Starting from a variationalformulation of the problem, two different approachesare proposed for the spatial discretization: a mixedFourier-Legendre and a full Legendre collocation. Thematching conditions at subdomain interfaces are care-fully analyzed, and the stability and efficiency of time-advancing schemes are investigated. The numerical val-idation with some significant test cases illustrates theaccuracy, flexibility, and robustness of our methods.These allow the treatment of complex geometries andheterogeneous media while retaining spectral accuracy.

INTRODUCTION

The Fourier method, successfully employed in recent yearsfor the solution of the dynamic elasticity equations (Gazdag,1981; Kosloff and Baysal, 1982; Kosloff et al., 1990; Faccioli,1991), suffers from drawbacks in handling nonperiodic bound-ary conditions, such as the free-surface condition, which isimportant in exploration geophysics as well as in seismology.The problem can be computationally tackled by using "zeropadding" (Kosloff and Baysal, 1982). Another approach re-places the expansion in trigonometric polynomials in thevertical direction by one in Chebychev polynomials (Kosloff etal., 1990; Priolo and Seriani, 1991), which are nonperiodic andcan easily accommodate the free-surface condition. However,algebraic polynomials (either Chebychev's or Legendre's) re-quire a nonuniform vertical grid, i.e., the nodes cluster towardthe edges of the domain and are rarefied in the interior(Canuto et al., 1988). Stability of explicit time marchingalgorithms requires (in this case) a time-step Ot that behaveslike N -2 (as opposed to N -1 for the Fourier method), where

N + 1 is the number of grid points used in each spatialdirection. This choice causes a single-domain application ofthe Chebychev method for solving realistic geophysical prob-lems in two or three spatial dimensions to be expensive. Whileretaining an expansion in algebraic polynomials in the verticaldirection, to overcome both the geometric and the time-steprestrictions, we resort to the alternative approach of dividingthe physical domain into subdomains (Canuto et al., 1988), asshown in Figure 1. In this way:

1) the spectral accuracy is maintained with fewer points ineach subdomain, so that exceedingly small values of Atcan be avoided;

2) a natural framework is provided to accommodate differ-ent material parameters in different regions of the model;and

3) the implementation of spectral methods on parallelcomputers is facilitated.

Spectral domain decomposition methods have proven to bevery effective for fluid dynamics applications (Maday andPatera, 1989; Quarteroni, 1991). The method that we advocateis a straightforward extension of these methods to the equa-tions of acoustics and elasticity. For wave propagation prob-lems, different domain decomposition approaches have beenused in Carcione (1991), Priolo and Seriani (1991), andTessmer et al. (1992), among others. The main differencesbetween our approach and the one of Priolo and Seriani residein the capability of our method to incorporate absorbingboundary conditions in the variational formulation, withoutresorting to absorbing boundary regions. Besides, when usingfinite-difference methods for advancement in time, our ap-proach yields naturally to diagonal mass matrices (because ofthe use of Gaussian integration), and therefore to fully explicitschemes. More generally, this approach (like that of Priolo andSeriani), which stems from a variational formulation, allows aflexible treatment of boundaries and interfaces, unlike thosebased on a more "point-wise" collocation form. In this paper,

Manuscript received by the Editor November 15, 1993; revised manuscript received May 22, 1995.*Dept. of Structural Engineering, Technical University of Milan, I-20133 Milan, Italy.$Centro di Ricerca, Sviluppo e Studi Superiori in Sardegna (CRS4), Cagliari, Italy.**Dept. of Mathematics, Technical University of Milan, Italy and Centro di Ricerca, Sviluppo e Studi Superiori in Sardegna (CRS4), Cagliari, Italy.§Dept. of Mathematics, Technical University of Milan, I-20133, Milan, Italy.© 1996 Society of Exploration Geophysicists. All rights reserved.

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Spectral Decomposition for Wave Modeling

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we apply the subdomain spectral technique to solve the 2-Dproblem of acoustic and elastic wave propagation, startingfrom the variational, or "weak," form of the equations ofmotion. The nonperiodic spatial discretization is performed byLegendre nodes rather than by Chebychev nodes, owing to thesimpler form obtained for the subdomain interface conditions.

In 2-D problems, one can assume periodic conditions on thelateral boundaries, and discretize by evenly spaced nodeshorizontally, so that a Fourier expansion can be used(Figure 2a). On the other hand, a full-Legendre approximationcan also be convenient when nonperiodic conditions (such asreflecting and absorbing conditions) apply on the wholeboundary (Figure 2b). It is emphasized that the discretizationby Legendre nodes does not entail more computational diffi-

'Top

F'

c1

F2

IF

F3

Q2

F4

FBot

FIG. 1. Decomposition of the computational domain fl into twosubdomains 11 1 and f1 2 .

a)

culty than that by Chebychev nodes (although the latter aredefined analytically). Efficient routines are available (Canutoet al., 1988) that provide both the Legendre nodes and theweights needed to construct the spectral differentiation matrixby which the spatial derivatives are computed.

Domain partitioning can be used at least in two ways. Let usdenote by H the typical size of each subdomain, by N thepolynomial degree of the spectral solution, and assume thatthe measure of the domain fi is normalized to 1. Then:

1) The first option is characterized by large values ofboth H and N, i.e., H is of the order of 1 and N islarge, say between 10 and 100.

2) The second option makes use of many small sub-domains and low degree polynomials inside eachsubdomain, i.e., H << 1 and N is small, saybetween 1 and 10.

When H is very small and N = 1 or 2, case (2) resembles theclassical h -version of the finite-elements method, whereas, case(1) is completely similar to the p-version of finite elements,see, e.g., Szabo and Babuska (1991) and Quarteroni and Valli(1994). When N is reasonably low, the spectral solution can beeasily handled on a matrix-by-vector basis. By that we meanthat one can compute the matrix required for polynomialdifferentiation by using the so-called pseudospectral derivativematrix, which allows the calculation of derivatives at theLegendre nodes, as illustrated in the Appendix [see in partic-ular equation (A-16), and Canuto et al., 1988]. Furthermore,domain-decomposition techniques [especially under approach(2)] make it possible to address fairly general geometriesbecause they allow a natural treatment of both physicalboundaries and subdomain interfaces, as well as absorbingboundaries.

The single-domain Legendre approximation for the scalarwave equation in heterogeneous media has been introduced inMaggio and Quarteroni (1994). In this paper we will illustrateits generalization to the multidomain case. Partitions like thoseillustrated in Figure 6 will be considered.

An outline of this paper is as follows. First we formulate ina variational form the 1-D problem and derive its multidomain

b)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0x-axis

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0x-axis

FIG. 2. Mixed Fourier-Legendre (a) and full Legendre spectral grids (b). Figures refer to the square [0, 11 2 discretized by 11 X 11gridpoints.

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version. Then, for the 2-D case, we address both the scalar(acoustic) and the vector (elastic) problem. The domain-decomposed problem is approximated by the generalizedGalerkin method based on the Legendre's expansion, or on acombined Fourier-Legendre approach, and the structure ofthe interface equations is investigated. Next, we discuss brieflytime-advancing procedures to be combined with the spatialequations derived previously. The last section contains somenumerical applications in which the flexibility and accuracy ofthe proposed method are demonstrated on structures withvarying degrees of complication. The Appendix provides fulldetails of the derivation of the generalized Galerkin methodthat we advocate. The issue of stability is addressed in thesection devoted to the numerical results.

MATHEMATICAL FORMULATION OF THE PROBLEM

For the sake of exposition, we first consider propagation ofwaves in one spatial dimension in the presence of two subdo-mains. The extension to 2-D propagation over several subdo-mains will be evident afterward.

The 1-D elastic wave equation for a material with constantdensity and variable elastic modulus can be written, in theabsence of body sources, as

a Iu a ( C2 aU)at , az

az , 0 `z `I 1, t > 0, (1)

where u is the displacement, and c = c(z) is the pertinentpropagation speed of the material. At the upper surface z = 0,the stress-free boundary condition reads

au=0 atz=0,t>_0. (2)

az

At the bottom boundary z = L 1 , a radiation condition (i.e.,first-order paraxial condition) can be conveniently imposed inthe form

au 1 auaz + c at =0 atz=L1,t>_0. (3)

This problem needs to be supplemented by initial conditionson both u and au/at at t = 0. To start with, we split thematerial domain I = (0, L 1 ) into I i U 1 2 , where I i = (0,y] and l2 = [y, L 1 ), for some y : 0 < y <L 1 . For t > 0,the original problem (1)–(3) can be formulated as

a 2u a au

at c^ az I c1 az^)in l i , (4a)

au l

az—=0 atz = 0, (4b)

azu--- = -

a auat^^ az ^ c2 az2)

in 1 2 , (4c)

au 2 1 au z

az C2 atat z = L i , (4d)

U 1= u 2 at z = y, (4e)

2 3u1 2 au 2c1 ---- =c2 - ---at z ='/, (4f)

where the notation u i = u1 1,, c, = cFI, is understood. This isequivalent to restricting (1) to both intervals I 1 and I2 andenforcing the prescribed boundary conditions (2) and (3), andrequiring continuity of both the displacement and the stressacross the interface point z = y (note that c may be differentwhen approaching z = y from different sides).

An equivalent mathematical formulation, more suitable forderiving our numerical approximation, can be obtained by a"variational argument." Without pretending to be rigorous, wewant, however, to outline the basic feature of the variationalformulation.

Let v = v(z) be a smooth test function. If we multiply bothsides of (1) by v and integrate over I l and I2 , we obtain(applying integration-by-parts)

2 ff a 2u,(t) fja aui(t)

v dz = — c,. ^ v dz

az azat2, i1 ,

2

(

Z alli(t) aV Z au1(t)v_ – c — — dz + –c 1 z=oaz az az

l 1 Ii

2 au 1 (t) v I 2 au 2 (t) v I 2 all 2 (t) v+c1 az

== Y — C2 az=1 + CZ az ==L i

In view of equations (4b) and (4f), the first three termsinside the brackets vanish. Moreover, because of equation(4d), it follows that

2 '22 au 2

C2v C2 at V at z = L 1.

We conclude that for each t, u 1 and u 2 satisfy the varia-tional problem

2a 2at2 i 8u;(t) —äv

atz v dz = –c^ az az dz

i=1 ii,

au 2(t)—C2at VIz=L,

u I (t)=u 2 (t) atz=y. (5)

Formally speaking, equation (5) has been obtained fromequation (1) after taking at each t > 0 the variation withrespect to all admissible test functions. The latter are preciselyprovided by all functions v:I – R that are square-integrablealong with their first derivative.

SPECTRAL APPROXIMATION BY LEGENDRE COLLOCATION

1-D model

Referring to the Appendix for details and notation, for eacht > 0 we look for a polynomial U N : (0, L 1 ) – II such thatu N I 1, = uN`O, i = 1, 2 satisfy the following set of equations.

1) Internal equations:

82u a rII/ all (z^ lIN — L J (`)( 2 N J =0

at 2 az LL N \ ` az

atzJ`);1<j<_N-1,i=1,2. (6)

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The { zf` ) , 0 <_ j < N } are the Gauss-Lobatto nodes ofthe interval I . Remember that JN` ) w is, for any functionw, its interpolant at the nodes {zf` ) , 0 < j < N}. Whenc i is constant in I ; , JN` ) can be disregarded inequation (6). Hence, (6) states that the differentialequation is satisfied (i.e., it is collocated) at all Gauss-Lobatto nodes internal to I.

2) Boundary conditions:

2au (1) a2u a

C1 az = ^ 0 { at2az [ j (1) (C 2 aU A

N1 az }

at z = 0 (7)

auN ) 1 au (2)a2UN) ac^ +— --

az c2 at )—w(2^

" at2 az

au (2)

I JNZ^( cz az) I atz=L1, (8)

where wo` ) and w 1;,' ) , i = 1, 2 are Legendre weightsdefined in the Appendix. These equations enforce, in aweak form, the boundary conditions (2) and (3).

3) Interface equations:

uN ) = uNZ) at z = y (9)

and

U N(1) au (2)2 N 2 N

—Cl az + C2 az

(ilN a t — JN^^^' a uN^)at az az

a 2u «^ a a

+ woz)^z — (10)at az az

These equations correspond to the matching conditions (4e)and (4f). We emphasize that the stress continuity (4f) has beenexpressed in a weak-form through equation (10). If the right-hand sides of equations (7), (8), and (10) were set to zero, thenthe problem could be regarded as a classical collocationapproximation to equations (4a)—(4f). In its actual form,equations (6)—(10) are indeed a variational form of the collo-cation problem. As outlined in the Appendix, this problemstems directly from formulation (5) of the original problemequations (1)—(3). The main advantage of this scheme overclassical collocation lies in its higher accuracy. Clear numericalevidence is reported for the case of a point source (Maggio andQuarteroni, 1994). Intuitively, this can be explained by theobservation that in equation (7), for instance, classical collo-cation would enforce the free-surface condition, while varia-tional collocation entails a linear combination between such acondition (on the left-hand side) and the differential equationitself (on the right-hand side). In other words, the differentialequation (1) is not only collocated at the internal points [seeequation (6)], but also at the boundary (although in a weakersense). Similar considerations apply for z = L 1 , as well as forz = y [see equations (8) and (10)]. Since both and wNZ Ibehave like 1/N 2 as N — 0 it follows that equations (2) and(3), as well as the stress continuity at the interface, are satisfied

as N goes to infinity. Collecting equations (6)—(10), the formu-lation of the problem in space is now closed, providing a systemof algebraic equations that should be discretized in time.Time-marching methods will be discussed in a later section.

2-D acoustic model

In this section, the mathematical models are derived for thescalar (acoustic) wave propagation equation in the domainf2 = {(x, z):—M x ^ M, 0 ^ z <— L 1 }. For a 2-D,isotropic, and homogeneous acoustic medium, we write themomentum conservation and pressure-displacement equationsin the form

(32u ap

P at z ax + fx(11)

a 2w ap— -P ate az +f,

all awp = —X ax + az , (12)

where p is the pressure, u and w are the displacementcomponents in the x- and z-directions, p(x, z) is the density,X(x, z) the Lame constant, and fx (x, z) and f0 (x, z) are bodyforces per unit volume. From equations (11) and (12), oneobtains the acoustic wave equation when both density andseismic velocity are spatially variable, i.e.,

a 2pla 1 1

Pczatz V. PVP/=—V.\Pf)-—V F, (13)

with c 2 = X./p and F = (fY , fZ )/p (this can be regarded as aparticular case of a more general acoustic equation in whichthe right-hand side is replaced by a source term g). Thesubdomain formulation for the 2-D case is straightforwardprovided we partition the physical domain fl into fl = (—M,M) X I and Sl 2 = (—M, M) x J. Denoting this time withp^, the restriction of p to (1^, i = 1, 2, with n the outwardnormal to afl , and referring to Figure 1 for notations, thevariational formulation gives

21 a 2P+2i c2 at 2 v+ i=1 0 Pi i i=1 Pi

J F•Vv=—^ J (F•nz)v

i=1 f2, (=1 adu,

z+ 1 ap2)v+ 1 1 ap` v (14)az az anr Pi P2 t=i ao,^r Pt t

for all sufficiently smooth test functions v that are continuousacross F. Since we are considering source terms f that areimpulses acting at interior points of f1 1 , the first term on theright-hand side of equation (14) vanishes. Moreover, thesecond term at the right-hand side must vanish to ensurecontinuity of displacement normal to the boundary. Further-more,

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,clan v=0,iF '0,

if a free-surface condition is set on F Top . Therefore, theright-hand side of equation (14) reduces to

1 appright-hand side = an

Vr, ur z p i i

+fF I ante. (15)

3Ur4UFBOr P2 Z

We show two alternative approaches for the solution ofproblem (14) with the right-hand side of equation (15),namely, a mixed Legendre-Fourier approximation, and a fullLegendre one.

1) Mixed Legendre-Fourier approximation.—We assumeM large enough for u to be considered periodic at x = —Mand x = M. Accounting for the periodicity on the verticalboundaries of fZ, the right-hand side in equation (15) reducesto

1 ap z— v.

pzc2 atf

j,ur4urxor

The spatial grid is given by the cartesian product of two 1-DGauss-Lobatto grids; see Figure 2b. The behavior of thesolution on the interface F is the same as in the previouscase—continuity of pressure and displacements are preserved.

The integrals are subsequently replaced by the Gauss-Lobatto quadrature formulas. Then the collocation version ofour scheme is obtained by repeating the process that has beenindicated before for the mixed Fourier-Legendre approxima-tion (for details see Maggio and Quarteroni, 1994).

2-D elastic model

For the in-plane elastodynamic problem (P-SV-waves), theequations of momentum conservation are given by

a 2u _ a^ 'x ao x,P ate ax + az + fx ,

(16)azw a(r, aU"

pare = ax + az + f^ ,

and therefore to

1 aP 2

pz azfF Bf,

where uxx , o, u, are the stresses.The stress-strain relationship for an isotropic elastic solid

expressed in terms of displacement derivatives reads

1 ap efjv,

P2C2 at Bor

since the paraxial condition

1 ap apcat az

holds on FBot . The spatial grid, represented in Figure 2a, isuniform in the horizontal direction and is the same as in the1-D case in the vertical direction.

The spectral approximation provides a numerical solution interms of a finite expansion of trigonometric (Fourier) polyno-mials in the horizontal direction, and of algebraic (Legendre)polynomials in the vertical one. The generalized Galerkinapproximation is then obtained by replacing all integrals in 1,in equation (15) by numerical integrations based on theFourier-Legendre nodes. Then, proceeding in a similar way tothat shown in the Appendix for the 1-D case, we can restate thegeneralized Galerkin problem as a Fourier-Legendre spectralcollocation problem. We omit the details for brevity. We noticethat the horizontal derivatives are computed by a fast Fouriertransform (Gottlieb and Orszag, 1977).

2) Full Legendre approximation.—The periodic conditionsat x = ±M are replaced by absorbing conditions of theparaxial type. On the bottom side, the same boundary condi-tions as in the previous case apply (paraxial condition), whileon the top boundary, either Neumann or Dirichlet conditionscan be set. The right-hand side in equation (15) now reads

right-hand side = — Jp ap vtr, urz

au awQ,=(X+2p.) ax +X—

az

au awu,z =x ax +(X+2µ)—,

az

=µ

(

au awaz + ax), (17)

where X and p. are the Lame's constants. Note that eachsubdomain is characterized by different values of X, p., p. For ahorizontal upper boundary, the free-surface condition at z = 0requires that

U, = 0, Q = 0. (18)

At the bottom boundary z = L 1 , a first-order radiationcondition, in the form proposed by Stacey (1988), can beconveniently imposed. On the vertical sides x = M and x =—M, we can either assume periodicity (in which case acombined Fourier-Legendre approximation will be used) orrequire that a radiation condition similar to the one at thebottom boundary holds (in that case a 2-D full Legendreapproximation needs to be adopted). Let us confine ourselvesto the mixed Fourier-Legendre approximation. After reformu-lating equations (16) in variational form, the spectral colloca-tion method gives the following set of equations, which com-pletely define the 2-D elastodynamic problem for twosubdomains:

1) Equations (16) enforced at the internal nodes of eachsubdomain;

2) The free-surface variational conditions at z = 0: —M <_x<_M

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Spectral Decomposition for Wave Modeling

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w — —^^^( 8 2u ate a^ (1 ) 'v _o Pi at e ax az

(1

^( a2w ao- auzz °) o P 1 ax az = o, ( 19 )

where Q(i l, ox^ ), o- (1 are the approximations in T, ofthe stresses given by equations (17);

3) the radiation conditions at z = L 1 , -M < x <- M,omitted here for brevity;

4) the interface conditions at z = y, -M <- x <- M

a 2u ao(l^ aU (1 ^(W'^P1 + wo^^P2) at^^ _ ` j, r ax + az

aQ (2) aQ (2)

+ (il (2^( + —azx U(1) — Q «) —(1^ — Q(2)

° ax(XX)

(Cr(X^ Xz ),

a 2w 8cr aQ(1^

N i PI(w +w o^)P2) at^r = WNi( ax + 8z )

a o- a 4((2) I

xz + zz) — cr — Q «) — Q (1) — (2)̂

+ ° ax az (z^ x^) (^^

(20)

where u r and w r indicate the u and w components oninterface F, respectively.

TIME MARCHING

In previous work (Maggio and Quarteroni, 1994) severalschemes, both explicit and implicit, were considered for ad-vancing the spectral approximation of the acoustic wave equa-tion in time. For the convenience of the reader, we resume themain conclusions of this investigation concerning accuracy,computational efficiency, and stability. Remember that forexplicit methods, Ot is subject to the stability restriction

kOt N z , (21)

for a suitable constant k > 0. On the other hand, whileimplicit methods of order 1 or 2 are expected to be uncondi-tionally stable, they should be used with small values of At toprevent loss of accuracy, especially for high-frequency tran-sient calculations.

A distinguishing feature of the problem at hand is thesimultaneous presence of second-order and first-order timederivatives (the latter stems from the first-order radiationcondition at absorbing boundaries). This demands the use ofdifferent schemes for a 2 /at 2 and for a/at, both possessingsecond-order accuracy, and whose combination should yield anoverall scheme preserving the stability of each individualscheme.

In this respect, it is found that the most suitable combinationis provided by the explicit LF2-LF2 scheme (leap frog 2ndorder for a 2 /at 2 - leap frog 2nd order for a/at). In itsdifferential form, this scheme reads

1 p n+1 - 2p n + pn-11Pet At2 — V ^P

Vpn )

/ 1_ - V • I fn at each internal point, (22)\P

p n+1 _ p n-1 apn

20t + an

= 0 at points on the absorbing boundary, (23)

where the superscript n means that the quantity is evaluatedfor t = n At. The experimental analysis on a single-domainapproximation shows that the combination of equations (22)and (23) is second-order accurate with respect to 1t. Besides,condition (21) holds with k 2.80 (for a theoretical analysissee Augenbaum, 1990, and Canuto et al., 1988).

An alternative is provided by the fully implicit B2-B2 scheme(backward 2 nd order for a 2 /at 2 - backward 2 nd order fora/at). In this case, equations (22) and (23) should be replacedby

1 2pn+^ _ 5p n + 4p n-' _ p n-2 1PC2 Ate -D.(Pvpn+l^

_ _V. (1 fn+1/(24)

P

3p n+1 - 4p n +, p n-1 apn+l

+c

This scheme turns out to be unconditionally stable [i.e., onecould virtually take k = +x in equation (21)]. Nevertheless,for good accuracy, the allowable time-step should be of thesame order of magnitude as that of the explicitLF2-LF2 scheme. Although each step requires the solution ofa linear system, this formulation has the same symmetricmatrix, which can be decomposed a la Cholesky at the begin-ning of the process.

NUMERICAL RESULTS

2-D acoustic propagation with full Legendre approximation

We provide two examples of full Legendre collocation forthe solution of the 2-D acoustic equation (13). Two structuresare considered to illustrate the performance of the proposedmethod with respect to the different domain partitioningcriteria ("h-version" and "p-version") we have described inthe Introduction.

1) Propagation in a single flat layer.—We consider a simplestructure containing a horizontal uniform layer overlying ahomogeneous half-space. This test problem, also used inprevious studies (Tal-Ezer et al., 1987), requires an accuratenumerical treatment because of the presence of post-critical-angle phenomena and fairly high velocity contrast. The modeladopted is shown in Figure 3, including the velocity values. Forthe sake of simplicity we assume that the density takes thevalue p = 1 g/cm 3 everywhere. The source term -0 • F inequation (13) is a delta function centered atx = 1250 m, z =800 m, and has a time-dependent part given by the wavelet

h(t) = cos [wfo(t - to)] exp [- Ifo(t - to) 2], (26)

20t an = 0. (25)

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with a high-frequency cutoff fo = 50 Hz, peak frequency fp =25 Hz, and t o = 0.060 s. We put three receivers on a line ata depth of 300 m with horizontal coordinates of 1250 m,1850 m, and 2000 m, respectively. All numerical values are thesame as those used in Tal-Ezer et al. (1987), or very close tothem. First-order paraxial conditions were used on the bottomand vertical sides of the boundary and a free-surface conditionon top. The numerical simulation is based on 1225 squaresubdomains of equal size, each containing a 10 x 10 Legendremesh giving an average of 5.6 points per shortest wavelength.This can be seen as an example of option (b) we advocated inthe introduction. Time-marching was performed by the previ-ously described LF2-LF2 scheme. In Figure 4 we show foursnapshots of the pressure field at t = 0.4 s, 0.8 s, 1.0 s, and1.2 s, respectively. The main arrivals, identified in the caption,are clearly portrayed in the snapshots and coincide with thosediscussed in detail in Tal-Ezer et al. (1987). Starting fromabout t = 1.0 s, there is evidence of weak spurious arrivalsfrom the upper two corners of the grid. For a more accurateevaluation of the results, the pressure time-histories calculatedat the receivers are compared in Figure 5 with the analyticalsolution obtained by ray-summation (see Paolucci et al., 1992)for the D, R, SM, and first RM arrivals identified in Figure 4.The agreement between the solutions is excellent, consideringthat the analytical solution does not include the head wavegenerated at the interface. This wave is not present at receiverR1 and reaches R2 and R3 at about the same time as theprecritical-angle reflection from the interface.

2) Irregular fault.—As an example of full Legendre collo-cation associated with option (a) in the Introduction,equation (13) was solved for the structure shown in Figure 6,which contains an irregular fault with simplified geometry. Asa source term, we considered the divergence of a single pointimpulse of the form

I

0 500 1000 1500 2000 2500

distance (m)

FIG. 3. Configuration of flat layer model (R1, R2, R3 =receivers).

f, =f = 6 (x — xs)h(t),

where XS is the point x = 300 m, z = 125 m. Thetime-dependent part is given by the wavelet

h(t) _ [ 1 — 0.5-rr 2fo(t — t o ) 2] exp [-0.25ir 2fo(t — t o ) 2],

with high-frequency cutoff fo = 40 Hz, peak frequency fp =20 Hz, and t o = 0.050 s. As in the previous case, we set auniform density p = 1 g/cm 3 in the whole domain. Threereceivers Rl, R2, and R3 have been placed, respectively, at thepoints:x = 250 m,z = 300 m;x = 350 m,z = 200 m;x =300 m, z = 400 m. A Neumann condition has been set on theupper boundary and paraxial condition on the rest. The timemarching scheme is the same as the previous case. Thesubstantial difference is that here we discretize using only ninetrapezoidal subdomains and polynomials of degree N = 30.Each subdomain is mapped into the square 51, E = [ - 1, 1] 2 ;

which allows a simple computation of the differential operatorsand the scalar products of the problem (see also Priolo andSeriani, 1991, for more details).

Four amplitude snapshots of the pressure field are illus-trated in Figure 7, where the direct wave, the transmitted wave,the reflection from the top boundary, and the diffraction fromthe upper corner of the fault can be recognized easily. For thesame problem, but using a delta function instead of —V • F asthe source term in equation (13), we have carried out acomparative analysis between the present method and thespectral element method (SPEM) (Patera, 1984). The resultsobtained are illustrated in Figure 8, where the pressure timehistories at the receivers are displayed. The excellent agree-ment between the two solutions should be noted for the twoleading events (direct and reflected arrival), while minordifferences visible for longer propagation times are partlycaused by the differences in the treatment of absorbing bound-aries.

Obviously, a domain decomposition method is well suitedfor codes that are run on parallel computers. When thenumber of subdomains is reasonably low (as in the present caseand when option (1) of the Introduction is adopted), one canperform computations on a massive parallel machine simplyusing a 1-to-1 mapping between subdomains and processors.This experiment has been done for the example at hand usingnine nodes of an IBM-SP1 computer, i.e., a parallel machinewith distributed memory. The efficiency of the parallel solverhas been measured in terms of speed-up, defined as theCPU-time required by the sequential (nonparallel) code di-vided by np times the CPU-time of the parallel computation,with u p denoting the number of processors. Results are shownin Table 1. A rough analysis of the observed speed-up is thefollowing. The overall cost of our algorithm implemented on aparallel machine with distributed memory is essentially causedby the solution of the problem in each subdomain (performedby the single processor) plus the message-passing operation(performed by the whole machine). The same algorithmimplemented on a sequential computer does not require themessage passing operation. The computational effort per sub-domain per time-step is proportional to N 3 (because of thecalculation of the pseudospectral derivatives), while the mes-sage-passing effort is proportional to N, the number of nodalvalues associated to each edge. Therefore, the speed-up ratetakes the following expression:

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Spectral Decomposition for Wave Modeling

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1 npN 3aspeed-up = N3 + aN =

1 a + NZ'

for some positive parameter a. This theoretical trend shows agood agreement with the observed rates and explains why thespeed-up rates approximate the ideal value of 1 when Nincreases.

2-D elastic propagation with mixed Legendre-Fourierapproximation

Dispersed Rayleigh waves in surface layer.—To test thecapability of the present method to handle sharp velocity

contrasts in elastic problems (P-SV waves), we considered thesimple structure consisting of a thin horizontal layer overlyinga uniform half-space. For the layer, a thickness of 60 m wasassumed, P-wave and S-wave velocities were set at 2000 m/sand 1155 m/s, respectively, and a density p = 1.2 g/cm 3 wasimposed. Velocities of 3000 m/s and 1500 m/s and a density of2.0 g/cm 3 were assigned to the half-space. A vertical pointforce in the form of a Ricker wavelet was applied at a depth of0.9 m (the horizontal position is not relevant), having the timedependence (26) with a high-cut frequency f0 = 42 Hz. Thegrid size was 512 in the horizontal direction, where a Fourierdiscretization with constant spacing of 10 m was adopted. In

FIG. 4. Amplitude snapshots for layer model at 0.4 s, 0.8 s, 1.0 s, and 1.2 s, respectively. Labeled wavefronts correspond to direct wave(D), reflected wave (R), transmitted wave (T), surface multiple (SM), reflected multiples (RM), and transmitted multiple (TM).

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the vertical direction, a partition into two Legendre subdo-mains was introduced with a total size of 60, i.e., 30 nodes forboth the layer and the half-space.

In this example, absorbing boundary regions (Kosloff andKosloff, 1986) were implemented instead of the radiationconditions. Except for the grid size, the input data for thisproblem are essentially the same as those of Example 3 inKosloff et al. (1990, 744). Figure 9 displays the horizontaldisplacement time section at the depth of the source. Thefundamental mode is the Rayleigh wave, which is preceded bystrongly dispersive surface wave multiples. Our time section issimilar to that produced by Kosloff et al. (1990, 741), but ismore accurate because of a much finer vertical discretization ofthe layer. The previous authors compared their results with an

numericalanalytical

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

time (s)

FIG. 5. Comparison between numerical and analytical pressuretime-histories at the receivers R1, R2, and R3 shown inFigure 3. The analytical solution does not include the headwave from the interface.

incomplete analytical solution (calculated by the propagatormatrix method) which includes only trapped surface wavesmodes and lacks some of the precursor multiply reflectedP-waves that are evident in their numerical solution as well asin ours.

In addition, we consider the single traces relative to a coupleof receivers RI and R2, placed at the same depth of the sourceand at a distance of 500 and 2000 m, respectively. The traces ofhorizontal and vertical displacements, reproduced in Figure 10and 11, show that the body waves preceding the main Rayleighwave pulse can reach significant amplitudes. Careful inspectionreveals that our traces agree well with the correspondingnumerical waveforms in Kosloff et al. (1990, 743-744). How-ever, the present method is practically unaffected by theuncertainty at the layer boundary positions common to dis-crete-grid methods.

Checking numerical stability

Since numerical instability may arise after long propagationtimes from the treatment of absorbing boundaries (Mahrer,1990), we have checked the behavior of the LF2-LF2 schemeby measuring the L 2 -norm of the spectral solution UN as afunction of time

E(t) = fn

^ U (x, z, t)1 2 dx dz.

In this computation we are enforcing first-order radiationconditions on the whole boundary. We expect E(t) to decay

Table 1. Speedup for the irregular fault solved in a parallelfashion.

grid-points perdimension per

subdomain 10 20 30 40

speedup 0.757 0.945 0.966 0.972

R1

R2

R3

Q4 Q, Q6

600

0

100 200 300 400 500 600 0 100 200 300 400

500 600

distance (m) x—axis

FIG. 6. (a) configuration of the irregular fault problem. (b) domain decomposition used.

a) b) o

loo

200

300

400'

500

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Spectral Decomposition for Wave Modeling

1169

monotonically in time (after reaching its maximum) as thewavefronts leave the computational domain. Plotted inFigure 12 is the function E(t) for the case of the irregular faultdescribed in Figure 6. After a propagation of 5.0 s (after 50 000time-steps), E(t) reduces by a factor 6400 and no spuriouseffects caused by numerical instability are detected. This can beregarded as a demonstration of the long-term stability of theLF2-LF2 scheme. The persistence of some residual energyafter about 5.0 s is probably because of the low order of theparaxial condition that is used.

A comparison with finite-difference techniques

It is clearly of interest to compare the performance of thepresent spectral method with that of finite-difference algo-

rithms, which represent the most popular numerical techniqueused for wave propagation. We note first that the accuracy ofthe spectral method can be improved by simply increasing thedegree N of the polynomial used in each spatial dimension.Unlike higher-order finite differences, this does not require aspecial treatment for boundary conditions. Besides, the accu-racy is potentially higher, as one can realize by comparing, e.g.,the stencil of 4th order-finite differences with the one of thepresent method using N = 4. In the former case, a 9-pointTaylor formula is used, while in the latter the exact derivativesof a 4th order polynomial are calculated, using all 25 Gaussiannodes included in the given subdomain.

To illustrate the previous points, we have considered asimple case in which the acoustic problem admits a smooth

FIG. 7. Snapshots of the pressure wavefronts generated by an impulsive source within a layer overlying a simplified irregular fault.

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analytic solution in a square domain; both the proposedalgorithm (single-domain version) and two finite-differencecodes, FD-4 and FD-10, were used to compute the numericalsolution. The truncation errors of the latter schemes areO(Ot 2 , 0x 4 ) and O(Ot 2 , Ax 10 ), respectively (for FD-10 see,for instance, Dablain, 1986). We used Ot = 10 -6 s and apropagation time T = 0.1 s. The accuracies achieved as afunction of n, number of grid-points per spatial dimension(n = N + 1), are displayed in Figure 13. For low n, both thespectral and the FD-4 approximations give relatively largeerrors, because of the insufficient sampling rate, while FD-10 isobviously inapplicable because of its larger stencil width. Forlarger n, the higher rate of convergence of the present methodstems from the fact that the spectral accuracy is stronger thanany finite power of 1/N, for infinitely smooth solutions.

The computational effort should be taken into account aswell. At each time step, finite-difference algorithms require anumber of operations approximately equal to N 2 times thestencil extension, i.e., 9N 2 for FD-4 and 21N 2 for FD-10,while our method needs approximately O(N 3 ) operations. Asa consequence the spectral code and a finite-difference onerequire essentially the same CPU-time when N is set equal tothe stencil extension, while for larger values of N the compu-tational demands clearly diverge in favor of the latter method.These considerations seem to be an indication of the existenceof drawbacks in setting N too large (i.e., in choosing a fullp-version of finite elements method). On the other hand, suchan approach is clearly recommended when an excellent accu-racy is needed disrespectfully of a severe computational cost.

The situation is more involved when speaking about domaindecomposition. In this case a further disadvantage of thefinite-difference method, besides the boundaries problem, is amore difficult interface treatment. As a matter of fact, a largeramount of information should be exchanged among subdo-mains with respect to the O(N) scalar quantities of ourmethod, and this makes implementations on parallel machinesless effective. A detailed analysis of accuracy and computa-tional cost of the method at hand, with particular emphasis on

FIG. 9. Horizontal displacement time section calculated for thedispersed Rayleigh wave test at a depth of 0.9 m.

FIG. 8. The pressure time histories for the receivers R1, R2,and R3 of Figure 6b.

FIG. 10. Dispersed Rayleigh waves test. Time histories of thehorizontal displacements for different receivers placed at thesame depth of the source (0.9 m) and at horizontal distances of500 m (Rl) and 2000 m (R2).

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Spectral Decomposition for Wave Modeling 1171

R1

R2

1.1

1.0

0.9

0.8

0.7

0.60C

0.5

0.4

0.3

0.2

0.1

0.00

time (s)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2

time (s) FIG. 12. The L norm (normalized to 1) of the spectralsolution for long propagation time.

FIG. 11. Same as Figure 10 for vertical displacement.

10 -2

10 3

10 4

105

0L

10 6

E

107

108

109

cc present method13 -o FD-10o -- o FD-4

1010 1 1 1 I 1 1 1 17 8 9 10 11 12 13 14

n

FIG. 13. The rms errors between analytical and numerical solutions, resulting from the present method and twofinite-difference codes. The test refers to the smooth solution p(x, y, t) = Po cos [2ar(7x + 7y + t)] in(0, 1) 2 X (0, T), with T = 0.1 s. The parameter n is the number of grid points per direction.

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the optimal choice of parameters N and H, will be the objectof a future paper.

CONCLUSIONS

The proposed method appears to provide accurate numeri-cal solutions to wave propagation problems encountered in the2-D analysis of heterogeneous structures. The multidomaindescription allows an effective reduction of the number ofdiscretization points, and thus alleviates the severe restrictionon the time-step size imposed by the explicit integrationscheme and by the clustering of points near the boundary. Onthe other hand, such clustering makes the algorithm accuratein the presence of strong jumps in velocity, a characteristic thatis not shared by other commonly used spectral methods. Westress the fact that the proposed multidomain approach canreadily cover a broad range of structure configurations, goingfrom the lower finite-element limit to the upper single-domainspectral discretization. Indeed, in its design principle it allowsthe choice of the two critical parameters (the subdomain sizeH and the polynomial degree N) without any constraint. Thespatial stability of the solution is always guaranteed, while itsaccuracy depends in a known way on the choice of the previousparameters. An inherent advantage of the method we advocateis its efficiency when implemented in a parallel computingenvironment, as shown by the preliminary results reported inthis paper. Further work appears desirable to clarify the choiceof the subdomains as well of the number of discretizationpoints when more complex heterogeneities are present.

ACKNOWLEDGMENTS

The authors are grateful to R. Paolucci for providing theanalytic solution of the first example and to E. Priolo forcomputing the SPEM numerical solution for the second exam-ple. F. Maggio and A. Quarteroni have been supported bySardinian Regional Authorities.

REFERENCES

Augenbaum, J. M., 1990, The pseudospectral method for limited-areaelastic wave calculations, in Fitzgibbon, W. E., Ed., Computationalmethods in geosciences, 1-14.

Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A., 1988,Spectral methods in fluid dynamics: Springer-Verlag.

Carcione, J., 1991, Domain decomposition for wave propagationproblems: J. Sci. Comput., 6, 453-472.

Dablain, M. A., 1986, The application of high-order differencing to thescalar wave equation: Geophysics, 51, 54-66.

Davis, P. J., and Rabinowitz, P., 1984, Methods of numerical integra-tion, 2nd Edition: Academic Press, Inc.

Faccioli, E., 1991, Seismic amplification in the presence of geologicaland topographic irregularities, in Prakash, S., Ed., Proc. 2nd Inter-nat. Conf. on recent advances in geotechnical earthquake engineer-ing, 1779-1797.

Gazdag, J., 1981, Modeling of the acoustic wave equation withtransform method: Geophysics, 46, 854-859.

Gottlieb, D., and Orszag, S. A., 1977, Numerical analysis of spectralmethods: theory and applications: SIAM-CBMS.

Kosloff, D., and Baysal, E., 1982, Forward modeling by a Fouriermethod: Geophysics, 47, 1402-1412.

Kosloff, D., Kessler, D., Filho, A. Q., Tessmer, E., Behle, A., andStrahilevitz, R., 1990, Solution of the equations of dynamics elastic-ity by a Chebychev spectral method: Geophysics, 55, 754-748.

Kosloff, R., and Kosloff, D., 1986, Absorbing boundaries for wavepropagation problems: J. Comput. Phys., 63, 363-376.

Maday, Y., and Patera, A. T., 1989, Spectral element method for theincompressible Navier-Stokes equations, in Noor, A. K., and Oden,T. A., Eds., State of the art surveys on computational mechanics:Am. Soc. Mech. Eng., 71-142.

Maggio, F., and Quarteroni, A., 1994, Acoustic wave simulation byspectral methods: East-West J. Numerical Math., 2, 129-150.

Mahrer, K. D., 1990, Numerical time step instability and Stacey's andClayton-Engquist's absorbing boundary conditions: Bull. Seis. Soc.Am., 80, 213-217.

Paolucci, R., Suarez, M., and Sanchez Sesma, F., 1992, Fast computa-tion of SH seismic response for a class of alluvial valleys: Bull. Seis.Soc. Am., 82, 2075-2086.

Patera, A. T., 1984, A spectral element method for fluid-dynamics:Laminar flow in a channel expansion: J. Comput. Phys., 54, 468-488.

Priolo, E., and Seriani, G., 1991, A numerical investigation ofChebychev spectral element method for acoustic waves propagation,in Vichnevetsky, R., Ed., Proc. 13th IMACS World Congress onComputational and Applied Mathematics: Criterion Press, 551-556.

Quarteroni, A., 1991, Domain decomposition and parallel processingfor the numerical solution of partial differential equations: Surveyson mathematics for Industry, 1, 75-118.

Quarteroni, A., and Valli, A., 1994, Numerical approximation ofpartial differential equations: Springer-Verlag.

Stacey, R., 1988, Improved transparent boundary formulations for theelastic-wave equation, Bull., Seis. Soc. Am., 78, 2089-2097.

Szabo, B. A., and Babuska, I., 1991, Finite-element analysis: JohnWiley & Sons, Inc.

Tal-Ezer, H., Kosloff, D., and Koren, Z., 1987, An accurate scheme forseismic forward modeling: Geophys. Prosp., 35, 479-490.

Tessmer, E., Kessler, D., Kosloff, K., and Behle, A., 1992, Multi-domain Chebychev-Fourier method for the solution of the equationsof motion of dynamic elasticity: J. Comput. Phys., 100, 355-363.

APPENDIX

DERIVATION OF THE APPROXIMATE EQUATIONS BY THE SPECTRAL COLLOCATION METHODIN WEAK (OR VARIATIONAL) FORM

Let I = (-1, 1) denote a reference interval, N a fixedpositive integer (the polynomial degree), and 2, 0 ^ j N,the roots of (1 - z 2 ) L (z), LN being the Legendrepolynomial of degree N (e.g., Davis and Rabinowitz, 1984).We order z^ from left to right in a way that i o = -1, fN =

1; the remaining points are symmetrically distributed aroundz = 0. Let us introduce the coefficients

2 1

N(N + 1) [L N(21)]2 7 = 0, ... , N. (A-1)

Then z^ and i provide, respectively, the nodes and weightsof the Gauss-Lobatto formula for numerical integration overthe interval I. Precisely, for any continuous function cp, itsintegral f 1 t cp(z) dz can be approximated by the finite sumESN o cp (2i ) iii. The accuracy of this approximation is outlinedby the following exactness property:

N

W(zi)wi = r ^P(z) dz (A-2)J f_ 1

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Spectral Decomposition for Wave Modeling

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for all cp E P2N_1 (hereby, P, denotes the vector space ofalgebraic polynomials of degree <_ n). For details see Davisand Rabinowitz (1984) or Canuto et al. (1988).

Defining for any pair of continuous functions cp, 4,

N

(p, )N = LJ PP(Zj)^(Zj) ( , (A-3)j=o

it follows that (cp,' ) N can be regarded as an approximation ofthe integral f 1 1 .p(z)4s(z) dz. Furthermore, in view ofequation (A-2), we deduce the exactness formula

1

(PP, )N = p(z)tl,(z) dz (A-4)

for all p, i such that 9i E P 2N_ 1 .Now let d2 be the interval (0, L 1 ), partitioned into I i =

(0, y) and 7 2 = (y, L 1 ). For any I ; we map zj into z )

according to the transformation I —' I , i = 1, 2 and set of= wj (meas(I i )/ 2), where meas(I i ) denotes the length ofinterval I. Moreover, we define

('P, c)N = I 'P(z)^(ZJ`^)wjl , i = 1, 2. (A-5)j=o

From the property of exactness (A-4) it follows that

(^^ i)N — J p^

dz if p i E P 2N-1 • (A-6)

We define the space of piecewise polynomials as

VN = {pP E Co([-1, 1]): j E P N , l = 1, 2}

The spectral approximation to the multidomain problem (4) isdefined as follows.

For each t > 0 we look for u fi E P N , = 1, 2, such that

uN ) = uN) at z = y, t > 0 (A-7)

2 a 2u (iji 2 av NNzI at 2 , "N—

(C 2 aU N

` az , azN i=1 N

auN^— C 2 at V N I z=L t, t? 0 (A-8)

for all test functions VN E VN . This is precisely the finite-dimensional version of equation (5) in which we have replacedu i by a polynomial and used the Gaussian formula (A-5) toapproximate the product integrals.

A differential interpretation of equation (A-8) can be pro-vided. For any function 9 E C°(I i ), let JN` ) <p E P N denotethe polynomial interpolating pat the Legendre points z j0) ,j = 0, ... , N. Clearly JN`Icp = p if cp E A[setting H ( ` ) = JN ) (c 2 au NOlaz) for i = 1, 2, we observe thatH ( ` ) = c i2 auN` ) /az whenever c 1 is constant in I. For the firstterm on the right-hand side of equation (A-8) we have

2 Z au (1) av 2 av—^ C N ' N=— ( H

^ aZ az

id' N

` N aZ ) N'i= i= i

(since H ( ' ) =JNi>(c?auN` ) laz) at all Legendre nodes {z1 })

2 avN

_ _ H (i) a z dz,

[owing to equation (A-6)]

2 aH ^i^_ UN dz + (H^2 — H (li)vN2

azi=1 /;

— H 2 VNj'=LI + H ( ' VNI Z=o,

(after integration by parts)

2 aH (i^

( , UN) + (H^2) — H 1))VNzi1 az N

— H (2 VNIZ=L, +H ( 'vNIZ=o, (A-9)

[owing to (A-6)].Now let us define

2NUN N — N ^^^^i^ l— 1, 2.u—a at N az JN( 2Caaz )^

(A-10)

For each i, RN` ) denotes the residual of the differentialequation in I . Thus, RN`s uN` ) provides a measure of the extentat which u N` ) satisfies the original differential problem.

Collecting equations (A-8) and (A-9) and using the defini-tion (A-10), we conclude that the spectral problem (A-8) isequivalent to

2

(2

N )(RN^u2, VN)N = C2 — UNIz=YC l

i=1az az

2 au (1 ^ au (2^ au (2^N N N

+ C] az UN) lz=o + c2( at c2 az ) VNl z=L, .

(A-11)

For each j, let ^Il`> E IA N be the Lagrangian polynomial suchthat J( ' ) (z ) ) = 8 kj for 0 < k <— N. Then for each i = 1,2 and j = 0, ... , N, 4cjt`i is the unique function of VN (i.e.,a polynomial of degree N in both I i and 12) that is equal toone at the node zj^ i) and vanishes at all remaining Legendre-Lobatto nodes. These functions provide a Lagrangian basis forthe space VN .

Taking now as a test function in (A-9) UN = for i = 1,2 and 1 <— j N — 1, and noticing that UN (y) = UN (0) _0, we easily deduce using the definition (A-5):

32U(l) a au (`^R (`)u (`) _ — — J^`^ c =0N N at2 aZ N aZ

atz,i=1,2,1<_j<_N-1. (A-12)

This expression is nothing but the usual collocation of thegiven wave equation (1) at the internal nodes. The presence ofthe interpolation operator J(J) (clearly, J (J) can be droppedwhenever c, is constant in I ; , as we have specified above)indicates that we are carrying out a pseudospectral derivativeof ci2 auj`)/az.

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1174 Faccioli et al.

If we now take VN = ^ro 1) in equation (A-11) we obtain Finally, the displacement continuity condition u = uNZ Iat z = y closes the system (A-12)—(A-15) actually imple-

ci auN̂^ = woiiRN1)uN ) at z =0, (A-13) mented in the calculations.az We conclude this Appendix by recalling that a routine to

which is the stress-free condition (2) in a weak form.computez1 andL N (zJ ) is supplied, e.g., in Canuto et al., 1988.

Similarly, taking vN = ^,N^ 1 in equation (A-11) we deduce With the previous quantities z^, L N (z1 ), and w^ so obtained,the pseudospectral derivative matrix D is constructed as fol-

au (2) 1 au 2)\ (Canuto et al., 1988)

C 2cz ( az cat ) = RN^uNZ^ at z = L 1, (A-14)

i.e., the radiation condition (3) in a weak form.Finally, taking as VN the function that coincides with 1, V )

and with in 1 2 in equation (A-11), and proceeding asbefore, we obtain

au (2) ^u (1)c 2 N

— c 1N = (1)

0R (1)u (1) + o (2)R (z)u (2)

2 aZ 1 aZN N N N N

LN(zf) 1 J 1L N (2 1 ) ze —z^N

D= (N+1) 4 f=j=0

(A-16)

N—(N+1) 4 Q=j=N

0 otherwise

at z = y. (A-15)

This relation translates (still in a weak form) the stresscontinuity at the interface as stated in equation (4f) for thephysical solution.

In this way the calculation of the spatial derivative aulaz at thefi nodes is transformed in the matrix-vector multiplicationD • u, where u is the vector of the gridvalues {u(21 ), j =0, ... , N}.

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