Research Article On the Accuracy and Efficiency of ...downloads.hindawi.com/journals/amp/2016/9431583.pdfResearch Article On the Accuracy and Efficiency of Transient Spectral Element

Research ArticleOn the Accuracy and Efficiency of Transient Spectral ElementModels for Seismic Wave Problems

Sanna Mönkölä

Department of Mathematical Information Technology, University of Jyvaskyla, Agora, P.O. Box 35,40014 University of Jyvaskyla, Jyvaskyla, Finland

Correspondence should be addressed to Sanna Monkola; [email protected]

Received 30 November 2015; Revised 11 April 2016; Accepted 13 April 2016

Academic Editor: Christian Engstrom

Copyright © 2016 Sanna Monkola. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This study concentrates on transient multiphysical wave problems for simulating seismic waves. The presented models cover thecoupling between elastic wave equations in solid structures and acoustic wave equations in fluids. We focus especially on theaccuracy and efficiency of the numerical solution based on higher-order discretizations. The spatial discretization is performedby the spectral element method. For time discretization we compare three different schemes. The efficiency of the higher-ordertime discretization schemes depends on several factors which we discuss by presenting numerical experiments with the fourth-order Runge-Kutta and the fourth-order Adams-Bashforth time-stepping. We generate a synthetic seismogram and demonstrateits function by a numerical simulation.

1. Introduction

Several formulations exist formodeling the seismic vibrationsas interaction between acoustic and elastic waves. Typically,the displacement is solved in the elastic structure. Thefluid can be modeled using finite element formulationsbased on fluid pressure, displacement, velocity potential, ordisplacement potential [1]. Two approaches, in which thedisplacement is solved in the elastic structure, predominate inmodeling the interaction between acoustic and elastic waves.Expressing the acoustic wave equation by the pressure in thefluid domain leads to a nonsymmetric formulation (see, e.g.,[2, 3]), while using the velocity potential results in a symmet-ric system of equations (see, e.g., [4–7]). Recently, a velocity-strain formulation has been considered by Wilcox et al.[8]. Essentially, the difference between the different modelsis only in the choice of the variables presenting the wavepropagation in different domains. Nevertheless, from thenumerical point of view, this choice determines the featuresof the coupled problem. Hence, it also gives the guideline forusing appropriate discretization and solution methods.

Discretization methods play a crucial role in the effi-ciency. The key factor in developing efficient solution meth-ods is the use of high-order approximations without compu-tationally demanding matrix inversions. We attempt to meet

these requirements by using the spectral element [9] method(SEM) for space discretization. The SEM was pioneered inthe mid 1980s by Patera [10] and Maday and Patera [11],and it combines the geometric flexibility of finite elementswith the high accuracy of spectral methods. The method iswidely used for simulating seismic waves, in both frequencyand time domains (see, e.g., [12, 13]). We have applied it totime-harmonic acoustoelastic equations in [14, 15], while thispaper concentrates on the accuracy and efficiency of the timedomain solutions.

In time domain simulations, the efficiency of the methoddepends also strongly on the time discretization.The second-order central finite difference (CD), or leap-frog, schemeis a low-order method which is easy to implement andthus in general use for discretizing in time domain. Thismethod is used with the SEM by Komatitsch et al. [16], anda modification for utilizing larger time steps in the fluid thanin the solid media is presented by Madec et al. [17]. Since themethod is only of second-order accuracy, higher-order timediscretizations are needed for efficient computer simulationsbased on higher-order space discretization.The fourth-ordercentral finite differences are not usable at this stage, sincethe scheme is unconditionally unstable. Instead, a second-order accurate time discretization scheme can be converted

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2016, Article ID 9431583, 15 pageshttp://dx.doi.org/10.1155/2016/9431583

2 Advances in Mathematical Physics

to a fourth-order accurate one, as is done by Tarnow andSimo [18]. This symplectic scheme is compared with theCD method by Nissen-Meyer et al. [19]. The importance ofhigher-order time discretizations, in the field of seismology,has recently been considered by, for example, De Basabe andSen [20], Peter et al. [21], and Liu et al. [22] and referencestherein.

Lax and Wendroff [23] introduced a higher-order gener-alization of the finite difference scheme.Themethod has beenapplied for acoustic wave equations to provide the fourth-order accuracy, with the finite difference space discretizationby Dablain [24] and with the spectral element space dis-cretization by Cohen and Joly [25]. However, this approachdoes not fit to a case, in which an absorbing boundarycondition is used for truncating the computational domain,unless velocity-stress or velocity-displacement formulation isused [26].

By Kubatko et al. [27], Runge-Kutta time discretiza-tion methods were used in conjunction with discontinuousGalerkin (DG) finite element spatial discretization, andAntonietti et al. [28] have applied the scheme for elasticwave propagation problems. The second- and third-orderRunge-Kutta methods are presented, with a fourth-orderHamiltonian-based space discretization for acoustic andelastic waves in separate domains by Ma et al. [29] and withℎ𝑝-adaptive discontinuous Galerkin method for simulatingtsunami propagation by Blaise and St-Cyr [30], respectively.Since there were no research results considering the accuracyand efficiency of the numerical simulation of fluid and solidwaves, in which the fourth-order Runge-Kutta (RK) time dis-cretization is combined with higher-order space discretiza-tion, we have applied it for acoustic [31] and elastic [32] waves.The method is used with the eighth-order finite differencespace discretization for solving an elastodynamic problemalso by Martin et al. [33].

With respect to the time step Δ𝑡, the CD method issecond-order accurate, while the RK method is fourth-order accurate. Although the computational effort of the RKmethod is approximately four times that of the CD schemeat each time step, the results of using higher-order time-stepping scheme were promising [31, 32]. Both methods leadto an explicit time-stepping scheme.The drawback is that theschemes need to satisfy the stability condition, which limitsthe length of the time step. For acoustic waves, Yang et al.[34] show that a fourth-order Runge-Kutta method is moreaccurate for time discretization than the fourth-order Lax-Wendroff method.

In this paper, we apply the CD and the RK methods tomultiphysical wave problems. We also present the fourth-orderAdams-Bashforth (AB) scheme to discuss the efficiencyfactors of the higher-order time discretization schemes. Inwhat follows, we first consider the mathematical models andboundary conditions in Section 2. Further, we discretize themodels with respect to space and time in Sections 2.2 and 2.3.Numerical examples are considered for the validation of theaccuracy of the approximations and for demonstrating thesynthetic seismogram in Section 3. The concluding remarksare presented in Section 4.

Γes

Γes

Γi

Γes

Γef

Γef

Γef

Ωs Ωf

x1 = 0

Figure 1: The domain Ω is divided into the solid part Ω𝑠and the

fluid part Ω𝑓.

2. Numerical Models

In what follows, we concentrate essentially on twomultiphys-ical linear models for simulating the transient propagation ofseismic waves. In bothmodels, the domainΩ ⊂ R2 is dividedinto the solid part Ω

𝑠and the fluid part Ω

𝑓(see Figure 1).

First, we present the model for elastic deformations in Ω𝑠

and use it as a starting point for deriving the different modelspresenting the propagation of acoustic vibrations in the fluiddomain Ω

𝑓. Then, we apply the coupling conditions to the

models and discretize the problems. We see that, despite thefact the difference between the models is only in the choiceof the variable presenting the wave propagation, it affects thestructure of the coupled problem at the discrete level. For thesake of the efficiency of the solving process and the accuracyof the numerical solution, the numerical methods, relying onthe model-specific features, are presented.

2.1. Coupled Problems. The deformation of the elastic struc-ture ismodeled by the displacement u in the solid domainΩ

𝑠,

such that

𝜌𝑠(x) 𝜕2u

𝜕𝑡2− ∇ ⋅ 𝜎 (u) = f , (1)

where 𝜌𝑠(x) is the density of the structure and f = (f

1, f2)𝑇 is

the source function. We concentrate on isotropic materials,for which the number of essential material constants reducesto the two Lame parameters, 𝜇 = 𝐸/(2(1 + ])) and 𝜆 =

𝐸]/((1+])(1−2])), where𝐸 is the Youngmodulus describingthe stiffness of the solid and ] is the Poisson ratio presentingthe compressibility of the solid as the ratio of lateral tolongitudinal strain in a uniaxial tensile stress, such that 0 <

] < 1/2. Hence, the stress tensor can be presented as 𝜎(u) =

𝜆(∇ ⋅ u)I + 2𝜇𝜀(u), where I is the identity matrix. Ingeneral, we assume the medium to be heterogeneous. Thus,the partial derivatives in ∇ ⋅ 𝜎 apply to 𝜆 and 𝜇 as well asto the displacement. The speed of pressure waves (P-waves)𝑐𝑝

= √(𝜆(x) + 2𝜇(x))/𝜌𝑠(x) and shear waves (S-waves) 𝑐

𝑠=

√𝜇(x)/𝜌𝑠(x) are presented as functions of the Lame param-

eters and density. The P-waves move in a compressionalmotion, while the motion of the S-waves is perpendicular tothe direction of wave propagation [35].

In principle, the wave equation in fluid media can bederived as a special case of (1), by taking into account thefact that there are no S-waves in fluids. That is because fluids

Advances in Mathematical Physics 3

are inviscid and, thus, have no internal friction and can notsupport shear stresses. Thus, taking the divergence of (1)leads to modeling acoustic waves in fluid domain Ω

𝑓by the

pressure field 𝑝 (see, e.g., [16, 36]),

1

𝜌𝑓(x) 𝑐 (x)2

𝜕2

𝑝

𝜕𝑡2− ∇ ⋅ (

1

𝜌𝑓(x)

∇𝑝) = 𝑓, (2)

where 𝜌𝑓(x) is the density of fluid, 𝑐(x) = √𝜆(x)/𝜌

𝑓(x) is the

speed of the wave, 𝑝 = −𝜆(x)∇ ⋅ u depends on the vector-valued displacement u in the fluid domain, and 𝑓 = −∇ ⋅ fis the source function. By using the velocity potential 𝜙, thecorresponding model is

1

𝑐 (x)2𝜕2

𝜙

𝜕𝑡2− ∇2

𝜙 = 𝑓𝜙, (3)

where 𝑓𝜙is the source function (see, e.g., [4, 5, 17]).

The equations need to be completed by the initial andboundary conditions to get a well-posed and physicallymeaningful problem. In this paper, we use the first-orderabsorbing boundary condition [37] for truncating the exte-rior domain on the boundaries Γ

𝑒𝑠and Γ𝑒𝑓. On the interface

Γ𝑖between fluid and solid domains, the normal components

of displacements and forces are balanced. That is presented,depending on the choice of variable in the fluid domain, by

𝜎 (u)n𝑠− 𝑝n𝑓

= 0,

𝜌𝑓(x) 𝜕2u

𝜕𝑡2⋅ n𝑠−

𝜕𝑝

𝜕n𝑓

= 0,

(4)

or

𝜎 (u)n𝑠+ 𝜌𝑓(x)

𝜕𝜙

𝜕𝑡n𝑓

= 0,

𝜕u𝜕𝑡

⋅ n𝑠+

𝜕𝜙

𝜕n𝑓

= 0,

(5)

where n𝑠and n

𝑓are the outward pointing unit normal

vector on the boundary Γ𝑖, with respect to the solid and fluid

domain, respectively. The interface conditions coupling thetwo domains are needed to model the following fact: whena wave coming from the fluid domain confronts the elasticdomain, it is not totally reflected, but part of it passes to theelastic domain and turns to elastic vibrations. The analogousaction is seen when the elastic wave propagates to the fluid,although usually the reflections from fluid to solid are minorwhen compared with the reflections from solid to fluid. Tobe more precise, the magnitude of the reflection depends onthe difference between the densities and the wave speeds ofthe materials. Thus, the reflections are more significant fromcomparatively stiff structures than formore flexible obstacles.

For the weak formulation of the system consisting of (1)-(2) and (4) we introduce the function spaces 𝑉 = {V ∈

𝐻1

(Ω𝑓)} and W = {w ∈ (𝐻

1

(Ω𝑠))2

}. By multiplying (1)with any test function w in the space W and (2) with anytest function V in the space 𝑉, using Green’s formula, andsubstituting the boundary conditions, we get the following

weak formulation: find (𝑝, u) satisfying (𝑝(𝑡), u(𝑡)) ∈ (𝑉×W)

for any 𝑡 ∈ [0, 𝑇] and

∫Ω𝑓

1


𝜕2

𝑝

𝜕𝑡2V 𝑑𝑥 + ∫

Ω𝑓

1

𝜌𝑓(x)

∇𝑝 ⋅ ∇V 𝑑𝑥

− ∫Γ𝑖

𝜕2u

𝜕𝑡2⋅ n𝑠V 𝑑𝑠 + ∫

Γ𝑒𝑓

1

𝑐 (x) 𝜌𝑓(x)

𝜕𝑝

𝜕𝑡V 𝑑𝑠

= ∫Ω𝑓

𝑓V 𝑑𝑥 + ∫Γ𝑒𝑓

1

𝜌𝑓(x)

𝑦extV 𝑑𝑠,

∫Ω𝑠

𝜌𝑠(x) 𝜕2u

𝜕𝑡2⋅ w 𝑑𝑥 + ∫

Ω𝑠

𝜎 (u) : 𝜀 (w) 𝑑𝑥 − ∫Γ𝑖

𝑝n𝑓

⋅ w 𝑑𝑠 + ∫Γ𝑒𝑠

𝜌𝑠(x)

⋅ (𝑐𝑝(x) 𝑛2𝑠1

+ 𝑐𝑠(x) 𝑛2𝑠2

𝑛𝑠1𝑛𝑠2

(𝑐𝑝(x) − 𝑐

𝑠(x))

𝑛𝑠1𝑛𝑠2

(𝑐𝑝(x) − 𝑐

𝑠(x)) 𝑐

𝑝(x) 𝑛2𝑠2

+ 𝑐𝑠(x) 𝑛2𝑠1

)𝜕u𝜕𝑡

⋅ w 𝑑𝑠 = ∫Ω𝑠

f ⋅ w 𝑑𝑥 + ∫Γ𝑒𝑠

gext ⋅ w 𝑑𝑠,

(6)

where 𝑦ext and gext = (gext1, gext2)𝑇 are the source term acting

on the artificial absorbing boundaries. Respectively, we candefine the weak formulation for the system of (1), (3), and (5).

2.2. Spatial Discretization. The spectral element method isobtained from the weak formulation of the coupled prob-lem by restricting the problem presented in the infinite-dimensional spaces into finite-dimensional subspaces. Thus,in order to produce an approximate solution for the problem,the given domain Ω is discretized into a collection of 𝑁

𝑒

quadrilateral elements Ω𝑖, 𝑖 = 1, . . . , 𝑁

𝑒, such that Ω =

⋃𝑁𝑒

𝑖=1Ω𝑖. Each element is associated with four nodes. For the

discrete formulation, we define the reference element Ωref =

[0, 1]2 and invertible affine mappings G

𝑖: Ωref → Ω

𝑖such

that G𝑖(Ωref ) = Ω

𝑖. The boundary of Ω

𝑖is a union of its

four edges, which are the images of the four edges of thereference element, obtained by the mappingG

𝑖. Respectively,

the vertex nodes of Ω𝑖are the images of the four vertices of

the reference element, obtained by the mapping G𝑖. These

properties are essential for the mappingsG𝑖to be sufficiently

smooth. Each of 𝑁𝑒elements is individually mapped to the

reference element, and we make use of the affine mappingto make transformations from the physical domain to thereference domain and vice versa. The mapping between thereference element and the 𝑖th element is defined such thatG𝑖(𝜉, 𝜁) = x = (𝑥

1, 𝑥2) ∈ Ωi, where 𝜉 and 𝜁 are the Gauss-

Lobatto points in the reference element (see, e.g., [9]).After the spatial discretization by the spectral elements,

the semidiscrete form of the coupled problem is stated as

M𝜕2u

𝜕𝑡2+ S

𝜕u𝜕𝑡

+ Ku = F, (7)

where u ∈ R�� is the global block vector containing the valuesof the variables in both fluid and solid domains at time 𝑡


at the Gauss-Lobatto points of the quadrilateral mesh. Thevariable which we use in the solid domain is the displacementu(x, 𝑡). Thus, the number of degrees of freedom (DOF) in thesolid domain, ��

𝑠, is in the two-dimensional domain twice

the number of discretization points in the solid domain 𝑁𝑠.

The total number of degrees of freedom, that is, ��, dependson the variable of the fluid domain. If the fluid domain ismodeled by using pressure𝑝(x, 𝑡) or velocity potential 𝜙(x, 𝑡),we have scalar values at each spatial discretization point,and the number of degrees of freedom in the fluid domain,expressed as ��

𝑓, is equal to the number of discretization

points in the fluid domain 𝑁𝑓.

In practice, the discrete counterparts of the variables areapproximated as linear combinations of the correspondingnodal values and the spectral element basis functions 𝜑

𝑖,

𝑖 = 1, . . . , 𝑁𝑓, and 𝜓

𝑖, 𝑖 = 1, . . . , 𝑁

𝑠, which are higher-order

Lagrange interpolation polynomials [9]. In the case of theformulation with pressure and displacement, entries of the�� × ��matricesM,S, andK and the right-hand side vectorF are given by the formulas

M = (

(M𝑠)11

0 00 (M

𝑠)22

0

(A𝑓𝑠)1

(A𝑓𝑠)2

M𝑓

),

S = (

(S𝑠)11

(S𝑠)12

0(S𝑠)21

(S𝑠)22

00 0 S

𝑓

),

K = (

(K𝑠)11

(K𝑠)12

(A𝑠𝑓)1

(K𝑠)21

(K𝑠)22

(A𝑠𝑓)2

0 0 K𝑓,

) ,

F = (f𝑠

f𝑓

) ,

(8)

where the 𝑁𝑓

× 𝑁𝑓matrix blocks and the 𝑁

𝑓-dimensional

right-hand side vector corresponding to the fluid domain are

(M𝑓)𝑖𝑗

= ∫Ω𝑓

1


𝜑𝑖𝜑𝑗𝑑𝑥,

(S𝑓)𝑖𝑗

= ∫Γ𝑒𝑓

1

𝜌𝑓(x) 𝑐 (x)

𝜑𝑖𝜑𝑗𝑑𝑠,

(K𝑓)𝑖𝑗

= ∫Ω𝑓

1

𝜌𝑓(x)

∇𝜑𝑖⋅ ∇𝜑𝑗𝑑𝑥,

(f𝑓)𝑖

= ∫Ω𝑓

𝑓𝜑𝑖𝑑𝑥 + ∫

Γ𝑒𝑓

𝑦ext𝜑𝑖𝑑𝑠,

(9)

where 𝑖, 𝑗 = 1, . . . , 𝑁𝑓. Respectively, the 2𝑁

𝑠× 2𝑁𝑠block

matrices and the 2𝑁𝑠-dimensional vector representing the

elastic waves are

M𝑠= (

(M𝑠)11

00 (M

𝑠)22

) ,

S𝑠= (

(S𝑠)11

(S𝑠)12

(S𝑠)21

(S𝑠)22

) ,

K𝑠= (

(K𝑠)11

(K𝑠)12

(K𝑠)21

(K𝑠)22

) ,

f𝑠= (

(f𝑠)1

(f𝑠)2

) ,

(10)

which have the components

((M𝑠)11)𝑖𝑗

= ∫Ω𝑠

𝜌𝑠(x) 𝜓𝑗𝜓𝑖𝑑𝑥,

((M𝑠)22)𝑖𝑗

= ∫Ω𝑠

𝜌𝑠(x) 𝜓𝑗𝜓𝑖𝑑𝑥,

((S𝑠)11)𝑖𝑗

= ∫Γ𝑒𝑠

𝜌𝑠(x) (𝑐𝑝𝑛2

𝑠1+ 𝑐𝑠𝑛2

𝑠2) 𝜓𝑗𝜓𝑖𝑑𝑠,

((S𝑠)12)𝑖𝑗

= ∫Γ𝑒𝑠

𝜌𝑠(x) (𝑐𝑝− 𝑐𝑠) 𝑛𝑠1𝑛𝑠2𝜓𝑗𝜓𝑖𝑑𝑠,

((S𝑠)21)𝑖𝑗

= ∫Γ𝑒𝑠

𝜌𝑠(x) (𝑐𝑝− 𝑐𝑠) 𝑛𝑠1𝑛𝑠2𝜓𝑗𝜓𝑖𝑑𝑠,

((S𝑠)22)𝑖𝑗

= ∫Γ𝑒𝑠

𝜌𝑠(x) (𝑐𝑝𝑛2

𝑠2+ 𝑐𝑠𝑛2

𝑠1) 𝜓𝑗𝜓𝑖𝑑𝑠,

((K𝑠)11)𝑖𝑗

= ∫Ω𝑠

𝜌𝑠(x) ((𝑐

2

𝑝− 2𝑐2

𝑠)𝜕𝜓𝑗

𝜕𝑥1

𝜕𝜓𝑖

𝜕𝑥1

+ 2𝑐2

𝑠(

𝜕𝜓𝑗

𝜕𝑥1

𝜕𝜓𝑖

𝜕𝑥1

+1

2

𝜕𝜓𝑗

𝜕𝑥2

𝜕𝜓𝑖

𝜕𝑥2

))𝑑𝑥,

((K𝑠)12)𝑖𝑗

= ∫Ω𝑠

𝜌𝑠(x) ((𝑐

2

𝑝− 2𝑐2

𝑠)𝜕𝜓𝑗

𝜕𝑥2

𝜕𝜓𝑖

𝜕𝑥1

+ 𝑐2

𝑠

𝜕𝜓𝑗

𝜕𝑥1

𝜕𝜓𝑖

𝜕𝑥2

)𝑑𝑥,

((K𝑠)21)𝑖𝑗

= ∫Ω𝑠

𝜌𝑠(x) ((𝑐

2

𝑝− 2𝑐2

𝑠)𝜕𝜓𝑗

𝜕𝑥1

𝜕𝜓𝑖

𝜕𝑥2

+ 𝑐2

𝑠

𝜕𝜓𝑗

𝜕𝑥2

𝜕𝜓𝑖

𝜕𝑥1

)𝑑𝑥,

((K𝑠)22)𝑖𝑗

= ∫Ω𝑠

𝜌𝑠(x) ((𝑐

2

𝑝− 2𝑐2

𝑠)𝜕𝜓𝑗

𝜕𝑥2

𝜕𝜓𝑖

𝜕𝑥2


+ 2𝑐2

𝑠(

1

2

𝜕𝜓𝑗

𝜕𝑥1

𝜕𝜓𝑖

𝜕𝑥1

+𝜕𝜓𝑗

𝜕𝑥2

𝜕𝜓𝑖

𝜕𝑥2

))𝑑𝑥,

((f𝑠)1)𝑖= ∫Ω𝑠

f1𝜓𝑖𝑑𝑥 + ∫

Γ𝑒𝑠

gext1𝜓𝑖𝑑𝑠,

((f𝑠)2)𝑖= ∫Ω𝑠

f2𝜓𝑖𝑑𝑥∫Γ𝑒𝑠

gext2𝜓𝑖𝑑𝑠,

(11)

where 𝑖, 𝑗 = 1, . . . , 𝑁𝑠.Thematrices arising from the coupling

between acoustic and elastic wave equations areA𝑓𝑠andA

𝑠𝑓,

for which it holds that

((A𝑓𝑠)1

)𝑖𝑗

= −∫Γ𝑖

𝑛𝑠1𝜓𝑗𝜑𝑖𝑑𝑠,

((A𝑓𝑠)2

)𝑖𝑗

= −∫Γ𝑖

𝑛𝑠2𝜓𝑗𝜑𝑖𝑑𝑠,

((A𝑠𝑓)1

)𝑖𝑗

= −∫Γ𝑖

𝑛𝑓1

𝜑𝑗𝜓𝑖𝑑𝑠,

((A𝑠𝑓)2

)𝑖𝑗

= −∫Γ𝑖

𝑛𝑓2

𝜑𝑗𝜓𝑖𝑑𝑠.

(12)

For A𝑓𝑠, 𝑖 = 1, . . . , 𝑁

𝑓and 𝑗 = 1, . . . , 𝑁

𝑠, whereas, for A

𝑠𝑓,

𝑖 = 1, . . . , 𝑁𝑠and 𝑗 = 1, . . . , 𝑁

𝑓.

The computation of the elementwisematrices and vectorsinvolves the integration over the elementwise subregions.Evaluating these integrals analytically is usually complicatedor even impossible. That is why a numerical integrationprocedure is used. In practice, we replace the integrals byfinite sums, inwhichweuseGauss-Lobattoweights andnodalpoints. The values of these sums are computed element byelement with the Gauss-Lobatto integration rule. Collocationpoints are now the nodes of the spectral element. All but oneof the shape functions will be zero at a particular node.Thus,for 𝑖 = 𝑗, (M

𝑓)𝑖𝑗

= 0 and (M𝑠)𝑖𝑗

= 0 implying that thematrices M

𝑓and M

𝑠are diagonal. Furthermore, the matrix

M is a lower triangular block matrix with diagonal blocks.Thus, the inverse of the matrix M is also a lower triangularblock matrix with diagonal blocks,

M−1

= (

(M𝑠)−1

110 0

0 (M𝑠)−1

220

−M−1𝑓

(A𝑓𝑠)1

(M𝑠)−1

11−M−1𝑓

(A𝑓𝑠)2

(M𝑠)−1

22M−1𝑓

),

(13)

and explicit time-stepping with central finite differencesrequires only matrix-vector multiplications.

In practice, the stiffness matrix K is assembled onceat the beginning of the simulation. It is stored by usingthe compressed column storage including only the nonzeromatrix elements. The other options would have been using amixed spectral element formulation [38].

In the case of the formulation with velocity potential anddisplacement, the entries of the ��×��matricesM,S, andKand the right-hand side vectorF are given by the formulas

M = (

(M𝑠)11

0 00 (M

𝑠)22

00 0 M

𝑓

),

S = (

(S𝑠)11

(S𝑠)12

(A𝑠𝑓)1

(S𝑠)21

(S𝑠)22

(A𝑠𝑓)2

(A𝑓𝑠)1

(A𝑓𝑠)2

S𝑓

),

K = (

(K𝑠)11

(K𝑠)12

0(K𝑠)21

(K𝑠)22

00 0 K

𝑓,

) ,

F = (f𝑠

f𝜙𝑓

) ,

(14)

where the 𝑁𝑓

× 𝑁𝑓matrix blocks and the 𝑁

𝑓-dimensional

right-hand side vector corresponding to the fluid domain are

(M𝑓)𝑖𝑗

= ∫Ω𝑓

𝜌𝑓(x)

𝑐 (x)2𝜑𝑖𝜑𝑗𝑑𝑥,

(S𝑓)𝑖𝑗

= ∫Γ𝑒𝑓

𝜌𝑓(x)

𝑐 (x)𝜑𝑖𝜑𝑗𝑑𝑠,

(K𝑓)𝑖𝑗

= ∫Ω𝑓

𝜌𝑓(x) ∇𝜑

𝑖⋅ ∇𝜑𝑗𝑑𝑥,

(f𝜙𝑓

)𝑖

= ∫Ω𝑓

𝜌𝑓(x) 𝑓𝜙𝜑𝑖𝑑𝑥 + ∫

Γ𝑒𝑓

𝜌𝑓(x) 𝑦𝜙ext𝜑𝑖𝑑𝑠,

(15)

for 𝑖, 𝑗 = 1, . . . , 𝑁𝑓. The 2𝑁

𝑠× 2𝑁

𝑠block matrices and

the 2𝑁𝑠-dimensional vector representing the elastic waves

are exactly the same as in the nonsymmetric formulation.The matrices arising from the coupling between acoustic andelastic wave equations areA

𝑓𝑠andA

𝑠𝑓, for which it holds that

((A𝑓𝑠)1

)𝑖𝑗

= ∫Γ𝑖

𝜌𝑓(x) 𝑛𝑠1𝜓𝑗𝜑𝑖𝑑𝑠,

((A𝑓𝑠)2

)𝑖𝑗

= ∫Γ𝑖

𝜌𝑓(x) 𝑛𝑠2𝜓𝑗𝜑𝑖𝑑𝑠,

((A𝑠𝑓)1

)𝑖𝑗

= ∫Γ𝑖

𝜌𝑓(x) 𝑛𝑓1

𝜑𝑗𝜓𝑖𝑑𝑠,

((A𝑠𝑓)2

)𝑖𝑗

= ∫Γ𝑖

𝜌𝑓(x) 𝑛𝑓2

𝜑𝑗𝜓𝑖𝑑𝑠.

(16)

For A𝑓𝑠, 𝑖 = 1, . . . , 𝑁

𝑓and 𝑗 = 1, . . . , 𝑁

𝑠, whereas, for A

𝑠𝑓,

𝑖 = 1, . . . , 𝑁𝑠and 𝑗 = 1, . . . , 𝑁

𝑓.

Using the vector-valued displacement u(x, 𝑡) in the fluiddomain doubles the number of degrees of freedom in thefluid domain. In other words, ��

𝑓= 2𝑁𝑓. Therefore also the


memory consumption for computing the pure displacement-displacement interaction is higher than in the case of theother couplings considered. Furthermore, if the formula-tion with displacement in both domains is considered, thedivergence of the variable in the fluid domain is involvedin the weak formulation, and we would need a schemethat approximates the functions in 𝐻(div, Ω

𝑓) better than

the spectral element method does. For instance, Raviart-Thomas finite elements, which are used for acoustic waveequations [39], could be used for this purpose. Furthermore,the Raviart-Thomas elements are not a good choice fordiscretizing the solid domain. Nevertheless, the couplingconditions should be satisfied at the interface of the twodomains. That is, we would need to change the discretizationapproach in the solid domain as well, to make the degreesof freedom coincide or fulfill the coupling conditions, forexample, by using Lagrange multipliers [40].

2.3. Time Discretization. After dividing the time interval[0, 𝑇] into 𝑁 time steps, each of size Δ𝑡 = 𝑇/𝑁, applying theappropriate time discretization into semidiscretized form (7),and taking into account the initial conditions, we obtain thematrix form of the fully discrete state equation. Despite theuse of higher-order space discretization, the time-stepping fortransient wave equations is usually performed by low-ordermethods like the central finite difference (CD) scheme. Byproceeding this way, in the case of the nonsymmetric formu-lation, at each time step 𝑖 we compute first the displacementu𝑖 and then the pressure 𝑝

𝑖 from the equations

M𝑠

u𝑖+1 − 2u𝑖 + u𝑖−1

Δ𝑡2+ S𝑠

u𝑖+1 − u𝑖−1

2Δ𝑡+ K𝑠u𝑖

+ A𝑠𝑓𝑝𝑖

= f 𝑖𝑠,

M𝑓

𝑝𝑖+1

− 2𝑝𝑖

+ 𝑝𝑖−1

Δ𝑡2+ S𝑓

𝑝𝑖+1

− 𝑝𝑖−1

2Δ𝑡+ K𝑓𝑝𝑖

+ A𝑓𝑠

u𝑖+1 − 2u𝑖 + u𝑖−1

Δ𝑡2= f 𝑖𝑓,

(17)

with the initial conditions

u0 = e𝑠0,

𝑝0

= e𝑓0

,

u1 − u−1

2Δ𝑡= e𝑠1,

𝑝1

− 𝑝−1

2Δ𝑡= e𝑓1

,

(18)

where u𝑖, 𝑝𝑖, f 𝑖𝑠, and f 𝑖

𝑓are the vectors u, 𝑝, f

𝑠, and f

𝑓at

𝑡 = 𝑖Δ𝑡. Because the matrix sums (M𝑠+ (Δ𝑡/2)S

𝑠) and

(M𝑓

+ (Δ𝑡/2)S𝑓) are diagonal, their inverses are obtained

simply by inverting each diagonal element. Respectively, the

state equation for the formulation with velocity potential anddisplacement is

M𝑠

u𝑖+1 − 2u𝑖 + u𝑖−1

Δ𝑡2+ S𝑠

u𝑖+1 − u𝑖−1

2Δ𝑡+ K𝑠u𝑖

+ A𝑠𝑓

𝜙𝑖+1

− 𝜙𝑖−1

2Δ𝑡= f 𝑖𝑠,

M𝑓

𝜙𝑖+1

− 2𝜙𝑖

+ 𝜙𝑖−1

Δ𝑡2+ S𝑓

𝜙𝑖+1

− 𝜙𝑖−1

2Δ𝑡+ K𝑓𝜙𝑖

+ A𝑓𝑠

u𝑖+1 − u𝑖−1

2Δ𝑡= f 𝑖𝜙𝑓

,

u0 = e𝑠0, 𝜙0

= e𝜙𝑓0

,u1 − u−1

2Δ𝑡= e𝑠1,

𝜙1

− 𝜙−1

2Δ𝑡= e𝜙𝑓1

,

(19)

where u𝑖, 𝜙𝑖, f 𝑖𝑠, and f i

𝜙𝑓are the vectors u, 𝜙, f

𝑠, and f

𝜙𝑓

at 𝑡 = 𝑖Δ𝑡. In this case, u𝑖+1 and 𝜙𝑖+1 need to be solved

simultaneously, and we need to invert (M+ (Δ𝑡/2)S) at eachtime step. If the boundaries of the computational domainconsist only of horizontal and vertical lines, the coefficientmatrix (M+ (Δ𝑡/2)S) needed for inversion at each time stepcan be implemented as a block matrix consisting of diagonalblocks.Then, u𝑖 and 𝜙

𝑖 can be solved simply by using matrix-vector multiplications from the formulas

u𝑖 = (I − D−1

𝑠

Δ𝑡

2A𝑠𝑓D−1

𝑓

Δ𝑡

2A𝑓𝑠)

−1

⋅ D−1

𝑠(𝑦1−

Δ𝑡

2A𝑠𝑓D−1

𝑓𝑦2) ,

𝜙𝑖

= D−1

𝑓(𝑦2−

Δ𝑡

2A𝑓𝑠u𝑖) , 𝑖 = 1, . . . , 𝑁 − 1,

(20)

where D𝑠= M𝑠+ (Δ𝑡/2)S

𝑠and D

𝑓= M𝑓

+ (Δ𝑡/2)S𝑓are

diagonal matrices and

𝑦1= (2M

𝑠− Δ𝑡2

K𝑆) u𝑖 + (

Δ𝑡

2S𝑆− M𝑠) u𝑖−1

+ Δ𝑡2f 𝑖𝑠+

Δ𝑡

2A𝑠𝑓𝜙𝑖−1

,

𝑦2= (2M

𝑓− Δ𝑡2

K𝑓) 𝜙𝑖

+ (Δ𝑡

2S𝑓− M𝑓)𝜙𝑖−1

+ Δ𝑡2f 𝑖𝜙𝑓

+Δ𝑡

2A𝑓𝑠u𝑖−1.

(21)

Remark 1. It would also be possible to uncenter one of thetwo first-order derivatives in time to uncouple the problem.That approach would involve first-order difference approx-imations and introduce some dissipation. That is why weneglect deeper considerations of the uncentered schemes.

Sate equation (7) can be presented as a system of differ-ential equations 𝜕y/𝜕𝑡 = 𝑓(𝑡, y(𝑡)), where y = (u, k)𝑇 isa vector of time-stepping variables u and k = 𝜕u/𝜕𝑡 andthe function 𝑓(𝑡, 𝑦(𝑡)) = (k, −M−1(Sk + Ku − F))

𝑇. To


this modified form, we can apply the fourth-order Runge-Kuttamethod, which is a Taylor seriesmethod. In general, theTaylor series methods keep the errors small, but there is thedisadvantage of requiring the evaluation of higher derivativesof the function 𝑓(𝑡, y(𝑡)). The advantage of the Runge-Kuttamethod is that explicit evaluations of the derivatives of thefunction 𝑓(𝑡, y(𝑡)) are not required, but linear combinationsof the values of 𝑓(𝑡, y(𝑡)) are used to approximate y(𝑡). In thefourth-order Runge-Kutta method, the approximate y at the𝑖th time step is defined as

y𝑖 = y𝑖−1 + Δ𝑡

6(𝑘1+ 2𝑘2+ 2𝑘3+ 𝑘4) , (22)

where y𝑖 = (u𝑖, 𝜕u𝑖/𝜕𝑡)𝑇 contains the global block vector u𝑖,including the values of the variables in both the fluid and thesolid domain at the 𝑖th time step, and its derivative k𝑖 = 𝜕u𝑖/𝜕𝑡at time 𝑡 = 𝑖Δ𝑡, 𝑖 = 1, . . . , 𝑁. The initial condition is given byy0 = e = (e

0, e1)𝑇, and 𝑘

𝑗= (𝑘𝑗1, 𝑘𝑗2)𝑇, 𝑗 = 1, 2, 3, 4, are the

differential estimates as follows:

(𝑘11

𝑘12

) = (𝑓1(𝑖Δ𝑡, u𝑖, k𝑖)

𝑓2(𝑖Δ𝑡, u𝑖, k𝑖)

) ,

(𝑘21

𝑘22

) = (

𝑓1(𝑖Δ𝑡 +

Δ𝑡

2, u𝑖 + 𝑘

11

2, k𝑖 +

𝑘12

2)

𝑓2(𝑖Δ𝑡 +

Δ𝑡

2, u𝑖 + 𝑘

11

2, k𝑖 +

𝑘12

2)

) ,

(𝑘31

𝑘32

) = (

𝑓1(𝑖Δ𝑡 +

Δ𝑡

2, u𝑖 + 𝑘

21

2k𝑖 +

𝑘22

2)

𝑓2(𝑖Δ𝑡 +

Δ𝑡

2, u𝑖 + 𝑘

21

2k𝑖 +

𝑘22

2)

) ,

(𝑘41

𝑘42

) = (𝑓1(𝑖Δ𝑡 + Δ𝑡, u𝑖 + 𝑘

31, k𝑖 + 𝑘

32)

𝑓2(𝑖Δ𝑡 + Δ𝑡, u𝑖 + 𝑘

31, k𝑖 + 𝑘

32)) .

(23)

In other words, in order to get differential estimates (23), thefunction 𝑓 is evaluated at each time step four times, and thenthe successive approximation of y is calculated by formula(22).

If the matrixM is diagonal, as is in the formulation withthe velocity potential, the only matrix inversion needed intime-stepping is computed simply by inverting each diagonalelement in the matrixM. This requires only �� floating pointoperations, which is the number of diagonal elements in thematrix M and known as the number of degrees of freedomin the space discretization. Since the matrix S containsonly diagonal blocks and coupling terms, the operationcount of the matrix-vector product Sk is of order ��. In thematrix-vector multiplication involving the sparse stiffnessmatrixK, only nonzero matrix entries are multiplied, whichrequires the order of 𝑟

2

�� operations, where 𝑟 is the orderof the polynomials in the spectral element basis. Besidesthese, 2�� additions and 3�� multiplications are needed for asingle evaluation of the function 𝑓. According to (22), thecomputation of y𝑖 needs 14�� floating point operations. Thus,the computational cost for each time step of the state equation

is of order 𝑂(𝑟2

��) also with the RK time-stepping. Althoughthe computational cost is of the same order for both theCD and the RK time-steppings, the number of floating pointoperators needed for the RK is nearly four times that of theCD.

Next, we make an effort to decrease the computing timeand stillmaintain the high accuracy provided by higher-ordertime discretizations. For this purpose, we present the fourth-order Adams-Bashforth (AB) method based on approximat-ing the functions 𝑓(𝑡, y(𝑡)) by interpolating polynomials. Itgives the solution y at the 𝑖th time step as

y𝑖 = y𝑖−1 + ℎ

24(55𝑓 ((𝑖 − 1) Δ𝑡, y𝑖−1)

− 59𝑓 ((𝑖 − 2) Δ𝑡, y𝑖−2) + 37𝑓 ((𝑖 − 3) Δ𝑡, y𝑖−3)

− 9𝑓 ((𝑖 − 4) Δ𝑡, y𝑖−4)) ,

(24)

where y𝑖 = (u𝑖, 𝜕u𝑖/𝜕𝑡)𝑇 contains the vector u𝑖 and itsderivative k𝑖 = 𝜕u𝑖/𝜕𝑡 at time 𝑡 = 𝑖Δ𝑡, 𝑖 = 4, . . . , 𝑁. Hence,it is a multistep method that requires information at fourprevious time steps implying that another method is neededfor starting the time marching. At this stage, we utilize thefourth-order Runge-Kutta method for computing the start-up values y0, y1, y2, and y3 to initialize the multistep method.

3. Numerical Examples

In this section, the computational accuracy and efficiency ofsolving seismic wave problems are considered. The focus ofSection 3.1 is on different formulations and implementationprocesses. In Section 3.2, the efficiency is further analysedby using higher-order time discretization. Finally, simulatedseismograms and two-dimensional illustrations presentingthe seismic wave motion are given in Section 3.3. Thenumerical experiments presented in this section are carriedout on an AMD Opteron 885 processor at 2.6GHz.

3.1. Comparison between Symmetric and Nonsymmetric For-mulations. We illustrate the computational cost of bothsymmetric and nonsymmetric formulations by solving atime-dependent problem both expressing the acoustic waveequation by the pressure and by using the velocity potential inthe fluid domain. The right-hand sides and initial conditionsare defined to satisfy the analytical solution𝑝 = 𝜔𝜌

𝑓(x)sin(𝜔⋅

x)cos(𝜔𝑡), 𝜙 = − sin(𝜔 ⋅ x) sin(𝜔𝑡), and u = (cos(𝜔 ⋅ x/𝑐𝑝(x)) cos(𝜔𝑡), cos(𝜔 ⋅ x/𝑐

𝑠(x)) cos(𝜔𝑡))

𝑇.The problem is solved in a domain, which consists of the

solid partΩ𝑠= [−1, 0]× [0, 1] and the fluid partΩ

𝑓= [0, 1]×

[0, 1] (see Figure 1). We use square-element meshes withmesh step size ℎ = 0.1, and the element order is increased inboth parts of the domain from 1 to 5.Themeshes arematchingon the coupling interface Γ

𝑖set at 𝑥

1= 0 for 𝑥

2∈ [0, 1].

On the other boundaries we have the absorbing boundaryconditions. The material parameters in the fluid domain are𝜌𝑓(x) = 1.0 and 𝑐(x) = 1.0. In the solid domain, we use the

values 𝑐𝑝(x) = 6.20, 𝑐

𝑠(x) = 3.12, and 𝜌

𝑠(x) = 2.7. The angular

frequency 𝜔 = 4𝜋 is the same for both media, and we set


1 2 3 4 5

CPU

tim

e

Order of the polynomial basis

PressureVelocity potential, Lapack LUVelocity potential, SuperLU

102

101

100

(a) CPU time with respect to the element order


1 2 3 4 5

Mem

ory

cons

umpt

ion

Order of the polynomial basis

106

105

(b) Memory consumption with respect to the element order

Figure 2: CPU time (in seconds) and memory (in kilobytes) consumed for solving a time-dependent problem with different formulations.The number of time steps is fixed to be 400, and square-element meshes with mesh step size ℎ = 0.1 are used in both media.

the propagation direction (1, 0) by the vector 𝜔 = (𝜔1, 𝜔2) =

(1, 0)𝜔. The time interval [0.0, 0.5] is divided into 400 steps,each of sizeΔ𝑡 = 0.00125, to guarantee the stability conditionalso for the higher element orders.

When the pressure formulation is considered, the timemarching involves only matrix-vector multiplications, andactual matrix inversions are not needed. In the implemen-tation of the velocity potential formulation, a “one-shot”method is required for solving the linear system includingu𝑖+1 and 𝜙

𝑖+1 at each time step 𝑖 = 1, . . . , 𝑁. In that case,the matrix which is needed to be inverted is stored either asa band matrix or by using the compressed column storageincluding only the nonzero matrix elements.

The Lapack LU decomposition routines dgbtrf anddgbtrs use the band matrix storage mode. With theseroutines the memory and CPU time consumption increasesrapidly when the element order is increased (see Figure 2).The SuperLU library routines perform an LU decomposi-tion with partial pivoting, and the triangular system solvesthrough forward and backward substitution. At this stage, wealso utilize the sparsity of the matrix by using the compressedcolumn storage mode. Since all the nonzero elements arenot near the diagonal of the matrix, the compressed columnstorage mode requires less storage and computing operationsthan the band storage mode. That is why the SuperLUlibrary gives a less demanding procedure for solving thelinear system than the Lapack library. This is because thematrices arising from the space discretization and includingthe coupling terms have, in general, a sufficiently largebandwidth. Thus, a remarkably larger amount of memory isneeded for storing the sparse coefficient matrix in the bandmatrix form used in conjunction with Lapack routines thanin the compressed column storage utilized with the SuperLU.Consequently, less time is needed when fewer elements are

employed in the solution procedure with the SuperLU. Theperformance of the Lapack routines could be improved, forinstance, by using an appropriate node numbering of themesh.

We conclude that the computational efforts are of thesame order of magnitude whether the problem in the fluiddomain is solved with respect to pressure or whether we usevelocity potential formulation in conjunction with the linearsolver provided by the SuperLU library.

The comparison between numerical and analytical solu-tion shows that in both media the accuracy improves whenthe element order grows until a certain error level is reached(see Figure 3). This error level, shown as a horizontal line inFigure 3, reflects the error level of time discretization. Sincewe use a fixed number of time steps at all element orders,the error of time discretization becomes dominant for higher-order elements. Hence, finer time steps or higher-order timediscretizations are needed in conjunction with higher-orderelements.

In principle, both symmetric andnonsymmetric formula-tion should lead to the same order of accuracy at each elementorder. The results obtained in the solid domain and depictedin Figure 3(a) are perfectly in balance with this hypothesis.However, in Figure 3(b) we see that in the fluid domainhigher accuracy is obtained with the velocity potential thanwith the pressure formulation at each element order. Inthe case of pressure formulation, the interface condition isderived by differentiating the displacement component twicewith respect to time. From the physical point of view, someinformation is lost in that procedure, and linear growth ofthe displacement with respect to time is not eliminated atthe interface. However, the solutions of the pressure formu-lation converge towards the analytical solution. Althoughthe pressure formulation gives less accurate results than the


1 2 3 4 5Order of the polynomial basis

Pressure

Max

imum

erro

r

Velocity potential, Lapack LUVelocity potential, SuperLU

10−2

10−3

10−4

(a) Solid domain

1 2 3 4 5Order of the polynomial basis


10−1

10−2

10−3

10−4

100

Max

imum

erro

r

(b) Fluid domain

Figure 3: Maximum errors, computed as 𝐿∞-norms, with respect to the element order. The number of time steps is fixed to be 400, and

square-element meshes with mesh step size ℎ = 0.1 are used in both media.

velocity potential formulation, it is still a plausible choice atsome point. For instance, the implementation process can behastened when the preexisting solvers of acoustic and elasticproblems can be harnessed in the implementation, and noadditional linear solvers are needed.

3.2. Higher-Order TimeDiscretizations. Wedemonstrate howthe efficiency of the method can be improved by usingthe fourth-order Runge-Kutta scheme instead of the centralfinite difference time discretization. In principle, the centralfinite difference scheme is second-order accurate, while thefourth-order Runge-Kutta method is fourth-order accurate.However, the error of space discretization limits the accuracyof the overall time-stepping scheme as well, and the stabilityissue restricts choosing a feasible length of the time step. Tobetter illustrate the error of time and space discretization, wecontinue with the example presented in Section 3.1, exceptthat we change the mesh step size to ℎ = 1/20 and usevarious time step lengths to observe stability and accuracyissues. The number of time steps needed for stability is firstdetermined numerically by using 50𝑖 time steps per timeperiod, for 𝑖 = 1, 2, 3, . . ., until a stable solution is achieved.From these results, we can define stability constant𝛼

𝑟for each

element order 𝑟 such that

Δ𝑡

ℎ=

𝛼𝑟

max {𝑐, 𝑐𝑝, 𝑐𝑠}√2

. (25)

The stability conditions corresponding to the largest stabletime step are given in Table 1.The stability region seems to beexactly the same with both the CD and the RK time-steppingfor the element orders 𝑟 = 1, 2, 3 and differs only slightlyfor the higher-order elements. The results reflect the well-known CFL condition and are in a good agreement with theexperiments presented for acoustic waves with the CD timediscretization by Cohen [9].

Table 1: Stability conditions for the CD and the RK time discretiza-tion schemes.

𝑟 1 2 3 4 5

Number oftime steps

CD 100 200 350 550 800

RK 100 200 350 600 900

𝛼𝑟

CD 0.8765 0.4383 0.2504 0.1594 0.1096

RK 0.8765 0.4383 0.2504 0.1461 0.0974

Westart the computationswith the largest stable time stepand then repeatedly add the number of time steps 𝑁 = 𝑇/Δ𝑡

by 300, until their number is larger than 3000. Proceedingin this way, for each element order we achieve a series ofnumerical results with various lengths of the time step. Themaximum errors between the numerical and the analyticalsolution with respect to Δ𝑡/ℎ are computed as 𝐿

∞-norms.Accuracy of the numerical solution is shown in Figure 4with respect to the ratio between the time step Δ𝑡 andthe mesh step size ℎ for both the CD and the RK time-steppings with five element orders 𝑟. Every curve representscomputations with a particular polynomial order which hasa characteristic discretization error. Naturally, the order ofthe space discretization error decreases when higher-orderelements are used. It is worth mentioning that errors aresomewhat smaller in the solid domain than in the fluiddomain.

It is seen that with the element orders 𝑟 = 1 and 𝑟 = 2 bothtime-stepping schemes give the same accuracy even whensufficiently large time steps are used. Moreover, this is theaccuracy of the space discretization since the error of spatialdiscretization dominates with low-order elements. In otherwords, the spatial error is larger than the temporal error, andthemaximum error with respect to the length of the time stepis not decreasing significantly even if smaller time steps areused.


10−1

10−2

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

CD, r = 1 RK, r = 1

CD, r = 2 RK, r = 2

CD, r = 3 RK, r = 3

CD, r = 4 RK, r = 4

CD, r = 5 RK, r = 5

Maximum error

Δt/h

(a) Solid domain

10−1

10−2

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

CD, r = 1 RK, r = 1

CD, r = 2 RK, r = 2

CD, r = 3 RK, r = 3

CD, r = 4 RK, r = 4

CD, r = 5 RK, r = 5

Maximum error

Δt/h

(b) Fluid domain

Figure 4: Comparison between maximum errors with the CD and the RK time discretizations. The mesh step size is fixed to be ℎ = 1/20,and the time step refinement gives a series of numerical results with various lengths of the time step for each element order.

Naturally, for each element order 𝑟 the solution withthe RK time-stepping is at least as accurate as the onecomputed with the CD time-stepping. When higher-orderelements are used, results computed with the RK timediscretization aremore accurate than the ones computedwiththe CD time discretization. Depending on the accuracy ofthe time discretization, the error of temporal discretizationmight be dominating with large time steps. This is shownespecially in the case of the CD time discretization with spacediscretization orders 𝑟 ≥ 3. In principle, also with thesehigher-order elements, the error curves turn to vertical lines,reflecting the accuracy of the space discretization, when thelength of the time step is refined enough. In the case of theRK time discretization, the error of space discretization isdominant even when long time steps are used. With the RKtime discretization, very fine time steps are needed only for𝑟 = 5 in the solid domain to achieve the error level of spatialdiscretization.

With the help of the results depicted in Figure 4 we canextrapolate that very fine time steps are needed with the CDtime discretization to keep the temporal error smaller thanthe spatial error when higher-order elements are involved.For 𝑟 = 3, we would already need thousands of time stepswith the CD time discretization to get the same accuracyas with the RK time discretization with 350 time steps.Tens of thousands of time steps are required to conceal thetemporal error with the CD time discretization with 𝑟 = 4.Respectively, for 𝑟 = 5 we need hundreds of thousands of

Table 2: Constants 𝑘𝑟concealing the temporal error with the CD

and the RK time discretizations.

𝑟 1 2 3 4 5

𝑘𝑟

CD 0.8765 0.1096 0.0167 0.0017 0.0003

RK 0.8765 0.4383 0.2191 0.0730 0.0266

time steps to achieve the error level of space discretization.The values of 𝑘

𝑟, satisfying

Δ𝑡

ℎ=

𝑘𝑟

max {𝑐, 𝑐𝑝, 𝑐𝑠}√2

(26)

and concealing the temporal error, are reported for differentelement orders in Table 2.

Since refining the time steps means more evaluations,we are also interested in measuring the CPU time neededin these simulations. The numerical results about CPU timeconsumption are seen in Figure 5. To conceal the temporalerror for 𝑟 = 1 and 𝑟 = 2, less computation time isneededwith theCD timediscretization thanwith theRK timediscretization.With higher-order elements, more remarkabletime saving occurs by using the RK time discretization toattain the accuracy of the space discretization. Althoughthe difference in time consumption is not significant with asufficiently small number of degrees of freedom, it can playan important role in large-scale real-life applications.


10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

CD, r = 1 RK, r = 1

CD, r = 2 RK, r = 2

CD, r = 3 RK, r = 3

CD, r = 4 RK, r = 4

CD, r = 5 RK, r = 5

Maximum error

101

102

103

100

CPU

tim

e

(a) Solid domain

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

CD, r = 1 RK, r = 1

CD, r = 2 RK, r = 2

CD, r = 3 RK, r = 3

CD, r = 4 RK, r = 4

CD, r = 5 RK, r = 5

Maximum error

101

102

103

100

CPU

tim

e

(b) Fluid domain

Figure 5: Accuracy with respect to CPU time consumptions (in seconds) with the CD and the RK time discretizations. The mesh step sizeis fixed to be ℎ = 1/20, and the time step refinement gives a series of numerical results with various lengths of the time step for each elementorder.

Next, we consider the Adams-Bashforth (AB) time-stepping scheme, with which only one evaluation of thefunction 𝑓(𝑡, y(𝑡)) is needed at each time step. The valuesare stored to be used in the four following time steps.Hence, the memory consumption is of the same order ofmagnitude as when the fourth-order RK method is utilized.In theory, employing the fourth-order AB time-steppingscheme, instead of the fourth-order RK method, saves time.Nevertheless, the stability region is smaller for the fourth-orderAB than the fourth-order RKmethod.The largest stabletime steps with the AB method, determined numerically inthe same way as is done in the previous example, are reportedin Table 3. This practical realization shows that the length ofthe time step that guarantees the stability conditions is muchsmaller with the AB method than with the RK method. Thecomputations are continued by carrying out the simulationswith the AB time discretization with smaller time steps. Thatis, for the element order 𝑟 the addition of 300𝑟 time steps isrepeated until the number of time steps is larger than 3000𝑟.The results are presented in Figure 6 as accuracy with respectto CPU time consumption. For comparison, the results of

the RK time discretization from the previous example arealso depicted there. Clearly, stability restrictions deterioratethe efficiency of the AB time-stepping scheme. The CPUtime used for the computation is in favor of the RK timediscretization; time that is more than an order of magnitudelonger is consumed to get the same accuracy with the ABtime discretization than with the RK time discretization.Thus, the efficiency of the method can not be improved byutilizing the AB time discretization. Furthermore, the ABtime discretization requires the solutions of the earlier timesteps to be stored and, hence, is not memory efficient.

3.3. Simulated Seismogram. In the last example, we considera Runge-Kutta time-simulation for the velocity potentialformulation with the element order 𝑟 = 4. We use thesame computational domain as in the previous examples withmatching square-elementmesheswith step size ℎ = 1/20.Thematerial parameters are 𝜌

𝑓(x) = 1.0, 𝑐(x) = 1.5, 𝑐

𝑝(x) = 4.0,

𝑐𝑠(x) = 2.0, and 𝜌

𝑠(x) = 2.5. A point source is set on the fluid

domain, such that

𝑓𝜙=

{

{

{

−1152𝜋(1 − 1152𝜋 (𝑡 −1

12)

2

exp(−576𝜋 (𝑡 −1

12)

2

)) , x = (0.9, 0.5)𝑇

,

0, x = (0.9, 0.5)𝑇

.

(27)


CPU

tim

e

Maximum error

101

102

103

AB, r = 1

AB, r = 2

AB, r = 3

AB, r = 4

AB, r = 5

RK, r = 1

RK, r = 2

RK, r = 3

RK, r = 4

RK, r = 5

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10−8

10−9

10−10

(a) Solid domain

CPU

tim

e

Maximum error

101

102

103

AB, r = 1

AB, r = 2

AB, r = 3

AB, r = 4

AB, r = 5

RK, r = 1

RK, r = 2

RK, r = 3

RK, r = 4

RK, r = 5

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10−8

10−9

10−10

(b) Fluid domain

Figure 6: Accuracy with respect to CPU time consumptions (in seconds) with the RK and the AB time discretizations. The mesh step size isfixed to be ℎ = 1/20, and the time step refinement gives a series of numerical results with various lengths of the time step for each elementorder.

Table 3: Stability conditions for the AB time discretization scheme.

𝑟 1 2 3 4 5

Number oftime steps AB 500 1550 3250 5450 8250

𝛼𝑟

AB 0.1753 0.0565 0.0270 0.0161 0.0106

It generates an impulse and is typical for modeling seismicwaves related to earthquakes or explosions (see, e.g., [29,41, 42]). The other source functions, 𝑦

𝜙ext, f , and, gext, aswell as the initial conditions, e

𝜙𝑓0, e𝑠0, and e

𝑠1, are zero-

valued. The simulation has been run from 𝑡 = 0 to 𝑡 =

2 with 3100 time steps, each of size Δ𝑡 = 0.000645. Theresults, observed as the values of the velocity potential at thepoint (0.7, 0.5), are presented in Figure 7. Respectively, thehorizontal and vertical components of the displacement atthe point (−0.3, 0.5) are seen in Figure 8. That is, we havesimulated results corresponding to measurements obtainedby a receiver, such as a seismometer, placed at a particularplace. Further, we present in Figure 9 a two-dimensionalsnapshot of both the domains at time step 1450.

4. Conclusions

We considered the spectral element space discretization fortime-dependent equations considering acoustic and elastic

wave propagation and their interaction. It is possible toconstruct the spectral element formulation such that it resultsin a global mass matrix that is diagonal by construction,which leads to efficient implementation.This is an advantagecompared with the classical finite element method.

The temporal discretization for time-dependent equa-tions was made by the second-order accurate central finitedifference scheme or the fourth-order accurate Runge-Kuttaor Adams-Bashforth approaches. The performances of thedifferent time discretization schemes were compared numer-ically. We found out that the Adams-Bashforth scheme hassuch strict stability conditions that it is not a feasible choice atthis stage.The CPU time used for the computation is in favorof the RK time discretization; namely, time that is more thanone order of magnitude longer is consumed to get the sameaccuracy with the AB time discretization than with the RKtime discretization.

We can conclude that, to make good use of higher-orderelements, also the time discretization should be done with ahigher-order scheme. As a rule of thumb we can say that theefficiency of the overall method suffers from the error of timediscretization if the order of the element is greater than theorder of the time discretization method used. The second-order central finite difference time discretization method isefficient with finite elements, but when high accuracy is


−0.0012

−0.0008

−0.0004

0.0000

0.0004

0.0008

300025002000150010005000

Am

plitu

de

Time step

Figure 7: Acoustic potential observed at the point (0.7, 0.5).

300025002000150010005000

Am

plitu

de

Time step

6e − 05

4e − 05

2e − 05

0

−2e − 05

−4e − 05

−6e − 05

−8e − 05

(a) Horizontal component

300025002000150010005000

Am

plitu

de

Time step

3e − 17

2e − 17

1e − 17

0

−1e − 17

−2e − 17

−3e − 17

−4e − 17

(b) Vertical component

Figure 8: Displacement observed at the point (−0.3, 0.5).

(a) Displacement amplitude inΩ𝑠

(b) Velocity potential inΩ𝑓

Figure 9: The snapshot of the wave propagation at time step 1450.


needed, it is best to use efficient higher-order time discretiza-tion methods such that the Runge-Kutta scheme.

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper.

Acknowledgments

The author would like to thank Professor Jari Toivanen andProfessor Tuomo Rossi for useful discussions. The researchwas supported by theAcademyof Finland,Grants 252549 and259925.

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