Bo Wang,Li-Lian Wang2,?College of Mathematics and Computer Science,Hunan Normal University,Changsha,Hunan 4008,China

2Division of Mathematical Sciences,School of Physical and Mathematical Sciences,Nanyang Technological University,637371,Singapore

Received 4 February 2014;Accepted 28 February 2014

OnL2-Stability Analysis of Time-Domain Acoustic Scattering Problems with Exact Nonreflecting Boundary Conditions

Bo Wang1,Li-Lian Wang2,?1College of Mathematics and Computer Science,Hunan Normal University,Changsha,Hunan 410081,China

2Division of Mathematical Sciences,School of Physical and Mathematical Sciences,Nanyang Technological University,637371,Singapore

Received 4 February 2014;Accepted 28 February 2014

.This paper is devoted to stability analysis of the acoustic wave equation exterior to a bounded scatterer,where the unbounded computational domain is truncated by the exact time-domain circular/spherical nonreflecting boundary condition(NRBC).Different from the usual energy method,we adopt an argument that leads toL2-a prioriestimates with minimum regularity requirement for the initial data and source term.This needs some delicate analysis of the involved NRBC.These results play an essential role in the error analysis of the interior solvers(e.g.,finite-element/spectral-element/spectral methods)for the reduced scattering problems.We also apply the technique to analyze a time-domain waveguide problem.

AMS subject classifications:65R10,65N35,65E05,65M70

Wave equation,time-domain scattering problems,exact nonreflecting boundary conditions,stability analysis,a priori estimates.

1 Introduction

In this paper,we consider the time-domain acoustic scattering problem:

whereDis a bounded obstacle(scatterer)with Lipschitz boundary ΓD,c＞0 is the wave speed,andn=x/|x|.Assume that the dataF,U0andU1are compactly supported in a 2D disk or a 3D ballBof radiusb,which contains the obstacleD.

The acoustic wave propagates in the free space exterior toD,so the first important issue is to reduce the unbounded domain to a bounded domain.One efficient way is to set up an artificial boundary and impose a transparent/non-reflecting boundary condition(NRBC)thereon(see e.g.,[8]).It is advantageous to use the exact NRBC,as it can be placed as close as possible to the scatterer,and the reduced problem,so as the discretized problem,can be best mimic to the continuous problem.Though such a NRBC is global in time and space in nature,fast and accurate numerical and/or semi-numerical means were developed for its evaluation and/or seamless integration with some solver in the reduced domain(see e.g.,[3,14,15]).

This paper is largely concerned with the analysis of the reduced scattering problem by the exact circular/spherical NRBC.We remark that in[5,15],the usual energy method(i.e.,testing the equation with?tU)was used to obtainH1-type estimates under strong regularity assumptions for the initial data and source term.Moreover,this approach did not lead to optimalL2-estimates.In this paper,we resort to an argument in[4,7],which,together with a delicate analysis of the involved NRBC,leads toL∞(L2)-andL2(L2)-a prioriestimates for the reduced problem with a minimum regularity requirement for the initial data and source term.With this at our disposal,we can also analyze a waveguide problem considered in[18].

The paper is organized as follows.We present the reduced problem and carry out thea prioriestimates in the forthcoming section.In the last section,we apply the argument to analyze a waveguide problem.

2 L∞(L2)-and L2(L2)-a priori estimates

2.1 The reduced problem

We first reduce the scattering problem(1.1)-(1.3)to a bounded domain via the exact circular/spherical NRBC(see e.g.,[3,8,15]),leading to

where the time-domain DtN boundary condition at the artificial boundary Γb:=?B,is given,in polar/spherical coordinates,by

Here,the kernel functions in the convolution are

whereKνis the modified Bessel function of the second kind of orderν(see e.g.,[1,17]),and L-1[h(s)]is the inverse Laplace transform of a Laplace transformable functionH(t):

Another useful alternative expression(cf.[15])of Td(U),where the temporal convolution is expressed in terms of expansion coefficients of?tU|r=b,is given as follows:

where ford=2,3,

It is clearly that

andω′ν(t)=cσν(t).SinceK-n(z)=Kn(z)(see[1,Formula(9.6.6)]),it suffices to considerωnandσnwithn≥0,ford=2.

Remark 2.1.It is seen from(2.4)that the NRBC is global in time and space,due to the involvement of the convolution and Fourier/spherical harmonic expansions.We refer to[3,14,15]for fast and accurate methods for dealing with the inverse Laplace transform and temporal convolution,and also refer to[11,13]for techniques to overcome globleness of the exact boundary conditions in the context of time-harmonic scattering problems.

2.2 A priori estimates

We now intend to derivea prioriestimates for the solution of the reduced problem(2.1)-(2.3).For this purpose,we first introduce some notation.Given a generic weight functionw,letHrw(Ω)be the usual weighted Sobolev space of complex-valued functions as in Admas[2]with the norm k·kHrw(Ω).As usual,L2w(Ω)=H0w(Ω)with inner productdenoted by(·,·)L2w(Ω).We drop the weight function,wheneverw≡1.To characterize the regularity in time,we also use e.g.,the spaceL∞(0,T;L2(Ω)),as defined in[2].

We formulate the equation(2.1)-(2.3)in a weak form(in space).For anyt＞0,it is to findU(·,t)∈V:={v∈H1(Ω):v|ΓD=0}such that

Remark 2.2.It is important to point out that using the standard energy argument(i.e.,takingV=?tUin(2.9),see[5,15]),we are able to derive the a priori estimates in the energy norm,that is,k?tUkL∞(0,T;L2(Ω))+ck?UkL∞(0,T;L2(Ω)).However,this requires strong regularity of the initial and boundary data,and does not lead to optimalL2-estimates.

Hereafter,we take a different route that will lead toL∞(L2)-a prioriestimates for the reduced problem(2.1)-(2.3),with a minimum requirement for the regularity of the inputs.We essentially employ an argument due to Dupont[7](also see Baker[4]),but significant care is needed to analyze the exact NRBCs.For this purpose,we first make necessary preparations.

Recallthe Plancherelor Parsevalidentity forthe Laplace transform(see e.g.,[6,(2.46)]).

Lemma 2.1.Let s=s1+is2with s1,s2∈R.If f,g are Laplace transformable,then

where γ is the absissa of convergence for both f and g,andˉg is the complex conjugate of g.

For notational convenience,we introduce the modified spherical Bessel function(cf.[17]):

Then,by(2.7),

We shall use the following property(see[5,15]).

Lemma 2.2.Let s=s1+is2with s1,s2∈R.Then we have

where Zn(z)=Kn(z)or kn(z).

The following lemma is indispensable for the forthcoming analysis.

Lemma 2.3.For any v∈L2(0,T)with v(0)=0,we have

for ν=n,n+1/2,where σνand ωνare the convolution kernel functions.

Proof.Using the property of the Laplace transform:

we have

where in the last step,we used the property of the Laplace transform and the definition:

Lettings1→0,we obtain

Moreover,using integrate by parts yields

which implies

Therefore,(2.14)follows from(2.18).

We now turn to the derivation of(2.15).By(2.8),

A direct calculation using integration by parts and the conditionv(0)=0,leads to

Therefore,by(2.14)and(2.19),

This implies(2.15).

With the above preparations,we are ready to derive theL∞(L2)-a prioriestimates by using an argument due to[4,7].

where C is a positive constant independent of T,c and any functions.

Proof.Let 0＜ξ≤T,and define

It is clear that

Moreover,for anyφ(x,t)∈L2((0,ξ)×Ω),we have

We show this identity below.Indeed,using integration by parts and(2.23),we have

Next,taking the test functionV=ψin(2.9),leads to

By(2.23),

Thus,integrating(2.25)fromt=0 toξand taking the real parts,yields

We derive from(2.22)and the Cauchy-Schwartz inequality that

Similarly,by(2.24),we have that for 0≤t≤ξ≤T,

For the NRBC term,we consider the 3D case(2D case is similar).Using Lemma 2.3,we obtain

Now,substituting(2.27)-(2.28)into(2.26),we have that for anyξ∈[0,T],

TakingL∞-norm with respect toξon both sides of(2.29),yields

Therefore,the estimate(2.20)follows directly from the Cauchy-Schwartz inequality.

Integrating(2.29)with respect toξover(0,T)and using the Cauchy-Schwartz inequality,leads to

Using the Cauchy-Schwartz inequality again,we derive theL2-bound(2.21).

2.3 Regular scatterers

The previous analysis applies to a general bounded scatterer with Lipschitz boundary.Accordingly,the results pave the way for analyzing finite-element/spectral-element approximations to the reduced problem.However,if the scatterer is a disk/ball,it is ideal to formulate the problem in the polar/spherical coordinates.Moreover,the NRBC turns out to be local in the space of Fourier/spherical harmonic coefficients.This allows us to further reduce the problem of interest to a sequence of decoupled one-dimensional problems(see(2.32)below).We refer to[15]for the fast spectral-Galerkin solver under this notion and[11]for the time-harmonic case coupled with an efficient technique for dealing with irregular scatterers.The previous results do not imply the estimates below,but the argument can be applied.

Consider the reduced problem(2.1)-(2.3)with a regular scatterer:

where Td(U)is the time-domain DtN map as before andb0＞0.We expand the solution and given data in Fourier/spherical harmonic series,e.g.,

Then the problem(2.30),after a polar(in 2-D)and spherical(in 3-D)transform,reduces to a sequence ofone-dimensionalproblems(forbrevity,we useuto denote the Fourier/spherical harmonic expansion coefficients ofU,and likewise,we useu0,u1andfto denote the expansion coefficients ofU0,U1andF,respectively):

whereβn=n2,n(n+1)andν=n,n+1/2 ford=2,3,respectively.

Hereafter,letI=(b0,b)andω=rd-1.We introduce the weighted spaceL2ω(I)(of complex-valued functions),and denote the(weighted)norm by k·kωand the inner product by(·,·)ω.The weak form for(2.32)is to find thatu(·,t)∈V:={φ∈H1ω(I):φ(b0)=0},such that for allt＞0 andw∈V,

Like Theorem 2.1,we derive thea priorestimates for(2.33).

Theorem 2.2.Let u be the solution of(2.33).If u0,u1∈L2ω(I)and f∈L1(0,T;L2ω(I)),then for all T＞0,and each mode n,

where C is a positive constant independent of T,c and any functions.

Proof.Like the proof of Theorem 2.1,takingw=Rξtu(r,τ)dτwith 0＜ξ≤Tin(2.33),and integrating the resulted equation from 0 toξ,we obtain that

where we used the property(2.24)to handle the integrals of the boundary terms.We now show the summation of three boundary terms is non-negative.By(2.14)and(2.19),

Using the Cauchy-Schwartz inequality leads to

and

Therefore,we obtain

This leads to the estimate(2.34).Moreover,integrating(2.37)from 0 toT,we obtain theL2-estimate(2.35)like Theorem 2.1.

Remark 2.3.The estimates in Theorem 2.2 are valid for each moden,which cannot be derived from Theorem 2.1.However,the converse statement is true.Indeed,using the Parseval’s identity of the Fourier/spherical harmonic series,we can claim Theorem 2.1 in the case of regular scatterers from Theorem 2.2 straightforwardly.

Remark 2.4.The stability results are essential for the analysis of numerical solvers for the reduced problem.We illustrate this in the forthcoming section.

3 Analysis of a waveguide problem

In this section,we apply the previous argument to analyze a waveguide problem considered in[18],which involves the exact planar non-reflecting boundary condition(cf.[8]).More precisely,let

and consider

wherec＞0 is the wave speed.Here,we assume that the given dataF,U0andU1are 2π-periodic iny,and are compactly supported(with respect tox),in an interval(0,a)for somea＞0.

We adopt the exact planar NRBC at the artificial boundaryx=a.This leads to the reduced problem in Ω:=(0,a)×[0,2π):

Note that the time-domain DtN map is given by

where the convolution kernelρmand Fourier coefficients{bUm}are given by(cf.[8]):

withJ1(·)being the Bessel function of the first kind of order 1(cf.[17]).Alternatively,we have(cf.[9,Table 19.1,Page 90]):

Sinceρ0=0 andρ-m=ρm,it suffices to considerm＞0 below.

3.1 A priori estimates

To derive theL2-a prioriestimates for the reduced problem(3.4)-(3.6),we first recall the following properties(see[18]).

Lemma 3.1.Let s=s1+is2with s1,s2∈R.Then for any integer m and s1＞0,

Like Lemma 2.3,the following result is very important for the analysis.

Lemma 3.2.For any v∈L2(0,T),we have

Lemma 2.1 that

Taking the real part of the above equation,we get

By Lemma 3.1,

By lettings1→0+in(3.12),a combination of(3.12)-(3.14)leads to(3.11).

DefineX:=?U∈H1(Ω):U|x=0=0?,and denote by 〈·,·〉L2(Γa)and k·kL2(Γa)the inner product and norm ofL2(Γa),respectively,where Γa={(a,y):0＜y＜2π}.The weak form of(3.4)-(3.6)is to findU∈Xfor allt＞0,such that

Theorem 3.1.Let U(∈X for t＞0)be the solution of(3.4)-(3.6).If U0∈L2(Ω),U1∈L2(Ω),and F∈L1(0,T;L2(Ω))for any T＞0,then we have U∈L∞(0,T;L2(Ω)),and there holds

where C is a positive constant independent of any functions and c.

Proof.Taking

in(3.15)and following the same lines as in the proof of Theorem 2.1,we have

According to the definition ofψ(x,y,t)and the Cauchy-Schwartz inequality,we have

We next show that for anyt＞0,

It follows from(3.7),Theorem 3.2 and the orthogonality of{eimy}that

Thus,the estimate(3.16)follows from(3.18)-(3.21)and the Cauchy-Schwartz inequality.According to(3.18)and(3.21),we have

Integrating this inequality from 0 toTw.r.tξand then using the Cauchy-Schwartz inequality we derive theL2-estimate(3.17).

3.2 Fourier-Legendre spectral-Galerkin approximation

We expand the solutionUand given dataU0,U1,Fin Fourier series:

Then(3.24)becomes

where the constant?c=2c/a.Then the weak form of(3.25)is to findv(·,t)∈V:={v∈H1(I):

v(-1)=0},such that for allw∈Vandt＞0

where(·,·)is the inner product ofL2(I).

We can derive the followinga prioriestimates for each modem.

Theorem 3.2.Let v(∈V for t＞0)be the solution of(3.26)-(3.27).If v0∈L2(I),v1∈L2(I),and f∈L1(0,T;L2(I))for any T＞0,then we have v∈L∞(0,T;L2(I)),and there holds

where C is a positive constant independent of any functions and c.

Therefore,we derive thea prioriestimates by using the Cauchy-Schwartz inequality.

LetVN:=?ψ∈PN:ψ(-1)=0?,wherePNis the set of all polynomials of degree at mostN.The semi-discretization Legendre spectral-Galerkin approximation of(3.25)is to findvN(?x,t)∈VNfor allt＞0 such that

where INis the interpolation operator on(N+1)Legendre-Gauss-Lobatto points,andv0,N,v1,N∈PNare suitable approximations of the initial values.

In what follows,we perform the error estimates for the scheme(3.30).For this purpose,we make some preparations.

Lemma 3.3.Let ρm(t)be the kernel function defined in(3.8).Then we have

for all integer m.

Proof.Recall the properties of the Bessel functions(see[1]):

By(3.8)and the above properties,we obtain that form≥1,

Sinceρ0=0,andρ-m=ρm,the upper bound is valid for allmandt＞0.

Consider the orthogonal projection:0π1N:0H1(I):={u∈H1(I):u(-1)=0}→0PN:=PN∩0H1(I),such that

Recall the Legendre-approximation results(see e.g.,[12]):for anyu∈0H1(I)∩Hs(I)with 1≤s≤N+1,

whereDis a positive constant independent ofN,sandu.

We also recall the approximation result on Legendre-Gauss-Lobbatto interpolation:for anyu∈Hs(I)with 1≤s≤N+1(see e.g.,[12]):

Moreover,we shall use the trace inverse inequality(see e.g.,[16]):for anyφ∈PN,

With the above preparations,we are now ready to carry out the error analysis.It is clear that by(3.26)and(3.30),

To this end,let

Then we derive from(3.30)and(3.34)that for anywN∈VN,

and

Thus it remains to deal with the other terms at the right-hand side of(3.40).We derive from the integration by parts and Cauchy-Schwartz inequality that

Using the integration by parts,we then infer from Lemma 3.3 and the inverse inequality(3.38)that

Finally,using the factvN-v=eN-?eN,the triangle inequality and(3.41)-(3.42),we obtain from the approximation results(3.35)-(3.37)the following error bound for each modem.

Theorem 3.3.Let v and vNbe respectively the solution of(3.26)and(3.30).If v0,v1∈0H1(I)∩Hs(I),g∈L1(0,T;Hs(I))v∈L∞(0,T;0H1(I)∩Hs(I))and?tv∈L1(0,T;Hs(I))with1≤s≤N+1,then

where D is a positive constant independent of N,T and any function.A similar error bound holds forkv-vNkL2(0,T;L2(I)).

Remark 3.1.The presence of the NRBC brings about significantly subtle issues for the analysis compared with the standard setting in[4,7].Moreover,the error bounds appear suboptimal.

Remark 3.2.We can further assemble the Fourier approximation and derive the error estimates for the full Fourier-Legendre spectral approximation with the aid of Theorem 3.3.This follows a standard procedure(cf.[12]),so we omit the details.

Acknowledgments

The first author is supported by NSFC Grant(11341002)and the second authors is partially supported by Singapore MOE AcRF Tier 1 Grant(RG 15/12),MOE AcRF Tier 2 Grant(MOE 2013-T2-1-095,ARC 44/13)and Singapore A?STAR-SERC-PSF Grant(122-PSF-007).

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?Corresponding author.Email addresses:bowanghn@gmail.com(B.Wang),LiLian@ntu.edu.sg(L.L.Wang)