COMPLETE RADIATION BOUNDARY CONDITIONS: MINIMIZING …faculty.smu.edu › thagstrom › HWcomplete.pdf · approximations. Numerical experiments conﬁrm that the eﬃciencies predicted

COMPLETE RADIATION BOUNDARY CONDITIONS: MINIMIZINGTHE LONG TIME ERROR GROWTH OF LOCAL METHODS

THOMAS HAGSTROM∗ AND TIMOTHY WARBURTON†

Abstract.We construct and analyze new local radiation boundary condition sequences for first order,

isotropic, hyperbolic systems. The new conditions are based on representations of solutions of thescalar wave equation in terms of modes which both propagate and decay. Employing radiationboundary conditions which are exact on discretizations of the complete wave expansions essentiallyeliminates the long time nonuniformities encountered when using the standard local methods (PMLor Higdon sequences). Precisely we prove that the cost in terms of degrees-of-freedom per boundarypoint scales with ln 1

ε· ln cT

δwhere ε is the error tolerance, T is the simulation time, and δ is

the separation between the source-containing region and the artificial boundary. Choosing δ ∼ λwhere λ is the wavelength leads to the same estimate which has been obtained for optimal nonlocalapproximations. Numerical experiments confirm that the efficiencies predicted by the theory areattained in practice.

Key words. Translational wave representations, radiation boundary conditions.

AMS subject classifications. 65M12, 35L50, 35C15

1. Introduction. A central issue in the development of efficient computationalmethods for simulating time-domain wave propagation is the imposition of accurate,near-field radiation boundary conditions. In recent years a number of new techniqueshave been proposed which are capable of providing arbitrary accuracy [15, 16, 20].Foremost among these are fast, low-memory algorithms for evaluating integral oper-ators appearing in exact formulations [1, 2, 28], the perfectly matched layer (PML)[6, 3], and arbitrary-order local radiation boundary condition sequences [22, 13, 17, 14].The integral equation formulations have the advantage of excellent long-time accu-racy, but require special boundary shapes and, in 3+1 dimensions, spherical harmonictransforms. The local methods, on the other hand, can be used on polygonal artificialboundaries fit tightly arond scatterers or other inhomogeneities, but become costly iflong-time accuracy is required [8, 9, 10].

The goal of this paper is to present a method which combines the excellent long-time accuracy of the nonlocal conditions with the geometric flexibility of the localmethods. Following the constructions in [22, 13, 17, 14], we focus on auxiliary variableformulations applicable on polygonal artificial boundaries. Precisely we prove thatthe number of auxiliary variables, P , required to obtain an accuracy ε up to time Tsatisfies:

P = O

(ln

1ε· ln cT

δ

)(1.1)

where δ > 0 is the distance from the artificial boundary to the domain containingsources, scatterers, or other inhomogeneities. For efficient discretization it is natural

∗Department of Mathematics, Southern Methodist University, PO Box 750156, Dallas, TX 75275-0156, [email protected]. Supported in part by NSF Grant DMS-06010067, ARO Grant DAAD19-03-1-0146, and AFOSR Contract FA9550-05-1-0473. Any conclusions or recommendations expressedin this paper are those of the author and do not necessarily reflect the views of NSF, ARO or AFOSR.

†Department of Computational and Applied Mathematics, Rice University, 6100 Main St. - MS134, Houston, TX 77005, [email protected]. Supported in part by AFOSR Contract FA9550-05-1-0473.

1

2 T. HAGSTROM AND T. WARBURTON

to choose δ ∼ λ where λ is the smallest wavelength of interest. Then the asymp-totic complexity matches that of the optimized spatially nonlocal approximationsconstructed in [2]. Thus, in our view, the proposed method provides a satisfactorysolution to the time-domain domain truncation problem for isotropic systems, andit has the potential for generalization to more complex models. In contrast, the ex-act reflection coefficients calculated by Diaz and Joly [9] show that if the standardparametrizations of local radiation boundary conditions are used the number of aux-iliary functions satisfies:

P = O

(ln

1ε· cTδ

). (1.2)

Our numerical experiments below and in [17, 21, 18] confirm the poor long time errorbehavior when conditions constructed using Pade or more general Higdon parameters[25] are used. The simple explanation is that the standard parametrizations are onlyaccurate for propagating modes. The new construction works by explicitly approxi-mating an exact representation of the solution, the complete wave representation.

We also note that, although our focus here is on auxiliary variable formulationsof local boundary condition sequences, it is now known that such sequences can beviewed as specific discretizations of perfectly matched layers [4, 14]. Thus our con-struction also suggests a certain combination of real and complex grid stretching tooptimize PML discretizations. This will be pursued elsewhere. We note that a uni-form discretization of a PML also suffers from long-time error growth. Diaz and Joly[10] have calculated the exact reflection coefficient which, for a layer of width L andaverage absorption coefficient σ, implies:

L2σ = O

(ln

1ε· c2T

). (1.3)

The remainder of the paper is organized as follows. In Section 2 we give anelementary derivation of the complete wave representation for the scalar wave equationand show how it can be easily extended to systems. In Section 3 we discuss how thecontinuous representation can be approximated by a finite one and derive an easily-implemented sequence of radiation boundary conditions (for first-order systems) whichare exact on the finite representation. In Section 4 we prove the well-posedness of theresulting initial-boundary value problems and establish an error estimate for generalchoices of the parameters. In Section 5 we display a specific choice of parameters whichwe prove leads to the favorable complexity estimates claimed above. We also carryout a numerical construction to optimize the error estimates for particular values ofthe dimensionless parameter cT/δ. In Section 6 we separately discuss the structure ofthe boundary system and we explain our construction of the corner closures in Section7. Finally, in Section 8 we demonstrate the implementation of the method in a finitedifference code, including the implementation of corner compatibility conditions, andpresent some simple numerical experiments demonstrating the proposed method’sexcellent accuracy. We note that additional experiments for Maxwell’s equations [33]will appear elsewhere. For the scalar wave equation it is more convenient to implementconditions based on an alternative representation involving the sum of propagatingand evanescent modes. Boundary conditions based on such alternate representationsare tested in [21, 18] and optimal parametrizations are developed in [11].

COMPLETE RADIATION CONDITIONS 3

2. Complete wave representations. Suppose u(x, y, t) satisfies the scalarwave equation in the half-space x > −δ for some δ > 0. That is:

∂2u

∂t2= c2∇2u, x > −δ, y ∈ Rd−1, t > 0, (2.1)

with u = 0 at t = 0. Here we suppose the field is produced by sources, scatterers,and other inhomogeneities located in the half-space x < −δ. These effects are allaccounted for by Dirichlet data:

u(−δ, y, t) = g(y, t). (2.2)

We then easily derive a formula for the evolution in x of the transverse-Fourier-Laplacetransforms of the solution of (2.1)-(2.2):

u(x, k, s) = u(−δ, k, s)e−(s2+|k|2)1/2(x+δ). (2.3)

Here k ∈ Rd−1 are the dual Fourier variables to the transverse spatial coordinates y,|k| is the Euclidean norm, s is the dual Laplace variable to time, and s = s

c . Thebranch of the square root is chosen to have positive real part for <s > 0.

Now invert the transform, integrating with respect to k over the real hyperplaneRd−1 and with respect to s over the contour, s = iω + 1

T with ω real. Here T > 0 isfixed and has units of time. For numerical methods T will measure the time over whichapproximations based on the exact representation are valid. We begin by representingthe square root. Setting

k =k

cT, ω =

ω

T, (2.4)

we have((1 + iω)2 + k2

)1/2

= a+ ib, (2.5)

where by squaring we deduce

b =ω

a, 1 + k2 − ω2 = a2 − ω2

a2. (2.6)

As the right-hand side of the second equation above is an increasing function of a2 andis not greater than the left-hand side when a2 = 1 we conclude that for the branch ofthe square root chosen a ≥ 1. Thus we have for some φ ∈ [0, π2 ):

a =1

cosφ, b = ω cosφ. (2.7)

Reintroducing s we finally have:

(s2 + |k|2)1/2 = s cosφ+1cT

sin2 φ

cosφ, (2.8)

where

s =1T

+ iω, |k| = 1cT

√tan2 φ+ sin2 φ

ω2

T 2. (2.9)


Going over to polar coordinates, ρ, θ ∈ Sd−2, in the dual spatial variables and replacingρ = |k| by ρ(φ, ω) we obtain:

u(x, y, t) = (2π)−(d+1)

2

∫ π2

0

∫ ∞

−∞

∫

Sd−2eψuρd−2 dρ

dφdA(θ)dωdφ, (2.10)

where

ψ =(iω +

1T

)(t− cosφ

c(x+ δ)

)+ iρ(θ · y)− 1

cT· sin2 φ

cosφ(x+ δ), (2.11)

u = u

(−δ, ρθ, 1

T+ iω

). (2.12)

Setting

Φ(t, y, φ) = (2π)−(d+1)

2

∫ ∞

−∞

∫

Sd−2e(iω+ 1

T )t+iρ(θ·y)uρd−2 dρ

dφdA(θ)dω (2.13)

we finally have our complete wave representation of the solution, valid for x > −δ:

u(x, y, t) =∫ π

2

0

Φ(t− cosφ

c(x+ δ), y, φ

)e−

1cT · sin

2 φcos φ (x+δ)dφ. (2.14)

Note that an alternative representation can be derived by inverting the Laplacetransform along the imaginary axis. Then we express the solution as a superpositionof propagating waves at all possible angles and evanescent waves at all possible decayrates. Such an expression has been analyzed in the time domain by Heyman [24] andplays an important role in the plane wave fast time-domain algorithm [12]. We havealso used it to derive radiation boundary conditions [18]. The expression given here issomewhat more efficient for first order systems, but the alternative expression givenin [18] has some advantages for second order formulations.

2.1. Extension to systems and other generalizations. To extend (2.14) toisotropic systems we simply note that the representation above holds for any field com-ponent which satisfies the scalar wave equation. For example, suppose the equationsof acoustics hold in x > −δ:

∂p

∂t+ c∇ · v = 0, (2.15)

∂v

∂t+ c∇p = 0. (2.16)

Then by our assumptions on the initial data ∇×v = 0 so we may introduce a velocitypotential, v = ∇q, cp = −∂q

∂t . Then q satsfies the scalar wave equation and thusp and v admit the representation (2.14). Similar considerations hold for Maxwell’sequations which are discussed in more detail in [33].

The representation may also apply to systems with multiple wave speeds. Con-sider, for example, Navier’s equations of linear elasticity:

ρ∂2w

∂t2= (λ+ µ)∇∇ · w + µ∇2w. (2.17)


Introducing the Helmholtz decomposition of w:

w = ∇q +∇× η, ∇ · η = 0, (2.18)

we derive scalar wave equations for q and the components of η:

∂2q

∂t2= c21∇2q,

∂2η

∂t2= c22∇2η, (2.19)

where

c21 =λ+ 2µρ

, c22 =µ

ρ. (2.20)

Thus q and η satsify (2.14), albeit with different wavespeeds, and hence w satsfies acombined representation:

wi(x, y, t) =∫ π

2

0

Φ1,i

(t− cosφ

c1(x+ δ), y, φ

)e−

1c1T · sin

2 φcos φ (x+δ)dφ

+∫ π

2

0

Φ2,i

(t− cosφ

c2(x+ δ), y, φ

)e−

1c2T · sin

2 φcos φ (x+δ)dφ. (2.21)

Lastly we consider an anisotropic example, the acoustics system in a subsonicmean flow:

Dp

Dt+ c∇ · v = 0, (2.22)

Dv

Dt+ c∇p = 0, (2.23)

where

D

Dt≡ ∂

∂t+ V · ∇, |V | < c. (2.24)

Now we introduce the Helmholtz decomposition of v:

w = ∇q +∇× η, cp = −DqDt

, (2.25)

leading to the equations:

D2q

Dt2= c2∇2q, (2.26)

Dη

Dt= 0. (2.27)

The solution of the transport equation (2.27) in the half-space is dependent on thesign of Vx. If Vx ≤ 0 then η = 0 but if Vx > 0 then:

η = Σ(Vxt− (x+ δ), Vyt− y). (2.28)


As for the convective wave equation (2.26) the Fourier-Laplace transform of the solu-tion of the Dirichlet problem analogous to (2.3) is:

q(x, k, s) = q(−δ, k, s)esVx−c(s2+(c2−V 2

x )|k|2)1/2

c2−V 2x

(x+δ), (2.29)

with

s = s+ iVy · k. (2.30)

Following the same decomposition of the inversion integrals as in the case of thestandard wave equation yields the slightly more complex formula:

q(x, y, t) =∫ π

2

0

Φ(t− c cosφ− Vx

c2 − V 2x

(x+ δ), y − Vyt, φ

)

×e−c

(c2−V 2x )T

· sin2 φcos φ (x+δ)

dφ. (2.31)

The formula clearly displays the possibility of incoming phase velocities for outgoingwaves at outflow. We have not as yet considered the problem of constructing stableboundary conditions based on this representation, but we plan to do so in the future.This issue is closely related to the problem of constructing stable perfectly matchedlayers for anisotropic systems, which is still not completely understood [5, 3].

3. Approximate local boundary condition sequences. We now use (2.14)to derive approximate local boundary condition sequences. We focus on first ordersystems; the treatment of second order formulations is discussed in [21].

By way of motivation, consider an approximation to (2.14) derived by replacingthe φ integral by a quadrature rule with nodes φj and weights hj :

u(x, y, t) ≈P∑

j=0

hjΦ(t− cosφj

c(x+ δ), y, φj

)e− 1

cT ·sin2 φjcos φj

(x+δ). (3.1)

Generalizing the construction of [22], we introduce a second set of angles φj and defineauxiliary functions uj(x, y, t) by setting u0 = u and recursively solving in x > −δ:

cos φjc

∂uj+1

∂t− ∂uj+1

∂x+

1cT

sin2 φjcos φj

uj+1 =cosφjc

∂uj∂t

+∂uj∂x

+1cT

sin2 φjcosφj

uj , (3.2)

subject to uj+1(x, y, 0) = 0. It is straightforward to see that the individual termsin (3.1) are annihilated by one of the differential operators on the right-hand side of(3.2). Thus if we replaced u0 by the approximate representation we would conclude:

uP+1 = 0. (3.3)

We now simply impose (3.3) on incoming normal characteristic variables. Precisely,consider the first order hyperbolic system:

∂w

∂t+A

∂w

∂x+d−1∑

j=1

Bj∂w

∂yj= 0, (3.4)


where we have put A in diagonal form:

A =

D+ 0 00 −D− 00 0 0

, (3.5)

and the diagonal matrices D± are positive. Block w according to the blocks of A:

w =

w+

w−wc

. (3.6)

As in [22], a simple induction argument shows that vector functions wj defined via(3.2) (with u replaced by w) will also satisfy (3.5). Multiplying the recursion relationsby A and eliminating x derivatives yields the following system which can be consideredalong the boundary x = 0:

(I +

cos φjc

A

)∂wj+1

∂t+d−1∑

k=1

Bk∂wj+1

∂yk+

1cT

sin2 φjcos φj

Awj+1 =

(−I +

cosφjc

A

)∂wj∂t

−d−1∑

k=1

Bk∂wj∂yk

+1cT

sin2 φjcosφj

Awj . (3.7)

Imposing the termination condition:

wP+1,− = 0, (3.8)

we see that (3.7) implicitly defines a relationship between the outgoing characteristicvariables w0,+ and the incoming characteristic variables, w0,−. Precisely, if we assumethe functions wj are given along the boundary x = 0 for some time t and in additionthat ∂w0,+

∂t is known, then the remaining time derivatives may be directly computed.For the auxiliary variables associated with the outgoing characteristics we may solvefor increasing j:

(I +

cos φjc

D+

)∂wj+1,+

∂t=

(−I +

cosφjc

D+

)∂wj,+∂t

−(d−1∑

k=1

Bk

(∂wj+1

∂yk

∂wj∂yk

))

+

(3.9)

+1cTD+

(sin2 φjcosφj

wj,+ − sin2 φjcos φj

wj+1,+

).

For the auxiliary variables associated with the incoming variables, on the other hand,we solve for decreasing j:

(I +

cosφjc

D−

)∂wj,−∂t

=(−I +

cos φjc

D−

)∂wj+1,−

∂t

−(d−1∑

k=1

Bk

(∂wj+1

∂yk

∂wj∂yk

))

−(3.10)

− 1cTD−

(sin2 φjcosφj

wj,− − sin2 φjcos φj

wj+1,−

).


Lastly the time derivatives of the wj,c may be computed using the hyperbolic systemitself:

∂wj,c∂t

= −(d−1∑

k=1

Bk∂wj∂yk

)

c

. (3.11)

We note that the implementation of this method within a standard time-steppingprocedure is straightforward. We will discuss below how this has been done for differ-ence methods, while in [33] discontinuous Galerkin methods are used. As presentedso far, the formulation may seem somewhat ad hoc. However, it will be shown in thenext section that it is equivalent to a rational interpolant of the symbol of nonlocaloperators arising in exact boundary condition representations, with the interpolationnodes determined by both φj and φj .

The formal connection between (3.2) and a discretized PML is also easily estab-lished in the special case φj = φj . Solving for the x-derivatives and reintroducing thetemporal Laplace transform we obtain:

12

(∂uj+1

∂x+∂uj∂x

)=uj+1 − uj

Hj

, (3.12)

where

Hj =2

cosφj

c s+ 1cT

sin2 φj

cosφj

. (3.13)

We clearly recognize a box scheme discretization with a frequency-dependent meshspacing, Hj . (See also [4].)

4. Analysis. To derive error estimates we must estimate the complex reflectioncoefficient. For simplicity we will carry out the analysis for the acoustic system withd = 3, but similar estimates for Maxwell’s equations and for other dimensions could bederived in the same way. (See [18] for a direct treatment of the scalar wave equation.)

We consider the system in diagonalized form and slightly change notation:

∂w+

∂t+ c

∂w+

∂x+ c∇tan · vtan = 0, (4.1)

∂w−∂t

− c∂w−∂x

+ c∇tan · vtan = 0, (4.2)

∂vtan∂t

+c

2∇tan(w+ + w−) = 0, (4.3)

where

w+ = p+ vx, w− = p− vx. (4.4)

Specialized to this sytem equations (3.9)-(3.11) become:

1 + cos φjc

∂w+,j+1

∂t=−1 + cosφj

c

∂w+,j

∂t−∇tan · (vtan,j+1 + vtan,j)

+1cT

(sin2 φjcosφj

w+,j − sin2 φjcos φj

w+,j+1

), (4.5)


1 + cosφjc

∂w−,j∂t

=−1 + cos φj

c

∂w−,j+1

∂t−∇tan · (vtan,j+1 + vtan,j)

+1cT

(sin2 φjcos φj

w−,j+1 − sin2 φjcosφj

w−,j

), (4.6)

∂vtan,j∂t

+c

2∇tan(w+,j + w−,j) = 0. (4.7)

Assume that the solution is produced by sources and Cauchy data in the regionx < −δ and that the artificial boundary is located at x = 0. The error then satisfiesthe homogeneous system with zero initial data driven only by reflections from theartificial boundary. These we estimate using the data at x = −δ, which is crucial ifpositive results are to be obtained. (Roughly, the derivation of finite time estimateswith separation between sources and the boundary excludes glancing modes.)

Performing a Fourier-Laplace transformation as before, the solution for x > −δcan be completely characterized by w+(−δ, k, s):

w+

w−vtan

= w+(−δ, k, s)e−γ(x+δ)

1s−γs+γ−iks+γ

, (4.8)

where we have introduced

γ =(s2 + |k|2)

12 . (4.9)

Similarly, the error is a reflected wave completely characterized by e−(0, k, s):

e+e−etan

= e−(0, k, s)eγx

s−γs+γ

1−iks+γ

. (4.10)

Now e−(0, k, s) can be directly calculated using the boundary recursion.Lemma 4.1. The reflection from the artificial boundary is given by:

e−(0, k, s) = R(k, s)w+(−δ, k, s), (4.11)

where

R(k, s) =

P∏

j=0

γ − s cosφj − 1

cTsin2 φj

cosφj

γ + s cosφj + 1cT

sin2 φj

cosφj

·

γ − s cos φj − 1

cTsin2 φj

cos φj

γ + s cos φj + 1cT

sin2 φj

cos φj

×(γ − s

γ + s

)e−γδ. (4.12)

Proof: We introduce the notation:

σj =1cT

sin2 φjcosφj

, σj =1cT

sin2 φjcos φj

, (4.13)


and use the transform of (4.7) to eliminate vtan,j . The transform of (4.5),(4.6) is thengiven by:

(s(cos φj + 1) + σj)w+,j+1 +|k|22s

(w+,j+1 + w−,j+1) =

(s(cosφj − 1) + σj)w+,j − |k|22s

(w+,j + w−,j), (4.14)

(s(cosφj + 1) + σj)w−,j +|k|22s

(w+,j + w−,j) =

(s(cos φj − 1) + σj)w−,j+1 − |k|22s

(w+,j+1 + w−,j+1). (4.15)

Assuming <s > 0 we have γ 6= 0 and may write:(w+,j

w−,j

)= aj

(1s−γs+γ

)+ bj

( s−γs+γ

1

). (4.16)

Comparing with (4.8), (4.10) we have:

a0 = w+(−δ, k, s)e−γδ, b0 = e−(0, k, s), (4.17)

so that

R = e−γδb0a0. (4.18)

Moreover the termination condition implies:

bP+1 =(γ − s

γ + s

)aP+1. (4.19)

Substituting (4.16) into (4.14)-(4.15), noting that |k|2 = γ2− s2, and introducing thenotation:

α±,j = γ ± (s cosφj + σj), α±,j = γ ± (s cos φj + σj), (4.20)

we find that:

(γ + s)(α+,jaj+1 + α−,jaj) + (γ − s)(α+,jbj + α−,jbj+1) = 0, (4.21)

(γ + s)(α+,jbj + α−,jbj+1) + (γ − s)(α+,jaj+1 + α−,jaj) = 0. (4.22)

We recognize a pair of independent homogeneous equations for the quantitiesα+,jaj+1 + α−,jaj and α+,jbj + α−,jbj+1 which must be zero. Hence

aP+1 = (−1)P+1

P∏

j=0

α−,jα+,j

a0, b0 = (−1)P+1

P∏

j=0

α−,jα+,j

bP+1. (4.23)

Applying (4.19) and (4.18) produces (4.12).¦


Using this explicit representation for the reflection coefficient we can directlyestablish the well-posedness of the system with the approximate boundary and writedown error estimates.

Theorem 4.2. The initial-boundary value problem for the acoustic system inthe half-space x < 0 with the complete radiation boundary conditions (3.7)-(3.8) isstrongly well-posed. Moreover there exists a universal constant C such that for x < 0:

‖e(x, ·, ·)‖L2(Rd−1×(0,T )) ≤ ρC‖w(−δ, ·, ·)‖L2(Rd−1×(0,T )), (4.24)

where

ρ = max0≤φ≤π

2

P∏

j=0

| cosφ− cosφj |(cosφ+ cosφj)

| cosφ− cos φj |(cosφ+ cos φj)

·

(1− cosφ1 + cosφ

)e−

δcT · 1

cos φ . (4.25)

Proof: To establish well-posedness we set s = η + iω with η > 0 and using (2.8)conclude from (4.12) that:

|R|2 =

P∏

j=0

(η

cosφ − η cosφj − 1cT

sin2 φj

cosφj

)2

+ ω2(cosφ− cosφj)2

(η

cosφ + η cosφj + 1cT

sin2 φj

cosφj

)2

+ ω2(cosφ+ cosφj)2

×

P∏

j=0

(η

cosφ − η cos φj − 1cT

sin2 φj

cos φj

)2

+ ω2(cosφ− cos φj)2

(η

cosφ + η cos φj + 1cT

sin2 φj

cos φj

)2

+ ω2(cosφ+ cos φj)2

(4.26)

×

(η

cosφ − η)2

+ ω2(cosφ− 1)2

(η

cosφ + η)2

+ ω2(cosφ+ 1)2

e−

2δηcos φ .

As 0 ≤ φ ≤ π2 we immediately conclude that |R| < 1.

To derive the error estimate we set η = 1cT . Then |R| simplifies to ρ. Recalling

Parseval’s relation for Fourier-Laplace transforms (e.g. [27, Ch. 7]) the result isimmediate.¦.

4.1. Equivalent rational approximant. From (4.25) we recognize that theapproximate boundary conditions are exact not only for terms in the complete waveexpansion with parameter φj but also with parameter φj . It is of interest to relate theapproximate conditions to some approximation to the symbol of nonlocal operatorsappearing in an exact boundary condition. We may take the latter to be:

w− =s− γ

s+ γw+ = − |k|2

(s+ γ)2w+. (4.27)

To compute the symbol of the approximate condition we simply repeat the calculationof the reflection coefficient noting that:

w− =s−γs+γ a0 + b0

a0 + s−γs+γ b0

w+

=(s− γ)a0 + (s+ γ)b0(s+ γ)a0 + (s− γ)b0

w+. (4.28)


We conclude that the approximation satisfies:

w− = −|k|2 P (−γ)− P (γ)(s+ γ)2P (−γ)− (s− γ)2P (γ)

w+, (4.29)

where

P (z) =P∏

j=0

(z − cosφj s− 1

cT

sin2 φjcosφj

)(z − cos φj s− 1

cT

sin2 φjcos φj

). (4.30)

Clearly we have written an even rational function of γ and thus a rational functionof s and |k|2. Moreover, the symbol of the exact condition is interpolated whereverP (γ) = 0. Rational approximation to the symbol of the exact condition is a standardapproach to deriving approximate radiation boundary conditions; see, for example,[23]. However, previous work was typically restricted to homogeneous rational approx-imants. Those constructed here are not homogeneous functions of s and |k|. Theydepend directly on the time scale T or, equivalently, the choice of contour where theexact symbol is interpolated.

5. Parameter selection. According to (4.25), the error estimate is optimizedby choosing angles, φj , to minimize ρ. Here we adapt Newman’s well-known con-struction of rational approximants to |x| [29] to derive an explicit set of angles whichachieve the tolerance with P = O(ln 1

ε · ln cTδ ). (See [30] for a further discussion of

optimal approximations to similar functions.)Choosing a tolerance, ε > 0, we note that both the exponential factor and the

parameter-dependent factor in (4.25) are bounded above by one. Thus we can satisfythe tolerance by making either less than ε. We begin by considering the regime wherecosφ is small. In particular we have:

e−δ

cT · 1cos φ < ε, (5.1)

whenever

cosφ <δ

cT

1ln 1

ε

≡ c0. (5.2)

Now restrict attention to the interval c0 ≤ cosφ ≤ 1 and consider the problem ofminimizing:

ρc ≡ maxc0≤c≤1

2P+1∏

j=0

|c− aj |c+ aj

·

(1− c

1 + c

), (5.3)

where now the parameters aj encompass both cosφj and cos φj . Following [29] set:

aj = gj+1, g = (c0)1

2P+2 . (5.4)

Suppose

gj+1 < c < gj . (5.5)


Then

ρc =

(j∏

k=0

(gk − c)(gk + c)

)

2P+2∏

k=j+1

(c− gk)(c+ gk)

≤(

j∏

k=0

(gk − gj+1)(gk + gj+1)

)

2P+2∏

k=j+1

(gj − gk)(gj + gk)

=

(j+1∏

k=1

(1− gk)(1 + gk)

)(2P+2−j∏

k=1

(1− gk)(1 + gk)

)(5.6)

≤2P+3∏

k=1

(1− gk)(1 + gk)

≤2P+3∏

k=1

exp (−2gk) = exp(−2g(1− g2P+3)/(1− g)

).

We thus must choose P sufficiently large that:

2g1− g2P+3

1− g> ln

1ε. (5.7)

Assuming for simplicity that ε¿ 1, cTδ À 1 it is sufficient to satisfy:

g > 1− 1ln 1

ε

(5.8)

which requires

P = O

(ln

1c0· ln 1

ε

). (5.9)

Using Theorem 4.2 we have thus proven:Theorem 5.1. Suppose ε < 1 and cT

δ > ln 1ε . Then there exists a universal

constant C such that if:

P > C lncT

δ· ln 1

ε, (5.10)

and the complete radiation boundary conditions (3.7)-(3.8) are imposed with parame-ters:

cosφj = c2j+12P+20 , cos φj = c

j+1P+10 , (5.11)

for j = 0, . . . , P and c0 defined by (5.2) the error satisfies:

‖e(x, ·, ·)‖L2(Rd−1×(0,T )) ≤ ε‖w(−δ, ·, ·)‖L2(Rd−1×(0,T )). (5.12)

Clearly this analysis, by considering only one of the factors in (4.25) at a time, doesnot produce optimal error estimates or optimal parameters. For practical purposes wewill compute parameters numerically by minimizing ρ for fixed values of δ

cT and P . We


employ the standard Remez algorithm (e.g.[30]) with initial approximations providedby the geometric distribution introduced above. We note that the computation isextremely rapid, so that in practice one could directly compute a minimal P -valueand the associated parameters for input values of ε and δ

cT . In Table 5.1 we list themaximum value of the reflection coefficient, ρ obtained for δ

cT = 10−2 − 10−6 and Pinreased in increments of 4. Note that we stopped increasing P once an accuracy ofε = 10−8 was attained. The computed values of the cosines may be accessed from thefirst author’s homepage: http://faculty.smu.edu/thagstrom/Optimal-cosines.

δcT P ρ

1.0e− 02 4 5.60e− 041.0e− 02 8 3.75e− 061.0e− 02 12 3.18e− 081.0e− 02 16 3.13e− 101.0e− 03 4 3.84e− 031.0e− 03 8 7.17e− 051.0e− 03 12 1.57e− 061.0e− 03 16 3.78e− 081.0e− 03 20 9.83e− 101.0e− 04 4 1.30e− 021.0e− 04 8 4.83e− 041.0e− 04 12 2.01e− 051.0e− 04 16 8.99e− 071.0e− 04 20 4.24e− 081.0e− 04 24 2.08e− 091.0e− 05 4 2.99e− 021.0e− 05 8 1.82e− 031.0e− 05 12 1.21e− 041.0e− 05 16 8.43e− 061.0e− 05 20 6.12e− 071.0e− 05 24 4.58e− 081.0e− 05 28 3.52e− 091.0e− 06 4 5.48e− 021.0e− 06 8 4.81e− 031.0e− 06 12 4.52e− 041.0e− 06 16 4.42e− 051.0e− 06 20 4.46e− 061.0e− 06 24 4.61e− 071.0e− 06 28 4.86e− 081.0e− 06 32 5.21e− 09

Table 5.1Reflection coefficients for optimized parameters as a function of δ

cTand P .

We first note that even for δ/cT = 10−4, which should be viewed as a very chal-lenging problem, an accuracy better than 0.05% is guaranteed with P = 8. Thus inmost practical circumstances the method will provide acceptable accuracy for negligi-ble effort. Second, the decay of ρ with increasing P is clearly spectral and consistentwith the analysis given above.


In Figure 5.1 we plot the optimal cosines for δcT = 10−3 and P = 4, 8 and in

Figure 5.2 we plot the corresponding reflection coefficients as a function of θ.

0 2 4 6 8 10 12 14 16 1810

−4

10−3

10−2

10−1

100

Cosines: δ/cT=10−3

Co

sin

e

P=4 P=8

Fig. 5.1. Optimal cosines for δcT

= 10−3.

6. Hyperbolicity of the boundary system. Although we have proven thatthe initial-boundary value problem for the acoustic system combined with the ap-proximate radiation boundary conditions is strongly well-posed, we would like toadditionally consider the well-posedness of the boundary system in isolation. Thisleads to confidence that it can be directly discretized as a hyperbolic system andalso motivates the form of the corner closures we will seek. To that end we consider(4.5)-(4.6) for j = 0, . . . , P , (4.7) for j = 0, . . . , P + 1, supplemented by (4.1), with∂w+∂x treated as an inhomogeneous term and w−,P+1 = 0. In three space dimensions

we then have a system of 4P + 7 first order equations on the boundary (3P + 5 intwo space dimensions). To analyze them we drop the zero order terms and set theinhomogeneous term, ∂w+

∂x , to zero. After Fourier transformation in space, we are leftto solve for the eigenvalues, ∂

∂t → cλ:

λw+,0 = −ik · vtan,0, (6.1)

(1 + cos φj)λw+,j+1 = (−1 + cosφj)λw+,j − ik · (vtan,j+1 + vtan,j) (6.2)

(1 + cosφj)λw−,j = (−1 + cos φj)λw−,j+1 − ik · (vtan,j+1 + vtan,j) (6.3)

λvtan,j = − i2k(w+,j + w−,j). (6.4)


10−5

10−4

10−3

10−2

10−1

100

10−20

10−15

10−10

10−5

100

Reflection coefficient: δ/cT=10−3

cos(θ)

R

P=4 P=8

Fig. 5.2. Reflection coefficients for δcT

= 10−3.

As in the calculation of the reflection coefficient we eliminate vtan,j and obtain:

λ2w+,0 = −|k|2

2(w+,0 + w−,0), (6.5)

(1 + cos φj)λ2w+,j+1 = (−1 + cosφj)λ2w+,j (6.6)

−|k|2

2(w+,j+1 + w+,j + w−,j+1 + w−,j) ,

(1 + cosφj)λ2w−,j = (−1 + cos φj)λ2w−,j+1 (6.7)

−|k|2

2(w+,j+1 + w+,j + w−,j+1 + w−,j) .

Following the calculation of the reflection coefficients we set γ =(λ2 + |k|2)

12 , α±,j =

γ ± λ cosφj , α±,j = γ ± λ cos φj and for γ, λ 6= 0 write:(w+,j

w−,j

)= aj

(1

λ−γλ+γ

)+ bj

( λ−γλ+γ

1

). (6.8)

Again we find that:

(γ + λ)(α+,jaj+1 + α−,jaj) + (γ − λ)(α+,jbj + α−,jbj+1) = 0, (6.9)

(γ + λ)(α+,jbj + α−,jbj+1) + (γ − λ)(α+,jaj+1 + α−,jaj) = 0, (6.10)


implying

α+,jaj+1 + α−,jaj = α+,jbj + α−,jbj+1 = 0. (6.11)

Now incorporating the termination condition and (6.5) we conclude that λ 6= 0,±i|k|is an eigenvalue if and only if:

(γ − λ

γ + λ

)2 P∏

j=0

((γ − cosφjλ)(γ − cos φjλ)(γ + cosφjλ)(γ + cos φjλ)

)= −1. (6.12)

This occurs only for λ imaginary such that γ is real. (See also Theorem 3 in [31].)We thus conclude:

Lemma 6.1. The boundary system is hyperbolic with characteristic speeds lessthan or equal to c.

7. Corner closures. To apply the boundary conditions for problems which arenot transversely periodic, we must impose boundary conditions on the auxiliary vari-able system at points where the artificial boundary meets another boundary, eitherphysical or artificial. From the preceding analysis we can view the auxiliary variablesystem as a hyperbolic system and seek boundary conditions in the form determinedby its characteristic structure. As the system is clearly isotropic in space we needsimply determine the number of characteristic directions which are tangential to theboundary of the artificial boundary.

As above, the analysis proceeds with lower order and inhomogeneous terms elimi-nated. Note that the difference of (4.5) and (4.6) already yields P+1 equations with nospatial derivatives. Now consider the equation determining eigenvalues, (6.12), with<λ > 0 and the component of k normal to the boundary of the artificial boundary avariable. If we define:

Q(z) = (z − λ)2P∏

j=0

(z − cosφjλ)(z − cos φj), (7.1)

we may rewrite it as:

Q(γ) +Q(−γ) = 0. (7.2)

This is a homogeneous equation in |k|2 and λ2 of degree P+2 in the squared variables.As there can be no solutions with k real, we conclude there are P + 2 solutions withimaginary parts of each sign. Thus P + 2 boundary conditions are required, both intwo and three space dimensions.

7.1. Physical boundaries. At physical boundaries the acoustic system requiresa single boundary condition. We suppose it takes the general form:

bpp+ bv · v = g, (7.3)

with g = 0 for x > −δ. For example, we may consider the solid wall boundarycondition v · n = 0 where n is normal to the boundary. As the linear combination ofvariables on the left-hand side of (7.3) applied to the auxiliary variables pj , vj satisfiesthe basic recursion relations (3.2) we conclude:

bppj + bv · vj = 0, j = 0, . . . , P + 1. (7.4)

We then impose these as our boundary conditions. Note that we have not yet triedto analyze the well-posedness of the boundary system terminated with (7.4), but wehave encountered no instabilities in numerical experiments.


7.2. Intersection of artificial boundaries. In this case we have no directway to translate the boundary conditions from the adjacent boundary to terminationconditions for the auxiliary variables. Instead, we attempt to generalize the closuresbased on compatibility conditions first proposed for homogeneous rational approxi-mants in [7, 32, 26]. Our construction is closely related to the one presented in [14],and in particular it can be easily interpreted via the analogy with PML mentionedearlier as a corner layer.

To minimize the algebraic complexity, we will focus first on the case of two spacedimensions. We also make the unnecessary assumptions that identical parametersand boundary condition orders are being used on each piece of the boundary and thatthe boundaries meet at a right angle. The more general case of boundaries meeting atangles greater than or equal to π

2 will be described in [33]. We identify the coordinatesx and y as the normals to the two boundaries.

We now define a doubly indexed collection of auxiliary variables which satisfyboth the recursions (3.2) associated with the two normals.

cos φkc

∂uj,k+1

∂t− ∂uj,k+1

∂x+

1cT

sin2 φkcos φk

uj,k+1 =

cosφkc

∂uj,k∂t

+∂uj,k∂x

+1cT

sin2 φkcosφk

ujk, (7.5)

cos φjc

∂uj+1,k

∂t− ∂uj+1,k

∂y+

1cT

sin2 φjcos φj

uj+1,k =

cosφjc

∂uj,k∂t

+∂uj,k∂y

+1cT

sin2 φjcosφj

ujk. (7.6)

Note that the variables u0,k can be identified with the auxilary variables for thepiece of the artificial boundary with normal coordinate x and uj,0 identified withvariables for the piece with normal coordinate y. Combining (7.5) with (7.6) for eachcomponent together with the acoustic system itself allows us to eliminate all spatialderivatives The only subtlety to the process comes from the fact that each directionis characteristic. The final forms we obtain are as follows. For j, k = 0, . . . , P :

∂pj+1,k+1

∂t+∂pj,k+1

∂t+∂pj+1,k

∂t+∂pj,k∂t

= cosφk∂vx,j+1,k

∂t− cos φk

∂vx,j+1,k+1

∂t

+ cosφk∂vx,j,k∂t

− cos φk∂vx,j,k+1

∂t

+sin2 φkT cosφk

vx,j+1,k − sin2 φkT cos φk

vx,j+1,k+1

+sin2 φkT cosφk

vx,j,k − sin2 φkT cos φk

vx,j,k+1

+ cosφj∂vy,j,k+1

∂t− cos φj

∂vy,j+1,k+1

∂t

+ cosφj∂vy,j,k∂t

− cos φj∂vy,j+1,k

∂t(7.7)

+sin2 φjT cosφj

vy,j,k+1 − sin2 φjT cos φj

vy,j+1,k+1

+sin2 φjT cosφj

vy,j,k − sin2 φjT cos φj

vy,j+1,k.


For j = 0, . . . P + 1, k = 0, . . . , P :

∂vx,j,k+1

∂t+∂vx,j,k∂t

= cosφk∂pj,k∂t

− cos φk∂pj,k+1

∂t

+sin2 φkT cosφk

pj,k − sin2 φkT cos φk

pj,k+1. (7.8)

And for j = 0, . . . P , k = 0, . . . , P + 1:

∂vy,j+1,k

∂t+∂vy,j,k∂t

= cosφj∂pj,k∂t

− cos φj∂pj+1,k

∂t

+sin2 φjT cosφj

pj,k − sin2 φjT cos φj

pj+1,k. (7.9)

Clearly we have written down 3(P +1)2 +2(P +1) ordinary differential equations for3(P + 2)2 variables. To close the system we require 4P + 7 additional equations. Weobtain 2P + 4 equations by specifying “outgoing characteristic” data from each edge.Precisely, let nx, ny = ±1 be the components of the normal vectors to the boundaries.For k = 0, . . . , P + 1:

∂p0,k

∂t+ ny

∂vy,0,k∂t

= x− normal edge value, (7.10)

and for j = 0, . . . , P + 1:

∂pj,0∂t

+ nx∂vx,j,0∂t

= y − normal edge value. (7.11)

The remaining 2P + 3 conditions follow from the termination conditions, which weaverage for j = k = P + 1. For k = 0, . . . , P :

∂pP+1,k

∂t− ny

∂vy,P+1,k

∂t= 0, (7.12)

for j = 0, . . . , P :

∂pj,P+1

∂t− nx

∂vx,j,P+1

∂t= 0, (7.13)

and

2∂pP+1,P+1

∂t− nx

∂vx,P+1,P+1

∂t− ny

∂vy,P+1,P+1

∂t= 0. (7.14)

Solving this sytem provides us with the time derivatives of all auxiliary variables fromeach edge at the corner point.

In three space dimensions an analagous set of 4(P +2)3 equations can be writtendown at a corner point where three boundary faces meet. On edges we derive equationssimilar to the two-dimensional corner case which also include spatial derivatives alongthe edge. We have not yet implemented the corner closures in three space dimensions,but plan to do so in the future. We also emphasize that while the constructiondescribed above is clearly consistent with the construction of the boundary conditionsand that numerical experiments show that it is stable and accurate, we have notanalyzed it.


8. Numerical experiments. To demonstrate the effectiveness of the proposedconditions we carry out two numerical experiments with the acoustic system in 2 + 1dimensions. In each case we employ 8th-order grid-stabilized difference methods inspace combined with the standard 4th order Runge-Kutta method in time. (See [19]for details.) The boundary system for the auxiliary variables is approximated withexactly the same methods and grid as the interior system.

8.1. Point source in a duct. Here we set c = 1 and consider for 0 ≤ t ≤ 100,x ∈ R, 0 < y < 1:

∂p

∂t+∂vx∂x

+∂vy∂y

= s(x, y, t), (8.1)

∂vx∂t

+∂p

∂x= 0, (8.2)

∂vy∂t

+∂p

∂y= 0, (8.3)

with

p(x, y, 0) = vx(x, y, 0) = vy(x, y, 0) = 0, vy(x, 0, t) = vy(x, 1, t) = 0. (8.4)

We take:

s(x, y, t) = g(t)δ(r), g(t) = e−150(t−0.75)2 . (8.5)

Here r is the distance to the point source,√

(x− xs)2 + (y − ys)2. We consider threedifferent choices for the latter: ys = 0.1 with xs = −1,−0.1,−0.01. The exact solutionis independently evaluated via quadratures. (We are confident that the solution hasbeen evaluated to more than 7 digits of accuracy.)

To impose the boundary condition on the duct wall we compute the time deriva-tive of the outgoing characteristic variable using 8th order one-sided differencing, asdiscussed in [19], and set the time derivative of the incoming variable so that ∂vy

∂t = 0.As discussed above, we use the same boundary conditions for the auxiliary variables,vy,j = 0, and impose them the same way.

We take our computational domain to be [0, L] × [0, 1]. At x = 0 we set theincoming characteristic equal to the exact solution and at x = L impose the approxi-mate radiation boundary condition. We generally choose L as small as possible whilemaintaing a uniform grid spacing in each dimension. Note that the closer the domainto the source location the finer the grid required to resolve the initial front.

We begin with the furthest separation of the source from the boundary, xs = −1,and take L = .02. We use boundary condition parameters computed for δ

cT = 10−2,setting T = 100. In Figure 8.1 we plot the L2-error relative to the maximum recordedL2-norm of the solution for P = 4, 8, 12. We clearly observe nearly uniform accuracy intime and spectral convergence with increasing P until the discretization error levelsare attained. Indeed, for P = 4 and P = 8 the observed maximum errors matchthe reflection coefficients displayed in Table 5.1, while for P = 12 the effects ofdiscretization error are measurable. Note that the computation was carried out with∆x = ∆y = 2 × 10−3 and ∆t = 3.3 × 10−4. By way of comparison in Figure 8.2 weplot the errors obtained using Pade parameters, cosφj = cos φj = 1 and the samevalues of P . The long time nonuniformity of the error, as predicted in [9], is evident,with all errors exceeding 10−2 for t > 20.


100

101

102

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

t

Re

lative

L2 E

rro

r

Duct Problem: xs=−1

P=4P=8P=12

Fig. 8.1. Relative errors for the complete conditions with xs = −1, L = .02.

Next we take xs = −0.1, L = .01, ∆x = ∆y = 10−3, ∆t = 2 × 10−4. Here wechoose parameters computed with δ

cT = 10−3 and T = 100. Results for P = 4, 8, 12, 16are displayed in Figure 8.3. Again we observe spectral convergence to discretizationerror levels, though as expected for this more difficult case the rates are somewhatdecreased in comparison to the first problem. Again the reflection coefficients correctlypredict the error levels.

In this case we also consider the effect of choosing parameters mismatched tothe actual source location. In particular we fix P = 8 and consider the choices:( δcT ≡ η = 10−2, T = 100), (η = 10−2, T = 10), (η = 10−4, T = 100). The

errors are plotted in Figure 8.4. Clearly the best overall results are obtained usingthe “correct” parameters, (η = 10−3, T = 100). With η chosen too large the shorttime error is comparitively large while with T chosen too small the long time error islarge. However, for η = 10−4 the results, though suboptimal, are reasonable. This isas expected as the error estimates for a particular choice of η and T hold so long asthe actual source locations and simulation times exceed those used to determine theparameters.

Finally we consider the most challenging configuration: xs = −0.01, L = .005,∆x = ∆y = 5 × 10−4, ∆t = 10−4. Here we choose parameters computed withδcT = 10−4 and T = 100. Results for P = 4, 8, 12, 16 are displayed in Figure 8.5. Againwe observe spectral convergence, though at a further reduced rate, with error levelspredicted by the reflection coefficients for P = 4 − 12. The effects of discretizationerror are noticeable for P = 16, particularly at early times.


100

101

102

10−10

10−8

10−6

10−4

10−2

100

t

Re

lative

L2 E

rro

r

Duct Problem: xs=−1 Pade Parameters

P=4P=8P=12

Fig. 8.2. Relative errors for Pade conditions with xs = −1, L = .02.

100

101

102

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

t

Re

lative

L2 E

rro

r

Duct Problem: xs=−0.1

P=4P=8P=12P=16

Fig. 8.3. Relative errors for the complete conditions with xs = −0.1, L = .01.


100

101

102

10−10

10−8

10−6

10−4

10−2

t

Re

lative

L2 E

rro

r

Duct Problem: xs=−0.1 P=8

η=10−3 T=100

η=10−2 T=100

η=10−4 T=100

η=10−2 T=10


100

101

102

10−10

10−8

10−6

10−4

10−2

t

Re

lative

L2 E

rro

r

Duct Problem: xs=−.01

P=4P=8P=12P=16



8.2. Free space problem with a distributed source. Our second exampleis the acoustic system in R2 × [0, 100] with the smooth source function:

s(x, y, t) = 100 · (100y2 − 1)8 · sin9 πx · sin9 πt, |y| ≤ 0.1, |x| ≤ 1. (8.6)

The computational domain is (−1.1, 1.1) × (−.2, .2). Now we use the complete radi-ation conditions on all four sides, applying the corner closure described above. Pa-rameters are based on the choice δ

cT = 10−3, T = 100. We record the pressureat three spatial locations: (.2571, .07143),(.8057, .07143),(1.0971, .07143). We take∆x = ∆y = 2.8571× 10−3, ∆t = 5× 10−4. As we have no exact solution, errors aremeasured against a solution computed with a PML of 500 points in each layer direc-tion. Precisely, we use the PML formulation of [3] with a complex frequency shiftparameter of α = .05 and hyperbolic tangent damping and grid stretching profileswith a maximum value, σmax = 20, of the damping parameter and a maximum valueof γmax = 3 of the grid stretching parameter. Comparisons with results obtained us-ing layers with 300 and 400 points and different absorption and stretching amplitudeslead to the conclusion that the infinite grid solution has been computed to more than13 digits of accuracy.

We display the results in Figure 8.6. Again the convergence is spectral with errorlevels commensurate with the reflection coefficients. In particular, we observe noadverse effects from the corner closures.

9. Conclusions. We have introduced a new parametrization of local radiationboundary conditions. We call them complete radiation boundary conditions, as theyare designed to absorb waves uniformly throughout a novel exact representation ofwaves in a half-space, which we call the complete wave representation. We prove thatthe new conditions satisfy very favorable complexity estimates relative to existinglocal treatments such as Higdon sequences or the perfectly matched layer. Onlycompressed nonlocal conditions are known to satisfy similar bounds, but unlike thesewe have demonstrated experimentally that the complete conditions can be applied ingeometrically flexible polygonal domains with no apparent loss of accuracy. Thus weargue that they represent a comprehensive solution to the longstanding problem ofdomain truncation for waves in isotropic media.

REFERENCES

[1] B. Alpert, L. Greengard, and T. Hagstrom, Rapid evaluation of nonreflecting boundarykernels for time-domain wave propagation, SIAM J. Numer. Anal., 37 (2000), pp. 1138–1164.

[2] , Nonreflecting boundary conditions for the time-dependent wave equation, J. Comput.Phys., 180 (2002), pp. 270–296.

[3] D. Appelo, T. Hagstrom, and G. Kreiss, Perfectly matched layers for hyperbolic systems:General formulation, well-posedness and stability, SIAM J. Appl. Math., 67 (2006), pp. 1–23.

[4] S. Asvadurov, V. Druskin, M. Guddati, and L. Knizherman, On optimal finite differenceapproximation of PML, SIAM J. Numer. Anal., 41 (2003), pp. 287–305.

[5] E. Becache, S. Fauqueux, and P. Joly, Stability of perfectly matched layers, group velocities,and anisotropic waves, J. Comput. Phys., 188 (2003), pp. 399–433.

[6] J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J.Comput. Phys., 114 (1994), pp. 185–200.

[7] F. Collino, High order absorbing boundary conditions for wave propagation models. Straightline boundary and corner cases, in Proceedings of 2nd Int. Conf. on Math. and Numer.Aspects of Wave Prop. Phen., R. Kleinman et al., ed., SIAM, 1993, pp. 161–171.


0 10 20 30 40 50 60 70 80 90 10010

−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

t

Re

lativ

e E

rro

r −

Sta

tion

1

Free Space Problem

P=4P=8P=12P=16

0 10 20 30 40 50 60 70 80 90 10010

−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

t

Re

lativ

e E

rro

r −

Sta

tion

2

P=4P=8P=12P=16

0 10 20 30 40 50 60 70 80 90 10010

−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

t

Re

lativ

e E

rro

r −

Sta

tion

3

P=4P=8P=12P=16

Fig. 8.6. Pressure errors for the free space distributed source problem.


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COMPLETE RADIATION BOUNDARY CONDITIONS: MINIMIZING …faculty.smu.edu › thagstrom › HWcomplete.pdf · approximations. Numerical experiments conﬁrm that the eﬃciencies predicted