Coupling discontinuous Galerkin methods and retarded potentials for

transient wave propagation on unbounded domains

Toufic Abboud Patrick Joly Jeronimo Rodrıguez Isabelle Terrasse

Abstract

This work deals with the numerical simulation of wave propagation on unbounded domains with localizedheterogeneities. To do so, we propose to combine a discretization based on a discontinuous Galerkin method inspace and explicit finite differences in time on the regions containing heterogeneities with the retarded potentialmethod to account the unbounded nature of the computational domain. The coupling formula enforces a discreteenergy identity ensuring the stability under the usual CFL condition in the interior. Moreover, the scheme allows touse a smaller time step in the interior domain yielding to quasi-optimal discretization parameters for both methods.The aliasing phenomena introduced by the local time stepping are reduced by a post-processing by averaging intime obtaining a stable and second order consistent (in time) coupling algorithm. We compute the numerical rateof convergence of the method for an academic problem. The numerical results show the feasibility of the wholediscretization procedure.

Keywords: coupling techniques, discontinuous Galerkin methods, retarded potentials, stability, hybrid methods,wave propagation, unbounded domains.

1 Introduction, motivation and main objectives

The context of this work, in the framework of a collaboration between INRIA and AIRBUS, is the numerical modelingin aeroacoustics. Typically, one aims at computing the propagation of sound around an airplane, taking into accountthe fluid flow around the airplane. The traditional mathematical models consist in linearizing the transient Eulerequations around a stationary solution of these equations that represents the reference flow (acoustic perturbationsare small perturbations). From the mathematical point of view, the model combines the difficulties linked to standardwave problems, in particular the fact that we have to solve a problem in an unbounded domain (such as the exterior ofthe airplane) with those linked to the presence of convection terms in the equations. For this reason, it is difficult touse standard numerical methods for solving the problem and that is why we have designed an original method tryingto conciliate the industrial context of the project (we wanted to privilege numerical methods developed at AIRBUS)with the specificities of our problem. Let us briefly expose the reasons that led to our choice: combine a discontinuousGalerkin approach for the discretization of the PDE’s inside the computational domain with time domain integralrepresentations on the exterior boundary for treating the unboundedness of the actual propagation domain

• In particular because of convection terms, there is no natural variational formulation for linearized Euler equa-tions, which “eliminates” standard and mixed finite element methods. That is why we have chosen to considerdiscontinuous Galerkin (DG) methods which have the advantage of easily handling convection terms and nat-urally lead to explicit schemes after time discretization (avoiding the problematic of mass lumping). This isa fundamental property especially in 3D: one wants to avoid inverting a big linear system at each time step.Moreover, these methods are well adapted to mesh refinement (conformity is not required) and variable accuracyin space. They benefit from an abundant research during the past ten years [1, 2, 3].

• In practical applications, the reference flow is uniform at infinity and often considered as uniform “far enough”(for instance far enough from the airplane). Choosing appropriately the boundary of the “interior domain”, onecan thus assume that the equations in the exterior domain have constant coefficients. In this situation, usingthe explicit knowledge of the Green’s function for these equations, we have to our disposal the so-called integralrepresentations for the solution of the exterior problem, that - as we shall see - can be exploited to construct exactor transparent boundary conditions for the interior problem. This is an alternative to other more traditionaltruncation domain methods such as absorbing boundary conditions [4] or perfectly matched layers that are not

1

exact. In 3D acoustics, these representations are often referred as retarded potentials (RP) formulas. Tradi-tionally considered as not very robust (because of the instabilities [5] that are met when collocation methods [6]are used for their discretization) as well as memory and cpu time consuming (because of their non-local naturein space and time), the use of these formulas has become attractive thanks to the development of space-timeGalerkin methods [7, 8, 9] that provide unconditional stability as well as multipole methods [10, 11, 12] thatprovide fast algorithms for the evaluation of the integral operators.

Independently of the industrial context of our work, coupling discontinuous Galerkin methods with time domainintegral equations has its own scientific interest, even in the simpler context of the standard scalar wave equation.This is the problematic we shall address in this paper. This type of question (coupling surface integral methodswith volume methods) has already been extensively treated in the literature for stationary problems [13]. When timedomain problems are concerned, one has to face the additional difficulty of the time discretization and of the stability ofthe coupled method, which is the major difficulty. One can find many contributions from the engineering community,mainly in the context of finite element methods for the interior equations and collocation methods for the boundaryequations [14], much less from the applied mathematicians community. The present work has been partially inspiredby a recent work [15] in the simpler context of a problem of vibro-acoustics, in which the question of a stable couplingbetween finite element methods and Galerkin discretization of time domain equations has been considered in a rigorousway. In addition, we have rapidly realized that, in order to preserve the efficiency of the coupled method, we hadto face a new difficulty induced by the fact that the two methods we wish to work with, which are of quite differentnature, have contradictory interest

(i) the integral equations methods are known to be very accurate in the sense that their (Galerkin) discretizationinduces much less numerical dispersion that volume methods, due to the fact that the explicit Green’s functionhandles propagation phenomena in an exact manner. As a consequence, accurate results are obtained with loworder (low degree polynomials) elements. Moreover, the unconditional stability of the method does not imposeany restriction on the time step and it is well known that the computational efficiency of integral equations di-minishes when the time step diminishes: that is why one would like to use the maximum possible time step, theonly limitation being the accuracy. Our experience on the subject suggests that, for the low-order discretization,the optimal choice for ∆t corresponds to a CFL number of the order of 1.

(ii) On the other hand, discontinuous Galerkin methods suffer from numerical dispersion, except when higher ordermethods are used. As the time discretization is explicit, they are constrained by a severe CFL stability condition,which is more and more severe when the order of the Galerkin method increases (the value of the maximum CFLis much less than 1).

These considerations led us to construct a method authorizing different time steps for the time discretization ofboundary equations and interior equations respectively: a small time step for the interior equation, and a big timestep for the boundary equations. For achieving this goal, we tried to adapt the ideas from [16, 17] for local meshrefinement and local time stepping. Once again, as in [18, 19], our main concern was to guarantee a priori the stabilityof our method via discrete energy identities.

The remaining part of the paper is organized as follows. In section 2 we introduce the equations of the problem andwe derive a coupled variational formulation well suited to the numerical schemes under consideration (DG and RP). Insection 3 we introduce the global discretization of the formulation obtained in the previous section. Special attentionis paid to the existence of the discrete solution and to the stability of the discrete problem that is ensured through adiscrete energy identity. In section 4 we present some numerical experiments where the coupling method is appliedto a model problem and the numerical rate of convergence is obtained. In section 5 we introduce the post-processingby averaging in time for the interior unknowns allowing to reduce some spurious phenomena due to the local-timestepping. In section 6 we present the numerical results for the new post-processed scheme and we compare them withthe previous results. In section 7 we present the conclusions and some future work.

2 The model problem and its coupled RP/DG formulation

2

2.1 Setting of the problem and reformulation as an artificial transmission problem

We are interested in the propagation of acoustic waves in R3. We write the governing equations as a first orderhyperbolic system whose unknowns are the pressure p(x, t) and the velocity field v(x, t):

∣∣∣∣∣∣∣∣∣∣∣

1

ρc2∂p

∂t+ div v = f, (x, t) ∈ R3 × [0, T ],

ρ∂v

∂t+ ∇p = 0, (x, t) ∈ R3 × [0, T ],

p(x, 0) = p0(x), v(x, 0) = v0(x) x ∈ R3.

(1)

In (1), ρ and c are respectively the density of the fluid and sound velocity: these are strictly positive and boundedfunctions of the space variable x. We suppose that there exists a smooth open set Ωi ⊂ R3, called the interior domainin the sequel, outside which ( i. e. in the exterior domain Ωe = R3 \ Ωi ) the fluid is homogeneous and does notsupport any souce term, i. e.

∀ (x, t) ∈ Ωe × [0, T ], ρ(x) = ρ0, c(x) = c0, v0(x) = p0(x) = f(x, t) = 0. (2)

We shall denote Γ := ∂Ωe ≡ ∂Ωi the interface between the interior and exterior domains (see figure 1 on the left). For

Ωe

Ωi

n

Γ

2c0T

OeT (Γ)

ΩT (Γ)

OiT (Γ)

Γ

Figure 1: On the left: geometry of the problem. On the right: definition of the sets ΩT (Γ) and OlT (Γ), l ∈ i, e.

developing our method, we first formulate an artificial transmission problem between Ωi and Ωe which is equivalentto (1). Denoting (pi,vi) (respectively (pe,ve) ) the restriction of (p,v) to Ωi × [0, T ] (respectively Ωe × [0, T ]), oneknows that

((pi,vi), (pe,ve)

)is solution of the transmission problem

∣∣∣∣∣∣∣∣∣∣∣

1

ρc2∂pi∂t

+ div vi = f, (x, t) ∈ Ωi × [0, T ],

ρ∂vi

∂t+ ∇pi = 0, (x, t) ∈ Ωi × [0, T ],

p(x, 0) = p0(x), v(x, 0) = v0(x), x ∈ Ωi,

(3)

∣∣∣∣∣∣∣∣∣∣∣

1

ρ0c20

∂pe∂t

+ div ve = 0, (x, t) ∈ Ωe × [0, T ],

ρ0∂ve

∂t+ ∇pe = 0, (x, t) ∈ Ωe × [0, T ],

p(x, 0) = 0, v(x, 0) = 0, x ∈ Ωe,

(4)

and the transmission conditions∣∣∣∣∣∣

ve · n = vi · n, on Γ,

pe = pi, on Γ,(5)

3

where n is the unit normal vector to Γ, outging with respect Ωi. Reciprocally, if((pi,vi), (pe,ve)

)is solution of the

transmission problem (3)-(5), the field (p,v) obtained by concatenating (pi,vi) and (pe,ve) is solution of (1).

Remark 2.1 Configurations with scatterers in Ωi (with a suitable boundary condition) could be easily handled. Sincein this work we are interested on the coupling between the exterior domain and the interior domain we have decidednot to consider these situations for the sake of clarity.

2.2 Orientation

Let us explain the main lines of how we obtain a new formulation of the continuous problem that will be the basisfor the construction of the numerical method. The idea is to treat as long as possible separately the interior and theexterior problems by giving a variational formulation for each of them that is adapted to the discretization procedurethat we wish to apply to each of them and then to use the transmission conditions at the last moment to obtain acoupled variational problem, namely (50), whose well-posedness is guaranteed through energy considerations. Moreprecisely, the steps of the our approach are the following:

• we reduce the solution of the exterior problem to equations satisfied by the traces

πe := pe|Γ and vne := ve · n|Γ. (6)

This can be done through integral representation formulas using the retarded potentials which is the object ofsection 2.3 and will lead to the variational formulation (37) for the exterior (surface) problem.

• we derive a DG variational formulation for the interior unknowns based on the use of centered fluxed in orderto guaranty energy conservation (see (44)). This is the object of section 2.4.

• we exploit the transmission conditions to finally derive a variational formulation for the coupled problem (see (50))which is done in section 2.5.

2.3 Reformulation of the problem for the exterior unknowns through integral equations

Although the content of this section can be rigorously justified from the mathematical point of view, we shall presentit in a rather informal way. The justification of what follows would require the introduction of a functional frameworkwhich is not the main concern of this paper and could affect, due to technical heaviness, the clarity of the exposition.

We prefer to refer to the references [7, 20, 21, 22, 23] in which the reader will find the definition and properties of theanisotropic Sobolev spaces of causal functions on Γ × R in which the operators that we introduce in the forthcomingsection are properly defined.

In the same way, we shall systematically write as integrals that should be written with appropriate duality brackets(see for instance [7, 20, 21, 22]) or understood in the principal value sense.

2.3.1 The retarded potential integral operators and their main properties

Let us first introduce some notation. If f denotes a function defined on R3, we set

fi := f |Ωi , fe := f |Ωe . (7)

For q : R3 7→ R such that q is smooth enough in Ωi and Ωe (for instance q ∈ H1(Ωi ∪Ωe), we introduce the jumps andmean values on Γ defined by

[[q]]Γ := qe|Γ − qi|Γ, qΓ :=1

2

(qi|Γ + qe|Γ

). (8)

Analogously, for w : R3 7→ R3 such that w is smooth enough in Ωi and Ωe (for instance w ∈ H(div; Ωi ∪ Ωe) ), weintroduce the normal jumps and mean values

[[w · n]]Γ := we · n|Γ − wi · n|Γ, w · nΓ :=1

2

(wi · n|Γ + we · n|Γ

). (9)

4

Let (ϕ, ψ) : Γ × R 7→ R × R such that ϕ(x, t) = ψ(x, t) = 0 for t < 0, we introduce (pϕ,ψ,vϕ,ψ) as the unique solutionof the transmission problem

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1

ρ0c20

∂pϕ,ψ∂t

+ div vϕ,ψ = 0, in R3 \ Γ × R+, (a)

ρ0∂vϕ,ψ

∂t+ ∇pϕ,ψ = 0, in R3 \ Γ × R+, (b)

[[pϕ,ψ]]Γ = ψ, on Γ × R+, (c)

[[vϕ,ψ · n]]Γ = ϕ, on Γ × R+, (d)

pϕ,ψ(x, 0) = vϕ,ψ(x, 0) = 0, in R3 \ Γ. (e)

(10)

From this we can define four space-time integral boundary operators, namely

YΓ : ψ(x, t) −→(YΓ ψ

)(x, t), x ∈ Γ, t > 0,

WΓ : ψ(x, t) −→(WΓ ψ

)(x, t), x ∈ Γ, t > 0,

ZΓ : ϕ(x, t) −→(ZΓ ϕ

)(x, t), x ∈ Γ, t > 0,

W∗Γ : ϕ(x, t) −→

(W∗

Γ ϕ)(x, t), x ∈ Γ, t > 0,

as follows:

ZΓ ϕ := pϕ,0Γ, WΓ ψ := p0,ψΓ,

W∗Γ ϕ := vϕ,0 · nΓ, YΓ ψ := v0,ψ · nΓ.

(11)

Note that by linearity of (ϕ, ψ) −→ (pϕ,ψ,vϕ,ψ) we have

vϕ,ψ = v0,ψ + vϕ,0, and pϕ,ψ = p0,ψ + pϕ,0, (12)

which implies [ZΓ WΓ

W∗Γ YΓ

][ϕ

ψ

]=

[pϕ,ψΓ

vϕ,ψ · nΓ

]. (13)

These operators are, in the time domain, the equivalent of the so-called Calderon-Zygmund operators [22] in thefrequency domain. They have many mathematical properties. The one which interests us is a positivity property inan a L2-like space-time framework. Given T > 0, we introduce the bilinear form bT

(·, ·)

acting on couples of functions(ϕ, ψ) : Γ × R+ 7→ R × R defined by

bT((ϕ, ψ), (ϕ, ψ)

):=

∫ T

0

∫

Γ

[ZΓ ϕ ϕ + WΓ ψ ϕ + W∗

Γ ϕ ψ + YΓ ψ ψ]

dγ dt. (14)

Proposition 2.1 For any T > 0, the bilinear form bT(·, ·)

is positive: for any (ϕ, ψ) : Γ × R+ 7→ R × R

bT((ϕ, ψ), (ϕ, ψ)

)=

1

2

∫

R3\Γ

[1

ρ0c20

∣∣pϕ,ψ(x, T )∣∣2 + ρ0 |vϕ,ψ(x, T )|2

]dx ≥ 0. (15)

Proof: Taking the inner product in R3 of (10).(b) with vϕ,ψ, we obtain after integration over R3 \ Γ × [0, T ]

∫ T

0

∫

R3\Γ

[ρ0

2

∂

∂t

∣∣vϕ,ψ∣∣2 + ∇pϕ,ψ · vϕ,ψ

]dx dt = 0. (16)

Next we multiply (10).(a) by pϕ,ψ and again integrate over R3 \ Γ × [0, T ]. After space integration by parts in eachsubdomain, we obtain

∫ T

0

∫

R3\Γ

[1

2ρ0c20

∂

∂t

∣∣pϕ,ψ∣∣2 − ∇pϕ,ψ · vϕ,ψ

]dx dt

−∫ T

0

∫

Γ

( (pϕ,ψ)e (vϕ,ψ)e · n − (pϕ,ψ)i (vϕ,ψ)i · n ) dγ dt = 0.

(17)

5

Standard algebraic manipulations lead to the identity

(pϕ,ψ)e (vϕ,ψ)e · n − (pϕ,ψ)i (vϕ,ψ)i · n = [[pϕ,ψ]]Γ vϕ,ψ · nΓ + pϕ,ψΓ [[vϕ,ψ · n]]Γ . (18)

Using (13) and (10 -(c)(d)) into the last equation we get

(pϕ,ψ)e (vϕ,ψ)e · n − (pϕ,ψ)i (vϕ,ψ)i · n = [W∗Γϕ+ YΓψ] ψ + [ZΓϕ+ WΓψ] ϕ, (19)

that we substitute into (17) to obtain

∫ T

0

∫

R3\Γ

[1

2ρ0c20

∂

∂t

∣∣pϕ,ψ∣∣2 − ∇pϕ,ψ · vϕ,ψ

]dx dt

−∫ T

0

∫

Γ

([W∗

Γϕ+ YΓψ] ψ + [ZΓϕ+ WΓψ] ϕ)

dγ dt = 0.

(20)

Finally, by adding (16) and (20) we get

∫ T

0

d

dt

∫

R3\Γ

[1

2ρ0c20

∣∣pϕ,ψ∣∣2 +

ρ0

2

∣∣vϕ,ψ∣∣2]

dx dt

−∫ T

0

∫

Γ

[ZΓ ϕ ϕ + WΓ ψ ϕ + W∗Γ ϕ ψ + YΓ ψ ψ ] dγ dt = 0.

(21)

which is nothing but (15) thanks to (10 -(e)).

Next we state a stronger result when T is small enough. For simplicity, we shall consider the case where Ωi is connected.The extension to the case where Ωi has a finite number of connected components is straightforward.

Let us introduce some notation. For T > 0, we introduce (see figure 1 on the right for a geometrical interpretation ofthese sets)

ΩT (Γ) :=

x ∈ R3 \ Γ / d(x,Γ) < c0 T, OT (Γ) := R3 \ ΩT (Γ), (22)

which obviously satisfy

ΩT (Γ) = ΩeT (Γ) ∪ ΩiT (Γ), ΩℓT (Γ) := ΩT (Γ) ∩ Ωℓ, ℓ ∈ i, e,

OT (Γ) = O eT (Γ) ∪ O i

T (Γ), OℓT (Γ) := OT (Γ) ∩ Ωℓ, ℓ ∈ i, e.

(23)

Since Ωe is unbounded, O eT (Γ) is non empty for all T > 0. For O i

T (Γ) this is only true for T small enough and that iswhy we introduce:

T ∗(Ωi) = supT > 0 / O i

T (Γ) 6= ∅ , that satisfies 0 < T ∗(Ωi) ≤diam (Ωi)

2 c0, (24)

(note that T ∗(Ωi) = diam (Ωi)/(2 c0) when Ωi is a ball).

Proposition 2.2 For any 0 < T < 2T ∗(Ωi), the bilinear form bT(·, ·)

is positive definite:

bT((ϕ, ψ), (ϕ, ψ)

)= 0 =⇒ ϕ(x, t) = ψ(x, t) = 0, x ∈ Γ, t ∈ [0, T ]. (25)

Proof: The idea of the following proof has been suggested to us by G. Lebeau. According to (15), we have to showthat

pϕ,ψ(x, T ) = 0, vϕ,ψ(x, T ) = 0, ∀ x ∈ R3 \ Γ, =⇒ ϕ(x, t) = ψ(x, t) = 0, x ∈ Γ, t ∈ [0, T ].

We first note that, due to the finite propagation velocity of the wave equation

supp pϕ,ψ(·, t) ∪ vϕ,ψ(·, t) ⊂ Ωt(Γ), ∀ t > 0.

6

Similarly, changing t in T − t and using pϕ,ψ(x, T ) = 0, vϕ,ψ(x, T ) = 0, ∀ x ∈ R3 \ Γ, we have

supp pϕ,ψ(·, t) ∪ vϕ,ψ(·, t) ⊂ ΩT−t(Γ), ∀ t ∈ ]0, T [.

Therefore∀ t ∈ ]0, T [, supp pϕ,ψ(·, t) ∪ vϕ,ψ(·, t) ⊂ Ωt(Γ) ∩ ΩT−t(Γ)

and in particular∀ t ∈ ]0, T [, supp pϕ,ψ(·, t) ∪ vϕ,ψ(·, t) ⊂ ΩT/2(Γ). (26)

Let us consider pϕ,ψ(x, t) : R3 \ Γ × R → R and vϕ,ψ(x, t) : R3 \ Γ × R → R3 defined by:

pϕ,ψ(x, t) = pϕ,ψ(x, t) if t ∈ ]0, T [, pϕ,ψ(x, t) = 0 if t ∈ ] −∞, 0[ ∪ ]T,+∞[ ,

vϕ,ψ(x, t) = vϕ,ψ(x, t) if t ∈ ]0, T [, vϕ,ψ(x, t) = 0 if t ∈ ] −∞, 0[ ∪ ]T,+∞[ .

The proof will be achieved if we show that pϕ,ψ and vϕ,ψ vanish identically.

Because of the zero initial (t = 0) and final (t = T ) conditions satisfied by (pϕ,ψ,vϕ,ψ), (pϕ,ψ, vϕ,ψ) satisfy equations(10)-(a) and (10)-(b) in R3 \ Γ × R. Thus, the Fourier transforms in time of pϕ,ψ and vϕ,ψ, namely

pϕ,ψ(x, ω) =

∫ +∞

−∞

pϕ,ψ(x, t) e−iωt dt, vϕ,ψ(x, ω) =

∫ +∞

−∞

vϕ,ψ(x, t) e−iωt dt,

satisfy for each ω ∈ R

div vϕ,ψ +iω

ρ0c20pϕ,ψ = 0, in R3 \ Γ ,

∇pϕ,ψ + iω ρ0 vϕ,ψ = 0, in R3 \ Γ.

(27)

In particular, pϕ,ψ is solution of the Helmholtz equation

∆pϕ,ψ +ω2

c20pϕ,ψ = 0, in R3 \ Γ .

On the other hand we deduce from (26) that

∀ ω ∈ R, supp pϕ,ψ(·, ω) ⊂ ΩT/2(Γ).

In particular pϕ,ψ vanishes in O eT/2(Γ) and O i

T/2(Γ) which are both nonempty open sets since T < 2T ∗(Ωi). As Ωiand Ωe are connected, we can use a unique continuation argument (Holmgren’s theorem) to assert that:

pϕ,ψ = 0, in R3 \ Γ , which implies (cf. (27)) vϕ,ψ = 0, in R3 \ Γ .

This concludes the proof.

Remark 2.2 The fact hat T must be small enough for ensuring that bT is positive definite in not only a technicalconvenience. This is also a necessary condition. Let us set:

P =T > 0 / bT (·, ·) is positive definite

. (28)

By definition of P

T ∈ R+ \ P ⇐⇒ N (bT ) :=

(ϕ, ψ)|[0,T ] / bT((ϕ, ψ), (ϕ, ψ)

)= 0

is different from 0 .

Thus, according to the proof of proposition 2.1, we see that

N (bT ) =

(ϕ, ψ)|[0,T ] / pϕ,ψ(x, T ) = 0, vϕ,ψ(x, T ) = 0, ∀x ∈ R3 \ Γ.

7

It is clear that for T2 > T1, N (bT2) ⊃ N (bT1) (if (ϕ, ψ) ∈ N (bT1), (ϕ, ψ), the extension of (ϕ, ψ) by 0 in ]T1, T2],belongs to N (bT2)). Thus, there exists Tmax(Ωi) > 0 such that

P = ] 0, Tmax(Ωi) [ . (29)

From proposition 2.2, we know thatTmax(Ωi) ≥ 2T ∗(Ωi). (30)

The question of obtaining a good upper bound for Tmax(Ωi) > 0 is clearly linked to the boundary controllability theoryfor the wave equation inside Ωi. Let us introduce

Tc(Ωi) = infT > 0, / the wave equation in Ωi is controlable in time T from Dirichlet data on Γ

. (31)

It is well-known that Tc(Ωi) is directly linked to the geometry of Ωi and that in particular Tc(Ωi) ≥ diam Ωi/c0 [24, 25].We claim that

Tmax(Ωi) ≤ Tc(Ωi). (32)

Indeed, let τ > 0 and ϕτ : Γ × [0, τ ] → R with ϕτ 6= 0. Let(p(ϕτ ),v(ϕτ )

)be the solution of equations (10)-(a)(b) in

Ωi × [0, τ ] with zero initial data and Dirichlet condition p(ϕτ ) = ϕτ on Γ × [0, τ ]. Let us set

pτ (x) :=[p(ϕτ )

](x, τ), vτ (x) :=

[v(ϕτ )

](x, τ), ∀ x ∈ Ωi .

By definition of Tc(Ωi), one can find ϕcτ : Γ × [τ, τ + Tc(Ωi)] → R such that(pcτ ,v

cτ

)defined as the solution of (10)-

(a)(b) in Ωi × [τ, τ + Tc(Ωi)] with initial data (pτ ,vτ ) at t = τ and Dirichlet condition pcτ = ϕcτ on Γ × [τ, τ + Tc(Ωi)]satisfies

pcτ(x, τ + Tc(Ωi)

)= 0, vcτ

(x, τ + Tc(Ωi)

)= 0, ∀ x ∈ Ωi .

One easily checks that (by construction !) (ϕ, ψ) : Γ × [0, τ + Tc(Ωi)] defined by:

ϕ|[0,τ ] = ϕτ , ϕ|[τ.τ+Tc(Ωi)] = ϕcτ ,

ψ|[0,τ ] = v(ϕτ ) · n|Γ, ψ|[τ.τ+Tc(Ωi)] = vcτ · n|Γ,is not identically 0 and belongs to N (bτ+Tc(Ωi)). This means that τ + Tc(Ωi) belongs to [Tmax(Ωi),+∞[. As this istrue for any τ > 0, we deduce (32).

2.3.2 The expression of the associated bilinear forms

Since the system (10) has constant coefficients, it can be solved analytically using the fundamental solution of thewave equation. As a consequence, for x ∈ R3 \Γ and t, one can get expressions of pϕ,ψ(x, t) and vϕ,ψ(x, t) as integralson Γ× [0, t] involving ϕ and ψ. One can then obtain an expression for ZΓ ϕ,WΓ ψ,W∗

Γ ϕ,YΓ ψ by taking the limits ofpϕ,ψ(x, t) and vϕ,ψ(x, t) when x tends to Γ from Ωi or Ωe. The obtained expressions may involve singular integrals thathave to be taken in the principal value sense. More interesting for the development of space-time Galerkin methodsare the expressions of the space-time bilinear forms associated to ZΓ,WΓ,W∗

Γ,YΓ. Let us state the correspondingformulas (we refer the reader to [22, 23, 7] for the derivation of these formulas) as a

Proposition 2.3 For any T ≥ 0 and smooth enough functions ϕ, ϕ, ψ, ψ : Γ × R+ −→ R, one has the formulas∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∫ T

0

∫

Γ

ZΓ ϕ · ϕ dγ dt = ρ0

∫ T

0

∫

Γ

∫

Γ

•

ϕ (y, τ)

4π|x − y| ϕ(x, t) dγy dγx dt,

∫ T

0

∫

Γ

WΓ ψ · ϕ dγ dt =

∫ T

0

∫

Γ

∫

Γ

ny · (x − y)

4π|x − y|

ψ(y, τ)

|x − y|2 +

•

ψ (y, τ)

c0|x − y|

ϕ(x, t) dγy dγx dt,

∫ T

0

∫

Γ

W∗Γ ϕ · ψ dγ dt =

∫ T

0

∫

Γ

∫

Γ

nx · (x − y)

4π|x − y|

(ϕ(y, τ)

|x − y|2 +

•

ϕ (y, τ)

c0|x − y|

)ψ(x, t) dγy dγx dt,

∫ T

0

∫

Γ

YΓ ψ · ψ dγ dt = − 1

ρ0c2

∫ T

0

∫

Γ

∫

Γ

ψ(y, τ)•

ψ (x, t)

4π|x − y| nx · ny dγx dγy dt

− c

ρ0

∫ T

0

∫

Γ

∫

Γ

∫ t0 rotΓψ(y, σ) ds · rotΓψ(x, t)

4π|x − y| dγx dγy dt,

(33)

8

where τ = t− |x − y|/c0 and σ = s− |x − y|/c0 are retarded times, rotΓ is the tangential curl operator (see [7, 22])

and•

ϕ holds for the time derivative of ϕ.

2.3.3 The integral equations for the traces of the exterior unknowns

Let (pe,ve) : Ωe × R+ 7→ R × R3 be any solution of (4), we define (p,v) : R3 \ Γ × R+ 7→ R × R3 by extension by 0in Ωi:

p =

pe, in Ωe,

0, in Ωi,, v =

ve, in Ωe,

0, in Ωi.

From the definitions (8), (9) and (6), it is clear that

[[p]]Γ = πe, [[v · n]]Γ = vne .

Therefore (being 0, (p,v) satisfies obviously the wave equation inside Ωi) one deduces that, by uniqueness of thesolution of (10)

(v, p) = (vϕ,ψ, pϕ,ψ) with (ϕ, ψ) = (vne , πe).

On the other hand, we have

∣∣∣∣∣∣∣∣

1

2vne = v · nΓ = vϕ,ψ · nΓ = v0,ψ · nΓ + vϕ,0 · nΓ,

1

2πe = pΓ = pϕ,ψΓ = p0,ψΓ + pϕ,0Γ.

Using the boundary integral operators introduced in (11), these equations can be rewritten as:

∣∣∣∣∣∣∣∣

1

2vne = YΓπe + W∗

Γvne , on Γ, i)

1

2πe = ZΓv

ne + WΓπe, on Γ. ii)

(34)

Remark 2.3 These equations are boundary identities satisfied by the traces of any solution of the exterior equations(4). They are not linearly independent. Any of them can be derived from the other. However we shall use both of themfor the formulation of our coupled problem.

These two equations can be put in a space-time variational form which is preparatory for the Galerkin discretization.More precisely, we multiply the first equation (resp. the second equation) by πe (resp. vn

e ) and we integrate onΓ × [0, T ] to obtain

∣∣∣∣∣∣∣∣∣

∫ T

0

∫

Γ

[YΓπe + W∗Γv

ne ] πe dγ dt − 1

2

∫ T

0

∫

Γ

vne πe dγ dt = 0,

∫ T

0

∫

Γ

[ZΓvne + WΓπe] vn

e dγ dt − 1

2

∫ T

0

∫

Γ

πe vne dγ dt = 0.

(35)

This can be put in a more compact form by setting

ue = [vne , πe]

t, ue =

[vne , πe

]t, (36)

and by adding the two previous equations which leads to

bT(ue, ue

)− 1

2

∫ T

0

∫

Γ

[vne πe + πe vn

e

]dγ dt = 0. (37)

9

2.4 Reformulation of the problem for the interior unknowns in a DG variational form

We derive a variational formulation for the interior equations. Even though we still speak of the continuous problem,a mesh in space is introduced prior to the discretization of the problem, which is a particularity of DG formulations.Let us consider a partition of the interior domain into a collection of disjoint elements (usually of simplex type)Ωi = ∪T∈ThT and we introduce the functional space

H1(Th) :=q ∈ L2(Ωi), such that q|T ∈ H1(T ), ∀ T ∈ Th

.

Let Eh := e = ∂T ∩ ∂T ′, (T, T ′) ∈ Th × Th be the skeleton of the mesh. We point out that this set is different to∂Th := ∂T, T ∈ Th since the later includes the internal faces twice. We decompose Eh into two disjoint subsets

Eh = E inth ∪ Eexth ,

these sets containing the internal faces and the boundary faces.

For a given function ηi defined in Ωi we will denote by ηT its restriction to an element T . The DG formulation isbased on the fact that the solution of the interior problem can also be characterized (we are assuming here additionalsmoothness for the sake of simplicity) as the element (vi, pi) ∈ (H1(Th))3 ×H1(Th) such that for all T ∈ Th

∣∣∣∣∣∣∣∣

1

ρc2∂pT∂t

+ div vT = f, ρ∂vT

∂t+ ∇pT = 0, in T ,

vT · nT = vT ′ · nT , pT = pT ′ , on ∂T ∩ E inth ,

vT · nT = vT · nT , pT = pT , on ∂T ∩ Eexth ,

(38)

where nT is the outward unit normal vector and T ′ ∈ Th is adjacent to T . Notice that for the moment the boundaryconditions at the interface Γ have not been considered, that is, the equations are not coupled with the solution in theexterior. We multiply the two first equations in (38) by the corresponding test functions (v, p) ∈ (H1(Th))3 ×H1(Th)and we integrate over element T to obtain

∫

T

[1

ρc2∂pT∂t

+ div vT

]p dx =

∫

T

f p dx,

∫

T

[ρ∂vT

∂t+ ∇pT

]· v dx = 0.

We will consider a classical DG formulation involving central fluxes [1, 26] that for some reasons that will be clarifiedin the next sections it is convenient to present in the following way. We rewrite each integral term involving differentialoperators in space as the sum of two identical terms and we integrate by parts one of them. This gives rise to twoterms involving surface integrals (one per equation) that we rewrite using the transmission conditions in (38) to obtain∣∣∣∣∣∣∣∣∣∣

∫

T

[1

ρc2∂pT∂t

p− 1

2vT · ∇p+

1

2div vT p

]dx +

1

2

∫

∂T∩Einth

vT ′ · nT p dγ +1

2

∫

∂T∩Eexth

vT · nT p dγ =

∫

T

f p dx,

∫

T

[ρ∂vT

∂t· v − 1

2div v pT +

1

2v · ∇pT

]dx +

1

2

∫

∂T∩Einth

pT ′ v · nT dγ +1

2

∫

∂T∩Eexth

pT v · nT dγ = 0.

(39)Note that the surface integrals are separated in two terms: the first one includes the internal faces, the second onecontains the external ones. Adding (39) over all elements in Th we obtain∣∣∣∣∣∣∣∣∣∣

∫

Ωi

[1

ρc2∂pi∂tp− 1

2vi · ∇hp+

1

2divh vi p

]dx +

1

2

∑

T∈Th

∫

∂T∩Einth

vT ′ · nT p dγ +1

2

∫

Eexth

vi · n p dγ =

∫

Ωi

f p dx,

∫

Ωi

[ρ∂vi

∂t· v − 1

2divh v pi +

1

2v · ∇hpi

]dx +

1

2

∑

T∈Th

∫

∂T∩Einth

pT ′ v · nT dγ +1

2

∫

Eexth

pi v · n dγ = 0,

where ∇h(·) (resp. divh(·)) denotes the local ∇(·) (resp. div(·)) operator.

At this stage, it is useful to introduce some notation. Let q be a scalar function defined over ∂Th, we define the meanq and the jump [[q]] on the internal faces as follows (in the lines below, e ∈ E inth denotes the internal face shared byT and T ′)

q : E inth −→ R, [[q]] : E inth −→ R,

q|e := 12 (qT + qT ′), [[q]]|e := qT nT + qT ′ nT ′ .

(40)

10

For a given vector function w defined over ∂Th we introduce

w : E inth −→ R, [[w]]N : E inth −→ R,

w|e := 12 (wT + wT ′), [[w]]N |e

:= wT · nT + wT ′ · nT ′ .(41)

Using the following identity

∑

T∈Th

∫

∂T∩Einth

wT · nT qT dγ =

∫

Einth

[[[qi]] · wi + qi [[wi]]N

]dγ,

one obtains∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∫

Ωi

[1

ρc2∂pi∂t

p− 1

2vi · ∇hp+

1

2divh vi p

]dx +

1

2

∫

Einth

[[[p]] · vi − p [[vi]]N

]dγ +

1

2

∫

Eexth

vi · n p dγ =

∫

Ωi

f p dx,

∫

Ωi

[ρ∂vi

∂t· v − 1

2divh v pi +

1

2v · ∇hpi

]dx −

1

2

∫

Einth

[[[pi]] · v − pi [[v]]N

]dγ +

1

2

∫

Eexth

pi v · n dγ = 0.

(42)

This can be put in a more compact form by setting

ui = [pi,vi]t , ui = [pi, vi]

t . (43)

After summation of both equations we get

m(∂ui

∂t, ui) + a(ui, ui) +

1

2

∫

Γ

[pi vi · n + pi vi · n] dγ = f(ui, t), (44)

where we have introduced

m(ui, ui) =

∫

Ωi

[1

ρc2pi pi + ρ vi · vi

]dx, (45)

a(ui, ui) =1

2

∫

Ωi

[divhvi pi − divhvi pi − vi · ∇hpi + vi · ∇hpi] dx + (46)

1

2

∫

Einth

[[[pi]] · vi − pi [[vi]]N + pi [[vi]]N − [[pi]] · vi

]dγ,

f(ui, t) =

∫

Ωi

f(t) pi dx. (47)

2.5 Derivation of a coupled variational problem

We now explain how we couple the variational equations (37) and (44) by using the transmission conditions

∣∣∣∣∣∣

vi · n = vne , on Γ,

pi = πe, on Γ,(48)

that are nothing but the transmission conditions (5) written with the new unknowns defined in (6). The procedure issimilar to the one used in the DG formulation with central fluxes, namely

i) in the equation (44) we replace (vi · n, pi) by (vne , πe) in the boundary integral terms,

ii) in the equation (37) we keep ue in the bilinear form bT and we replace (vne , πe) by (vi · n, pi) in the remaining

integral term.

11

Doing so, and introducing the new bilinear form

c(ue,ui) =1

2

∫

Γ

[pi vne + πe vi · n] dγ, (49)

we get the following coupled variational problem

∣∣∣∣∣∣∣∣

m(∂ui

∂t, ui) + a(ui, ui) + c(ue, ui) = f(ui, t),

bT (ue, ue) −∫ T

0

c(ue,ui) dt = 0.

(50)

This formulation is the weak form of the coupled problem (3), (34), (48). The interest of this formulation is thatthe coupling terms are such that, thanks to the choice described above (points i) and ii)) are skew which ensures theenergy preserving nature of the formulation. More precisely, if (ui,ue) is a solution of (50) then replacing (ui, ue) by(ui,ue) in (50) and integrating in time on the interval [0, T ] the first equation, one easily obtains

1

2m (ui(T ),ui(T )) + bT (ue,ue) =

1

2m (ui(0),ui(0)) +

∫ T

0

f(ui(t), t) dt. (51)

This shows the uniqueness of the solution of the problem (50) (or equivalently (3), (34) and (48)). Then it is easy toprove from the equivalence between the original problem and the transmission problem (3)-(5) the following result

Proposition 2.4 Let (p,v) be the solution of the original problem (1), then((pi,vi), (πe, v

ne ))

defined by

(pi,vi) = (p,v)|Ωi , (πe, vne ) = (p,v · n)|Γ,

is solution of the coupled problem (50).

Reciprocally, if((pi,vi), (πe, v

ne ))

is solution of (50), the field (p,v), obtained by concatenation of (pi,vi) with thesolution (pe,ve) of the exterior problem (4) completed with one of the two boundary conditions pe|Γ = πe or ve ·n|Γ =vne ), is the solution of (1).

3 An energy preserving discretization scheme for the DG-RP problem

Our goal is to obtain a numerical method using a DG discretization in space and finite differences in time in Ωi and a(Galerkin) RP method for the approximation of the unknowns on the exterior boundary Γ. This will be done in foursteps:

• First of all we will proceed with the variational part of the discretization; that is, a semi-discretization in spacein Ωi and a space-time discretization in Ωe × [0, T ].

• The second step consists of discretizing in time the semi-discrete scheme in Ωi using the second order leap-frog(finite differences) scheme. At this step, the discretization of the coupling term will not be addressed.

• The coupling terms will be constructed considering energy considerations. We will derive a equation thatshould be satisfied by the coupling terms (not determined yet at this step) in order to enforce a discrete energyconservation similar to the one shown in section 2.5 at the continuous level (see (51)).

• Finally we will provide coupling terms satisfying the equality derived on the previous step. This will provide thestability of the global discretization procedure.

3.1 Construction of an energy preserving coupling scheme

12

First step: Discretizing through a Galerkin approach First of all we introduce the finite dimensional space wewill use for the volume unknowns. We assume that the partition of Ωi we have considered in section 2.4 is composedby tetrahedra and we introduce the space

H1h(Th) :=

q ∈ L2(Ωi), such that q|T ∈ Pk(T ), ∀ T ∈ Th

,

where Pk(T ) is the set of polynomials of degree k in T . In this way, the interior solution uhi = (vhi , phi ) and the

corresponding test functions uh = (vh, ph) belong to the space

Xhi := H1

h(Th)3 ×H1h(Th). (52)

To introduce the approximation space for the surface unknowns it is useful to set some notation. We consider atriangular mesh of the surface Γ = ∪S∈ShS and we define the following spaces

P1h(Γ) :=

g ∈ C0(Γ), / g|S ∈ P1(S), ∀S ∈ Sh

, P0

h(Γ) :=g ∈ L2(Γ), / g|S ∈ P0(S), ∀S ∈ Sh

. (53)

For a given time step ∆t and a Hilbert space X we introduce the space of piecewise constant functions of time withvalues in X :

P0∆t(R

+;X) :=g ∈ L2(R+, X) / g|[tn,tn+1] = gn+ 1

2∈ X, ∀ n ≥ 0

, (54)

where tn = n∆t. The RP unknowns uh,∆te = (vn,h,∆te , πh,∆te ) (resp. the test functions u

h,∆te = (vn,h,∆t

e , πeh,∆t)) will

be sought (resp. be taken) in the space

Xh,∆te := P0

∆t(R+;P0

h(Γ)) × P0∆t(R

+;P1h(Γ)). (55)

We obtain the discrete version of the variational formulation in (50) by introducing the discrete functions:∣∣∣∣∣∣∣∣∣∣∣∣∣

Find (uhi (t),uh,∆te ) ∈ Xh

i ×Xh,∆te such that for all (uhi ,

˜uh,∆te ) ∈ Xh

i ×Xh,∆te

m

(∂uhi∂t

, uhi

)+ a

(uhi , u

hi

)+ c

(uh,∆te , uhi

)= f

(uhi , t

),

bT

(uh,∆te ,

˜uh,∆te

)−∫ T

0

c

(˜uh,∆te ,uhi

)dt = 0.

(56)

Remark 3.1 With the discretization space (55) we have restricted our cells to the lowest order elements both in spaceand time. There is absolutely no obstruction to using higher order polynomial degrees. However, the expression of thematrices issued from the discretization (see section 3.2 and A) would be much more complicated and the implementationof the corresponding code much more involved. Moreover we dispose only of the lowest order and they already provideexcellent results. It seems to be more important to use a higher order discretization in the interior.

Second step: Finite differences discretization We use a centered explicit second order finite difference schemefor the time discretization of the interior equations. Contrary to what happens with the RP method (that in absenceof coupling is unconditionally stable regardless the time step size), the interior discretization is only stable under a(rather severe) CFL condition. For this reason we use in Ωi a time step p ∈ N \ 0 times smaller than the oneused by the RP method. Each time interval [tn, tn+1] (where tn = n∆t) will be divided into p subintervals of length∆tp := ∆t/p centered at times

tn+ 2k+12p =

(n+

2k + 1

2p

)∆t, k ∈ 1, . . . , p,

and we shall chose to compute the interior unknowns denoted by un+ 2k+1

2p

i at these instants. This is illustrated infigure 2 when p = 3. This leads to the following set of equations∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

Find un+ 2k+1

2p

i ∈ Xhi , and uh,∆te ∈ Xh,∆t

e such that for all (uhi ,˜uh,∆te ) ∈ Xh

i ×Xh,∆te

m

u

n+ 2k+12p

i − un+ 2k−3

2p

i

2∆tp, uhi

+ a

(un+ 2k−1

2p

i , uhi

)+ c

(⟨⟨uh,∆te

⟩⟩n+ 2k−12p , uhi

)= f

(uhi , t

n+ 2k−12p

),

bT

(uh,∆te ,

˜uh,∆te

)−⟨⟨∫ T

0

c

(˜uh,∆te ,uhi

)dt

⟩⟩= 0.

(57)

13

The terms between double angles, namely,

⟨⟨uh,∆te

⟩⟩n+ 2k−12p and

⟨⟨∫ T

0

c

(˜uh,∆te ,uhi

)dt

⟩⟩,

are quantities that are supposed to be consistent with

uh,∆te

(tn+ 2k−1

2p

)and

∫ T

0

c

(˜uh,∆te (t),uhi (t)

)dt,

and that will be chosen in the next step in order to ensure the stability of the numerical scheme.

Third step: Derivation of a discrete energy identity For the discretization of the coupling terms we proceedwith (discrete) energy considerations. We start by choosing

uhi =un+ 2k+1

2p

i + un+ 2k−3

2p

i

2,

as the test function in (57) to obtain

1

∆tp

(En+ k

p

i − En+ k−1p

i

)+ c

⟨⟨uh,∆te

⟩⟩n+ 2k−12p ,

un+ 2k+1

2p

i + un+ 2k−3

2p

i

2

= f

u

n+ 2k+12p

i + un+ 2k−3

2p

i

2, tn+ 2k−1

2p

,

(58)

where the discrete energy in the interior domain at time t = tn+ kp = (n+ k

p ) ∆t is defined by

En+ kp

i =1

4

[m

(un+ 2k+1

2p

i ,un+ 2k+1

2p

i

)+ m

(un+ 2k−1

2p

i ,un+ 2k−1

2p

i

)− 2∆tp a

(un+ 2k+1

2p

i ,un+ 2k−1

2p

i

)]. (59)

Adding (58) for k ∈ 1, . . . , p and n ∈ 0, . . . , N − 1 where N = T/∆t, one gets

ENi − E0i + ∆tp

N−1∑

n=0

p∑

k=1

c

⟨⟨uh,∆te

⟩⟩n+ 2k−12p ,

un+ 2k+1

2p

i + un+ 2k−3

2p

i

2

= ∆tp

N−1∑

n=0

p∑

k=1

f

u

n+ 2k+12p

i + un+ 2k−3

2p

i

2, tn+ 2k−1

2p

.

(60)

Choosing˜uh,∆te by uh,∆te in the second equation in (57) we obtain

bT (uh,∆te ,uh,∆te ) −⟨⟨∫ T

0

c(uh,∆te ,uhi ) dt

⟩⟩= 0. (61)

Adding the last two equations we get

ENi + bT (uh,∆te ,uh,∆te ) = E0i + ∆tp

N−1∑

n=0

p∑

k=1

f

u

n+ 2k+12p

i + un+ 2k−3

2p

i

2, tn+ 2k−1

2p

+

⟨⟨∫ T

0

c(uh,∆te ,uhi ) dt

⟩⟩− ∆tp

N−1∑

n=0

p∑

k=1

c

⟨⟨uh,∆te

⟩⟩n+ 2k−12p ,

un+ 2k+1

2p

i + un+ 2k−3

2p

i

2

.

(62)

We should compare this equation with its continuous counterpart (51). Our goal is to build a coupling not introducingor removing any energy at the interface Γ. The two terms in the second line in (62) should thus cancel.

Fourth step: Discretization of the coupling terms Since uh,∆te (t) is constant in the time interval [tn, tn+1] itis natural to choose (see figure 2 for a graphical representation of this choice when p = 3)

⟨⟨uh,∆te

⟩⟩n+ 2k−12p := uh,∆te (tn+ 2k−1

2p ) = uh,∆te (tn+ 12 ), k ∈ 1, . . . , p, (63)

14

which enforces the following discretization of the coupling term on the second equation of (57)

⟨⟨∫ T

0

c(˜uh,∆te ,uhi ) dt

⟩⟩:= ∆tp

N−1∑

n=0

p∑

k=1

c

˜

uh,∆te (tn+ 1

2 ),un+ 2k+1

2p

i + un+ 2k−3

2p

i

2

, (64)

in order to ensure a perfect balance of the discrete energy.

The coupling algorithm is thus given by

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

Find un+ 2k+1

2p

i ∈ Xhi , and uh,∆te ∈ Xh,∆t

e such that for all (uhi ,˜uh,∆te ) ∈ Xh

i ×Xh,∆te

m

u

n+ 2k+12p

i − un+ 2k−3

2p

i

2∆tp, uhi

+ a

(un+ 2k−1

2p

i , uhi

)+ c

(uh,∆te (tn+ 1

2 ), uhi

)= f

(uhi , t

n+ 2k−12p

),

bT

(uh,∆te ,

˜uh,∆te

)− ∆tp

N−1∑

n=0

p∑

k=1

c

˜

uh,∆te (tn+ 1

2 ),un+ 2k+1

2p

i + un+ 2k−3

2p

i

2

= 0.

(65)

The discrete energy identity satisfied by this formulation, namely,

ENi + bT (uh,∆te ,uh,∆te ) = E0i + ∆tp

N−1∑

n=0

p∑

k=1

f

u

n+ 2k+12p

i + un+ 2k−3

2p

i

2, tn+ 2k−1

2p

, (66)

will allow us to show the existence and uniqueness of solution of the discrete problem and the stability of the discretesystem under the usual CFL condition on the interior domain.

3.2 Algebraic formulation of the discrete problem

Let us introduce a basis ϕj , j ∈ 1, . . . , Nϕ (resp. ψj , j ∈ 1, . . . , Nψ) of P1h(Γ) (resp. P0

h(Γ)) and letηn− 1

2(t), n ∈ N∗ be functions defined by

ηn− 12(t) =

1, if t ∈ [tn−1, tn],

0, if not.

For any (ϕh,∆t, ψh,∆t) ∈ Xh,∆te , there exists Φn− 1

2 ∈ RNϕ , n ∈ 1, . . . , N and Ψn− 12 ∈ RNψ , n ∈ 1, . . . , N such

that

ϕh,∆t(x, t) =

N−1∑

n=1

Nϕ∑

j=1

Φn− 1

2

j ϕj(x) ηn− 12(t), ψh,∆t(x, t) =

N−1∑

n=1

Nψ∑

j=1

Ψn− 1

2

j ψj(x) ηn− 12(t), (67)

and we introduce the vector Un− 1

2e := [Φn− 1

2 ,Ψn− 12 ] ∈ RNRP where NRP := Nϕ +Nψ.

In a similar way, we introduce a basis Ξj , j ∈ 1, . . . , NΞ of the space H1h(Th). In consequence, for all (ph,vh) ∈ Xh

i

there exists P ∈ RNΞ and V k ∈ RNΞ , k ∈ 1, 2, 3 such that

ph(x) =

NΞ∑

j=1

P j Ξj(x), vh(x) =

3∑

k=1

NΞ∑

j=1

(V k)j ek Ξj(x), (68)

and we introduce the vector containing the interior unknowns, U i := [P ,V 1,V 2,V 3] ∈ RNDG , where NDG = 4NΞ.

Next, we introduce the NΞ ×NΞ matrices defined by

(Mp)i,j =

∫

Ωi

1

ρc2Ξi Ξj dx, (Mv)i,j =

∫

Ωi

ρ Ξi Ξj dx,

(Dk)i,j =

∫

Ωi

∂Ξj∂xk

Ξi dx, (Sk)i,j =

∫

Einth

[Ξj ek · [[Ξi]] − Ξi [[Ξj ek]]N

]dγ,

15

and the coupling matrices Ckψ ∈ MNΞ×Nψ , k ∈ 1, 2, 3 and Cϕ ∈ MNΞ×Nϕ given by

(Ckψ)i,j :=

∫

Γ

ψj Ξi ek · n dγ, (Cϕ)i,j :=

∫

Γ

ϕj Ξi dγ.

Then, the first equation in (65) is equivalent to

MUn+ 2k+1

2p

i − Un+ 2k−3

2p

i

2∆tp+ A U

n+ 2k−12p

i + C Un+ 1

2e = F n+ 2k−1

2p , k ∈ 1, . . . , p, (69)

where

M :=

Mp 0 0 0

0 Mv 0 0

0 0 Mv 0

0 0 0 Mv

, A :=

1

2

0 D1 − D∗1 + S1 D2 − D∗

2 + S2 D3 − D∗3 + S3

D1 − D∗1 − S∗

1 0 0 0

D2 − D∗2 − S∗

2 0 0 0

D3 − D∗3 − S∗

3 0 0 0

,

C :=1

2

Cϕ 0

0 C1ψ

0 C2ψ

0 C3ψ

,

(F n+ 2k−1

2p

)i

:=

∫

Ωi

f(tn+ 2k−12p ) Ξi dx, if i ∈ 1, . . . , NΞ,

0, if i ∈ NΞ + 1, . . . , 4NΞ.

The second equation in (65) is equivalent to the following matrix formulation (see the lemma in A)

n∑

m=0

Bn−mT (∆t) Um+ 1

2e − C∗

p∑

k=1

Un+ 2k+1

2p

i + Un+ 2k−3

2p

i

2∆tp = 0, (70)

where

BkT (∆t) =

BkT,Z(∆t) BkT,W(∆t)

BkT,W∗(∆t) BkT,Y(∆t)

, for k ≥ 0. (71)

Introducing the sets

C∆tk :=

(x,y) ∈ Γ2 such that c0 k ∆t ≤ |x − y| ≤ c0 (k + 1) ∆t

,

D∆tk :=

(x,y) ∈ Γ2 such that |x − y| ≤ c0 k ∆t

,

16

the matrices BkT,Z(∆t) ∈ MNϕ×Nϕ , BkT,W (∆t) ∈ MNϕ×Nψ , BkT,W∗(∆t) ∈ MNψ×Nϕ and BkT,Y(∆t) ∈ MNψ×Nψ in (71)are defined by (in the following expressions τ = |x − y|/c0)

(BkT,Z (∆t)

)i,j

= ρ0

[∫∫

C∆tk

Φj(y) Φi(x)

4π|x − y| dγx dγy −∫∫

C∆tk−1

Φj(y) Φi(x)

4π|x − y| dγx dγy

],

(BkT,Y(∆t)

)i,j

=1

ρ0c2

[∫∫

C∆tk

Ψj(y) Ψi(x)

4π|x − y| nx · ny dγx dγy −∫∫

C∆tk−1

Ψj(y) Ψi(x)

4π|x − y| nx · ny dγx dγy

]−

c

ρ0

[∫∫

C∆tk

rotΓΨj(y) · rotΓΨi(x)

8π|x − y|(tk+1 − τ

)2dγx dγy +

∫∫

C∆tk−1

rotΓΨj(y) · rotΓΨi(x)

8π|x − y|[∆t2 − (tk − τ)2 + 2∆t(tk − τ)

]dγx dγy +

∫∫

D∆tk−1

rotΓΨj(y) · rotΓΨi(x)

4π|x − y| ∆t2 dγx dγy

],

(BkT,W (∆t)

)i,j

=

∫∫

C∆tk

Ψj(y) Φi(x)

4π|x − y|3 ny · (x − y) (tk+1 − τ) dγx dγy +

∫∫

C∆tk−1

Ψj(y) Φi(x)

4π|x − y|3 ny · (x − y) (τ − tk−1) dγx dγy +

1

c0

[∫∫

C∆tk−1

Ψj(y) Φi(x)

4π|x − y| ny · (x − y) dγx dγy −∫∫

C∆tk

Ψj(y) Φi(x)

4π|x − y| ny · (x − y) dγx dγy

],

(BkT,W∗(∆t)

)i,j

= −(BkT,W(∆t)

)j,i.

Remark 3.2 Since the boundary Γ is assumed to be bounded, there exists an integer k0 = k0(∆t) such that for k ≥ k0

the set C∆tk is empty and the set D∆t

k coincides with Γ2. As a consequence, for k ≥ k0, the matrices BkT,Z , BkT,W and

BkT,W∗ are zero, whereas the matrix BkT,Y is k independent.

In consequence, the matrix formulation of the totally discretized coupled problem given by (65) can be written asfollows ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

Find Un+ 2k+1

2p

i and Un+ 1

2e such that

MUn+ 2k+1

2p

i − Un+ 2k−3

2p

i

2∆tp+ A U

n+ 2k−12p

i + C Un+ 1

2e = F n+ 2k−1

2p , k ∈ 1, . . . , p,

n∑

m=0

Bn−mT (∆t) Um+ 1

2e − C∗

p∑

k=1

Un+ 2k+1

2p

i + Un+ 2k−3

2p

i

2∆tp = 0.

(72)

See figures 2 and 3 for a schematic representation of the coupling mechanism.

3.3 Analysis of the discrete problem

3.3.1 Existence and uniqueness of the discrete solution

The existence and uniqueness of the discrete solution will be proven by induction on the discrete time interval [tm, tm+1]under the usual CFL condition for the interior problem. Let us assume that the discrete solution has been computedup to time tn, more precisely that the quantities

Um+ 2k+1

2p

i , (m, k) ∈ 0, . . . , n− 1 × 1, . . . , p, Um+ 1

2e , m ∈ 0, . . . , n− 1, (73)

17

Ωe

Ωi

t0 t13 t

23 t1 . . . tn−1 tn−

23 tn−

13 tn tn+ 1

3 tn+ 23 tn+1

Figure 2: Schematic representation of the coupling on the interior equations proposed by the energy preserving method(see the p first equations in (72)). All the values for the external trace appearing on the interior equations in the interval(tn, tn+1) are equal to a same value. In the figure, p = 3.

Ωe

Ωi

t0 t13 t

23 t1 . . . tn−1 tn−

23 tn−

13 tn tn+ 1

3 tn+ 23 tn+1

Figure 3: Schematic representation of the coupling on the exterior equations proposed by the energy preserving method(see the last equation in (72)). The value of the internal trace at tn+ 1

2 is given by a centered convex combination ofinternal traces at different time steps. In the figure, p = 3.

are known. We want to show that equations (72) allow us to determine the “next” unknowns, namely

Un+ 2k+1

2p

i , k ∈ 1, . . . , p, Un+ 1

2e . (74)

This will be done by showing that Un+ 1

2e is obtained by solving a linear system of the form

Mp(∆t) Un+ 1

2e = Gn+ 1

2 , (75)

where the right hand side Gn+ 12 is expressed in terms of known quantities in (73) and the matrix Mp(∆t) is invertible

under the CFL condition.

Then, once Un+ 1

2e is known, the quantities U

n+ 2k+12p

i , k ∈ 1, . . . , p, are computed “explicitly” (remember that M

is block diagonal) using the first p equations in (72). The reader will note that this process provides an algorithm forthe computation of the discrete solution, that we shall use for the implementation (see section 4.1).

Derivation of equation (75)

The computation consists in eliminating in the last equation in (72) the interior unknowns, using the first p equationsin (72). This is done through a technical lemma:

Lemma 3.1 Assume that Un+ 2k+1

2p

i , k = 1, · · · , p and Un+ 1

2e satisfy the first p equations in (72). Then

p∑

k=1

(Un+ 2k+1

2p

i + Un+ 2k−3

2p

i

)= Rp(X) U

n− 12p

i + Sp(X) Un+ 1

2p

i − ∆tp Pp(X) M−1C Un+ 1

2e + 2∆tp F

n+ 12 , (76)

18

where X = ∆tp M−1A ,Pk(·), Rk(·), Sk(·), k ≥ 0

are polynomials defined inductively by

Rk+1(x) = 2 + Sk(x), R0(x) = 0,

Sk+1(x) = Rk(x) − 2x (1 + Sk(x)) , S0(x) = 0,

Pk+1(x) = Pk(x) + 2 (1 + Sk(x)) , P0(x) = 0,

(77)

and where

Fn+ 1

2 :=

p∑

k=1

(I + Sp−k(X)) M−1F n+ 2k−12p . (78)

Proof: The idea is more generally to compute the sums

p∑

k=p+1−ℓ

(Un+ 2k+1

2p

i + Un+ 2k−3

2p

i

), ℓ = 1, · · · , p (79)

in function of Un+ 2(p+1−ℓ)−3

2p

i , Un+ 2(p+1−ℓ)−1

2p

i , Un+ 1

2e and

F n+ 2k−1

2p , p+ 1 − ℓ ≤ k ≤ p.

Note that the left hand side of (76) is nothing but (79) when ℓ = p.

The first p equations in (72) can be rewritten as

Un+ 2k+1

2p

i = Un+ 2k−3

2p

i − 2X Un+ 2k+1

2p

i − 2∆tp M−1C Un+ 1

2e + 2∆tp M−1 F n+ 2k−1

2p , k = 1, · · · , p. (80)

From (80) with k = p, we compute (79) for ℓ = 1, namely

Un+ 2p+1

2p

i + Un+ 2p−3

2p

i = 2Un+ 2p−3

2p

i − 2X Un+ 2p+1

2p

i − 2∆tp M−1C Un+ 1

2e + 2∆tp M−1 F n+ 2p−1

2p . (81)

Next, we make the ansatz that there exists polynomials Pk(·), Rk(·), Sk(·), Tk(·), k ≥ 0

such that for ℓ ∈ 2, · · · , pp∑

k=p+1−ℓ

(Un+ 2k+1

2p

i + Un+ 2k−3

2p

i

)= Rℓ(X) U

n+2(p+1−ℓ)−3

2p

i + Sℓ(X) Un+

2(p+1−ℓ)−12p

i

− ∆tp Pℓ(X) M−1C Un+ 1

2e

+ 2∆tp

p∑

k=p+1−ℓ

Tp+1−k(X) M−1F n+ 2k−12p ,

(82)

which we are going to prove by induction over ℓ. First note that (81) coincides with (82) for ℓ = 1 if we set:

P1(x) = 2x, R1(x) = 1, S1(x) = −2x, T1(x) = 1. (83)

Assume that (82) holds for some ℓ ∈ 1, · · · , p− 1. Let us show it for ℓ+ 1. Since

p∑

k=p−ℓ

(Un+ 2k+1

2p

i + Un+ 2k−3

2p

i

)=

p∑

k=p+1−ℓ

(Un+ 2k+1

2p

i + Un+ 2k−3

2p

i

)+

(Un+ 2(p−ℓ)+1

2p

i + Un+ 2(p−ℓ)−3

2p

i

),

using (82), we get, since 2(p+ 1 − ℓ) − 1 = 2(p− ℓ) + 1,

p∑

k=p−ℓ

(Un+ 2k+1

2p

i + Un+ 2k−3

2p

i

)= U

n+ 2(p−ℓ)−32p

i +Rℓ(X) Un+ 2(p+1−ℓ)−3

2p

i +(I + Sℓ(X)

)Un+ 2(p−ℓ)+1

2p

i

− ∆tp Pℓ(X) M−1C Un+ 1

2e + 2∆tp

p∑

k=p+1−ℓ

Tp+1−k(X) M−1F n+ 2k−12p .

(84)

19

Next, we get rid of Un+ 2(p−ℓ)+1

2p

i by using (80) with k = p− ℓ. Using 2(p+ 1 − ℓ) − 3 = 2(p− ℓ) − 1, we get

p∑

k=p−ℓ

(Un+ 2k+1

2p

i + Un+ 2k−3

2p

i

)=

(2I + Sℓ(X)

)Un+ 2(p−ℓ)−3

2p

i +(Rℓ(X) − 2X

(I + Sℓ(X)

))Un+ 2(p−ℓ)−1

2p

i

− ∆tp

(Pℓ(X) + 2

(I + Sℓ(X)

))M−1C U

n+ 12

e

+ 2∆tp

p∑

k=p+1−ℓ

Tp+1−k(X) M−1F n+ 2k−12p + 2∆tp

(I + Sℓ(X)

)M−1 F n+ 2(p−ℓ)−1

2p

(85)which is nothing but that (82) with ℓ replaced by ℓ+ 1 if we set Tℓ+1(x) = 1 + Sℓ(x) and :

Rℓ+1(x) = 2 + Sℓ(x), Sℓ+1(x) = Rℓ(x) − 2x (1 + Sℓ(x)) , Pℓ+1(x) = Pℓ(x) + 2 (1 + Sℓ(x)) . (86)

This concludes the proof since, if we assume that (86) also holds for ℓ = 0, (83) gives P0 = R0 = S0 = 0.

Substituting (76) into the last equation of (72) immediately leads to the

Corollary 3.1 Let

(Un− 2k+1

2p

i ,Un+ 1

2e

)be a solution of (72). Then at each time step, U

n+ 12

e is solution of (75) with

Mp(∆t) := B0T (∆t) +

∆t2p2

C∗Pp(X) M−1C, (87)

and where the right hand side Gn+ 12 depends linearly of U

n± 12p

i and (78).

The explicit expression of Gn+ 12 is not important for our purpose in this section. It could be useful for the implemen-

tation of the method but we shall proceed differently (see section 4.1).

Invertibility of the matrix Mp(∆t)

It will be proven provided that the CFL condition for the interior equations is satisfied. We first remark that, sincefor any integer k, (M−1A)k M−1 = M−1/2(M−1/2AM−1/2)k M−1/2, we can write

Pp(X) M−1 = M−1/2Pp(T) M−1/2, where T := ∆tp M−1/2 AM−1/2. (88)

and consequently

Mp(∆t) := B0T (∆t) +

∆t2p2

C∗ M−1/2 Pp(T) M−1/2 C. (89)

The invertibility of Mp(∆t) appears as a consequence of propositions 3.2 and 3.3 which express respectively that thematrices B0

T (∆t) and C∗ M−1/2 Pp(T) M−1/2 C are repectively positive definite and positive.

In what follows, we shall use the notation

u · v :=n∑

j=1

uj vj , ∀ (u,v) ∈ CN × CN , (90)

for the usual inner product in Cn and | · | for the associate euclidean norm.

Proposition 3.2 As soon as ∆t < 2T ∗(Ωi), where T ∗(Ωi) is defined by (24), the matrix B0T (∆t) is positive definite,

namely∀ V e ∈ RNRP , V e 6= 0, B0

T (∆t) V e · V e > 0. (91)

Proof: Let V e 6= 0 and vh,∆te the element of Xh,∆te whose vector of degrees of freedom is V e on the time interval

[0,∆t] and is zero elsewhere. Applying (119) for T = ∆t we have:

B0T (∆t)V e · V e = b∆t(v

h,∆te ,vh,∆te ). (92)

20

The result is then a direct consequence of the definition of T ∗(Ωi).

For our next result, it will be useful to use an explicit expression for the polynomials Pk(x). We begin with thepolynomials Sk(x). For simplicity of notation, we introduce:

r+(x) := − x+√

1 + x2, r−(x) := − x−√

1 + x2. (93)

Lemma 3.2 The polynomials Sk(·) defined in (77) are given by

Sk(x) =1 − x

x− 1

2x

[ (1 + r−(x)

)r+(x)k +

(1 + r+(x)

)r−(x)k

], ∀ x 6= 0. (94)

Proof: Eliminating the Rk’s using the first two equations of (77), we get:

∣∣∣∣∣∣

Sk(x) = −2xSk(x) + Sk−1(x) + 2(1 − x), (a)

S1(x) = − 2x, S0(x) = 0. (b)(95)

A particular solution of the recurrence equation (95).(a) is given by the constant sequence

Spk(x) =1 − x

x.

On the other hand, the general solution of the homogeneous recurrence relation associated with (95)(a) is

Shk (x) = A r+(x)k + B r−(x)k.

Thus, Sk(·) is of the form

Sk(x) =1 − x

x+ A r+(x)k + B r−(x)k.

Using the initial conditions (95)(b) allow us to compute A and B and conclude the proof.

Lemma 3.3 The polynomials Pk(·) defined in (77) are given by

Pk(x) =2k

x− 2

x2+

1

x2

[ (1 +

√1 + x2

)r+(x)k −

(1 −

√1 + x2

)r−(x)k

], ∀ x 6= 0. (96)

Proof: Using the last equation in (77) as well as (94) leads to

Pk(x) = 2k−1∑

j=0

(1 + Sj(x)

)= 2

k−1∑

j=0

1

x− 1

2x

[ (1 + r−(x)

)r+(x)j +

(1 + r+(x)

)r−(x)j

]

One easily concludes using

k−1∑

j=0

r±(x)j =1 − r±(x)k

1 − r±(x)and

r+(x) + r−(x) = −2x, r+(x) r−(x) = −2, r+(x) − r−(x) = 2√

1 − x2.

The details are left to the reader.

We shall also use the following general result

Lemma 3.4 Let P (x) be a polynomial with real coefficients and T be a real skew matrix of size N . Then P (T) ispositive in the sense that

P (T) v · v ≥ 0, ∀v ∈ RN ,

if and only ifℜ (P (λ)) ≥ 0, ∀λ ∈ σ(T) (the spectrum of T). (97)

21

Proof: We consider T as a linear operator in CN . From the fact that T is real and skew we know that, Ker T andR(T) denoting respectively the kernel and the range of T,

CN = Ker T ⊕ R(T), with dim R(T) = 2q

and that there exist q vectors

ek ∈ CN , k = 1, · · · , q

and q non zero real numbersµk, k = 1, · · · , q

such that:

e1, · · · , eq, e1, · · · , eq

is an orthonormal basis of R(T),

T ek = iµk ek, T ek = −iµk ek, k = 1, · · · , q.(98)

Let v ∈ RN , there exists v∗ ∈ Ker T and q complex numbersvk, k = 1, · · · , q

such that:

v = v∗ +

q∑

k=1

(vk ek + vk ek

)and P (T)v = P (0)v∗ +

q∑

k=1

(P (iµk)xk ek − P (−iµk) vk ek

)

Therefore, using P (−iµk) = P (iµk) since P has real coefficients,

P (T)v · v = P (0) |v∗|2 + 2

q∑

k=1

ℜP (iµk) |vk|2.

On concludes easily since σ(T) =0∪± i µk, k = 1, · · · , q

.

Proposition 3.3 Under the usual CFL condition for the interior problem, namely

Ccfl := supV i∈RNi

supW i∈RNi

∆tp A V i · W i

(M V i · V i)1/2 (M W i · W i)1/2

(= sup

vhi ∈Xhi

supwhi ∈X

hi

∆tp a(vhi ,w

hi )

m(vhi ,vhi )

1/2 m(whi ,w

hi )

1/2

)< 1,

(99)for any p ≥ 1, the matrix M−1/2Pp(T) M−1/2 with Pp(·) given by (96) and T given by (88), is positive.

Proof: It suffices to show the positivity of Pp(T), which can be done using lemma 3.4, since T is real and skew.

Since A V i · V i = T V i · V i and M V i · V i = V i · V i (resp. M W i · W i = W i · W i ) for V i = M−1/2 V i (resp.

W i = M−1/2 W i ) , we have, as T is normal

supV i∈RNi

supW i∈RNi

A V i · W i

(M V i · V i)1/2 (M W i · W i)1/2= sup

V i∈RNi

supW i∈RNi

T V i · W i

|V i| |W i|= max

λ∈σ(T)|λ| . (100)

Therefore, the CFL condition (99) means that, since T is real skew:

σ(T) ⊂iµ, µ ∈ (−1, 1)

.

Thus, according to lemma 3.4, it suffices to show that

ℜ (Pp(iµ)) ≥ 0, ∀ µ ∈ (−1, 1),

to conclude that Pp(T) is positive. By continuity, it suffices to consider µ = sin θ 6= 0. Using (96), we get

Pp(i sin θ) = − i2p

sin θ+

2

sin2 θ− 1

sin2 θAp(θ), Ap(θ) := (1 + cos θ) e−ipθ + (−1)p(1 − cos θ) eipθ,

which implies

ℜ(Pp(i sin θ)) =2 −ℜAp(θ)

sin2 θ. (101)

22

1. If p is even, Ap(θ) = (1 + cos θ) e−ipθ + (1 − cos θ) eipθ = 2 cos pθ − 2i cos θ sin pθ, which implies

ℜ(Pp(i sin θ)) = 21 − cos pθ

sin2 θ≥ 0.

2. If p is odd, Ap(θ) = (1 + cos θ) e−ipθ − (1 − cos θ) eipθ = − 2i sinpθ + 2 cos θ cos pθ, which yields

ℜ(Pp(i sin θ)) = 21 − cos θ cos pθ

sin2 θ≥ 0,

which concludes the proof.

As a direct consequence of propositions 3.2 and 3.3, we can state the

Corollary 3.4 Assuming that the interior CFL condition (99) is satisfied, for any p ≥ 1, the Mp(∆tp) defined by(87) (or (89)) is invertible. Thus, the discrete problem (72) admits a unique solution.

3.3.2 Stability of the discrete problem

The algorithm introduced in section 3.1 satisfies the energy balance in (66). We can now derive the following stabilityresult

Proposition 3.5 Let us assume that the time step ∆tp satisfies the CFL condition (99) and let

Un+ 2k+1

2p

i , (n, k) ∈ 0, . . . , N − 1 × 1, . . . , p, Un+ 1

2e n ∈ 0, . . . , N − 1,

be the solution of the discrete problem (72). We introduce (vne , πe) ∈ X ,h,∆t

e the (discrete) functions associated to the

degrees of freedom Un+ 1

2e and (pvn

e ,πe ,vvn

e ,πe) the solution of system (10) for this data on the interface Γ. Then, wehave

1

4

[m

(uN+ 1

2p

i ,uN+ 1

2p

i

)+ m

(uN− 1

2p

i ,uN− 1

2p

i

)]+

1

2

∑

l∈i,e

∫

Ωl

[1

ρ0c20p2vn

e ,πe(tN ) + ρ0|vvn

e ,πe |2(tN )

]dx ≤

1

1 − Ccfl

E0

i + ∆tp

N−1∑

n=0

p∑

k=1

f

u

n+ 2k+12p

i + un+ 2k−3

2p

i

2, tn+ 2k−1

2p

,

(102)where E0

i is the interior discrete energy at the initial time (see (59)). In particular, in absence of external forces, thebound in (102) is independent of N .

Proof: The numerical scheme has been built in such a way that (66) is satisfied. Using (15) we obtain that

bT (uh,∆te ,uh,∆te ) =1

2

∑

l∈i,e

∫

Ωl

[1

ρ0c20p2vn

e ,πe(T ) + ρ0|vvn

e ,πe|2(T )

]dx, (103)

From (59) and using the definition on (99) we get

ENi =1

4

[m

(uN+ 1

2p

i ,uN+ 1

2p

i

)+ m

(uN− 1

2p

i ,uN− 1

2p

i

)− 2∆tp a

(uN+ 1

2p

i ,uN− 1

2p

i

)]

≥ 1

4

[m

(uN+ 1

2p

i ,uN+ 1

2p

i

)+ m

(uN− 1

2p

i ,uN− 1

2p

i

)− 2 Ccfl m

(uN+ 1

2p

i ,uN+ 1

2p

i

) 12

m

(uN− 1

2p

i ,uN− 1

2p

i

) 12

]

≥ 1

4

[m

(uN+ 1

2p

i ,uN+ 1

2p

i

)+ m

(uN− 1

2p

i ,uN− 1

2p

i

)](1 − Ccfl) ,

(104)where we have also used the inequality 2ab ≤ a2 + b2. One easily obtains (102) from (103), (104) and (66).

Remark 3.3 Even if the bound in (102) blows up when Ccfl approaches the unity, in practice, one can chose ∆tp suchthat this constant is very close to one.

23

4 Numerical aspects on the coupling algorithm

4.1 Implementation of the energy coupling algorithm

In remark 3.2 we show that the matrices BkT (∆t) are non null for all positive values of k. In consequence, the numberof terms in the convolution in (72) increases with n. However, these matrices are k-independent for k large enough,

which allows to simplify the computations. Indeed, let us introduce the new unknownΠle

nl=0

defined by

Π0e := 0, Πn

e :=

n−1∑

l=0

Ul+ 1

2e ∆t, that implies U

n+ 12

e =Πn+1e − Πn

e

∆t. (105)

This provides

n∑

m=0

Bn−mT (∆t) Um+ 1

2e =

n∑

m=0

Bn−mT (∆t)Πm+1e − Πm

e

∆t=

n+1∑

m=1

Bn−mT (∆t) Πme ,

where

B0T (∆t) :=

B0T (∆t)

∆t, BlT (∆t) :=

BlT (∆t) − Bl−1T (∆t)

∆t, l ∈ 1, . . . , n. (106)

Note that, this time, the matrices BkT (∆t) are zero for k large enough. For this reason, in practice, we replace (72) bythe following equivalent system

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

Find Un+ 2k+1

2p

i and Πn+1e such that

MUn+ 2k+1

2p

i − Un+ 2k−3

2p

i

2∆tp+ A U

n+ 2k−12p

i + CΠn+1e − Πn

e

∆t= F n+ 2k−1

2p , k ∈ 1, . . . , p,

n+1∑

m=1

Bn−mT (∆t) Πme − C∗

p∑

k=1

Un+ 2k+1

2p

i + Un+ 2k−3

2p

i

2∆tp = 0.

(107)

We now provide an implementation of this coupling algorithm. Let us assume that

Um+ 2k+1

2p

i , (m, k) ∈ 0, . . . , n− 1 × 1, . . . , p, Πm+1e , m ∈ 0, . . . , n− 1,

have already been computed. The computations will be performed using a predictor-corrector algorithm.

We start introducing the (decoupled) unknowns Un+ 2k+1

2p

i,d , k ∈ −1, . . . , p that are the solution of the following

decoupled problem (which corresponds to the first set of equations in (107) when Πn+1e is removed)

∣∣∣∣∣∣∣∣

MUn+ 2k+1

2p

i,d − Un+ 2k−3

2p

i,d

2∆tp+ A U

n+ 2k−12p

i,d − CΠne

∆t= F n+ 2k−1

2p , k ∈ 1, . . . , p,

Un± 1

2p

i,d := Un± 1

2p

i .

(108)

Then, by linearity, the solution of problem (107) can be written as the addition of a correction term, that is,

Un+ 2k+1

2p

i = Un+ 2k+1

2p

i,d + Un+ 2k+1

2p

i,c , k ∈ −1, . . . , p.

These new unknowns Un+ 2k+1

2p

i,c , k ∈ −1, . . . , p are the solution of the following set of equations

∣∣∣∣∣∣∣∣∣∣∣

MUn+ 2k+1

2p

i,c − Un+ 2k−3

2p

i,c

2∆tp+ A U

n+ 2k−12p

i,c + CΠn+1e

∆t= 0, k ∈ 1, . . . , p,

n+1∑

m=1

Bn−mT (∆t) Πme − C∗

p∑

k=1

Un+ 2k+1

2p

i,c + Un+ 2k−3

2p

i,c

2∆tp = C∗

p∑

k=1

Un+ 2k+1

2p

i,d + Un+ 2k−3

2p

i,d

2∆tp,

(109)

24

with zero initial conditions, that is, Un± 1

2p

i,c := 0. It is important to notice that the right hand side of the p firstequations is identically equal to zero. Applying lemma 3.1 to the correction unknowns we get

p∑

k=1

(Un+ 2k+1

2p

i,c + Un+ 2k−3

2p

i,c

)= Rp(X) U

n− 12p

i,c︸ ︷︷ ︸= 0

+ Sp(X) Un+ 1

2p

i,c︸ ︷︷ ︸= 0

− ∆tp Pp(X) M−1CΠn+1e

∆t.

Replacing this expression into the last equation in (109) we get

B0T (∆t) Πn+1

e − ∆t2p2

C∗Pp(X) M−1CΠn+1e

∆t= C∗

p∑

k=1

Un+ 2k+1

2p

i,d + Un+ 2k−3

2p

i,d

2∆tp −

n∑

m=1

Bn−mT (∆t) Πme .

In consequence Πn+1e is the solution of the system

Mp(∆tp) Πn+1e = C∗

p∑

k=1

Un+ 2k+1

2p

i,d + Un+ 2k−3

2p

i,d

2∆tp −

n∑

m=1

Bn−mT (∆t)Πme , (110)

where Mp(∆tp) = Mp(∆tp)/∆t. Note that all the terms that are on the right hand side have been already computed.

Then we can organize the computations in the following three steps:

Prediction First of all we compute the interior decoupled unknowns Un+ 2k+1

2p

i,d , k ∈ 1, . . . , p, solution of (108).Note that those are fully explicit computations since the matrix M is diagonal or block diagonal.

Exterior solution We compute Πn+1e solving the linear system (110). In practice, the matrix Mp(∆tp) can be

computed, for example, using (87) and recursion (77).

Correction We finally compute Un+ 2k+1

2p

i,c , k ∈ 1, . . . , p, using the p first equations in (109) with zero initial con-ditions. This is a corrective term for the interior solution accounting for the coupling with the exterior. Note thatthis computation should be only done on the neighborhood of the boundary Γ which could accelerate this computations.

Remark 4.1 The correction step could be replaced by the computation of Un+ 2k+1

2p

i , k ∈ 1, . . . , p, solution of (107)since the exterior unknown has been already computed.

4.2 Numerical experiments for the energy preserving algorithm

To test the coupling algorithm we consider a model problem where the exact solution is known explicitly. We consider(1) (that is, free propagation without any obstacle) with T = 5, ρ = 1, c = 1 and f = 0. The system is perturbed byan initial condition on the pressure (v0 = 0) given by

p0(x, y, z) ≡ p0(r) = H

((r/r0) + 1

2

),

where r =√

(x− xc)2 + (y − yc)2 + (z − zc)2, r0 = 0.75, xc = (xc, yc, zc) = (0, 0, 0) and the (minimum 4-term)Blackman-Harris function [27], that is known to have a spectrum well concentrated around the origin, is defined fors ∈ [0, 1] (being zero elsewhere) by

H(s) = A0 −A1 cos(2πs) +A2 cos(4πs) −A3 cos(6πs), (111)

with A0 = 0.35875, A1 = 0.48829, A2 = 0.14128 and A3 = 0.01168. Note that the initial condition support is the ballcentered on xc with radius 0.75. The exact solution of this problem (that is radial) is

∣∣∣∣∣∣∣∣∣∣∣∣

p(x, y, z, t) ≡ p(r, t) =

1

2

(1 − t

r

)p0(r − t) +

1

2

(1 +

t

r

)p0(r + t), if r 6= 0,

p0(t) + t p′0(t), if r = 0,

v(x, y, z, t) ≡ v(r, t) = −∫ t

0

∇p((x, y, z), s) ds.

(112)

25

(a) Cube-Coarse: # RP Nodes = 218,# RP Triangles = 432, # DG Nodes =343, # DG Tetrahedra = 1296.

(b) Cube-Medium: # RP Nodes = 866,# RP Triangles = 1728, # DG Nodes =2197, # DG Tetrahedra = 10368.

(c) Cube-Fine: # RP Nodes = 3458,# RP Triangles = 6912, # DG Nodes= 15625, # DG Tetrahedra = 82944.

Figure 4: Meshes of the computational domain Ωi = (−1, 1)3 considered to compute the numerical rate of convergence.

The interior domain is Ωi = (−1, 1)3. For a given triangulation of the surface and a time step ∆t we discretize theexterior unknowns in the space Xh,∆t

e introduced in (55). We mesh the interior domain with tetrahedra. To simplifythe implementation of the coupling, we have considered a volume mesh whose trace coincides with the surface mesh(the general case being simply more technical). The interior approximation will be done on the space Xh

i introducedin (52) (k-th order polynomials). We remind that the time step in the interior is ∆tp = ∆t/p, p ∈ N \ 0 (p timessmaller than the one used by the exterior discretization). In this way we have four discretization parameters that wecan select: i) h, the surface mesh elements size (and in consequence the volume mesh elements size), ii) k the order ofthe polynomials in the interior discretization, iii) ∆t the time step for the exterior unknowns, iv) p the ratio betweenthe time discretization step on the exterior and in the interior. For every triple (h, k,∆t) this last discretizationparameter p will be chosen as the smallest integer such that ∆tp satisfies the usual CFL condition (99).

In order to compute the convergence rate of the method we have considered three meshes dividing the computationaldomain into n3 smaller cubes that we split into 6 tetrahedra (see figure 4) for n ∈ 6, 12, 24 corresponding toh ∈

√2/3,

√2/6,

√2/12. We measure the error for the interior unknowns on the interval [0, 5] using the maximum

of the energy norm over all the time steps on that interval, that is,

Er := maxtn+2k+1

2p ∈[0,5]

√√√√∫

Ωi

[(pn+ 2k+1

2p

i − p(tn+ 2k+12p )

)2

+

∣∣∣∣vn+ 2k+1

2p

i − v(tn+ 2k+12p )

∣∣∣∣2]

dx. (113)

We consider the two following discretization configurations:

• ∆t/h = 0.3 and k = 1 (first order polynomials). In this case, in order to satisfy the CFL condition in theinterior we choose p = 6,

• ∆t/h = 0.3 and k = 2 (second order polynomials). In this case, in order to satisfy the CFL condition in theinterior we choose p = 10.

We show the numerical results in figure 5. On the left, we plot (in log-log scale) the quatity Er maximum over timeof the energy norm of the error in Ωi (see (113)) versus the space discretization step in both cases (first and secondorder polynomials in the interior). In both configurations the method behaves like a first order method. This is notsurprising since we are using a low order approximation for the boundary equations (see remark 3.1).

To exhibit the (approximately) transparent nature of the artificial boundary we present (in figure 5 on the right) theenergy (in logarithmic scale) in Ωi when time evolves when ∆t/h = 0.3 and k = 2. For this computation we define

the energy at time step tn+ 2k+12p by

1

2

∫

Ωi

[∣∣∣∣pn+ 2k+1

2p

i

∣∣∣∣2

+

∣∣∣∣vn+ 2k+1

2p

i

∣∣∣∣2]

dx. (114)

As expected, when the mesh is refined, the energy of the numerical solution in Ωi converges for small times (less than1.75 for example) and rapidly goes to zero for a time large enough (larger than 2.5). We remind that the energy of the

26

10−0.9

10−0.7

10−0.5

10−2

h

Max

of

the

erro

r in

en

erg

y n

orm

Max of the error in energy norm. ∆ t/h = 0.3.

∆ t/h = 0.3, P1.

∆ t/h = 0.3, P2.

h

0 1 2 3 4 510

−8

10−6

10−4

10−2

Time

En

erg

y in

Ωi

Energy decay in Ωi. ∆ t/h = 0.3. P

2 polynomials.

h = 1/3 * 21/2

h = 1/6 * 21/2

h = 1/12 * 21/2

Figure 5: Conservative scheme. On the left we present the maximum of the energy norm of the error in Ωi for thethree meshes in figure 4 for k = 1 and k = 2 when ∆t/h = 0.3. Both configurations provide first order convergencerate. On the right we present the energy decay in Ωi for the three meshes in figure 4 when k = 2.

exact solution in Ωi for a time larger than 0.75 +√

3 ≈ 2.4821 is identically equal to zero since the initial condition iscompactly supported on the sphere centered at the origin with radius 0.75.

In order to have a better knowledge of the error committed we present four plots comparing the exact and the numericalsolutions. More precisely, we have:

(a) the evolution in time of (phi , πh,∆te , p) at P = (1, 0, 0), located on the boundary,

(b) the evolution in time of (vhi · n,v · n) at P ,

(c) the evolution in time of (phi , p) at Q = (2/3, 0, 0), located in the interior,

(d) the evolution in time of ((vhi )x, (v)x) at Q.

We represent the interior (DG) unknowns in blue, the surface (RP) unknowns in green and the exact solution in red.

In the first set of plots shown in figure 6.(a) the discretization parameters are h =√

2/6, ∆t/h = 0.3, p = 10 and k = 2.The potential corresponding to the trace of the pressure at the interface (which is a RP unknown and it is displayedin green) seems to be a reasonable approximation of the exact solution (represented in red). However, the DG part ofthe solution (in blue) seems to be less accurate. A closer look to the time signals allow to identify two sources of error:i) low-medium frequency oscillations (that might come from a too coarse discretization), ii) high frequency oscillations(that might come from the local time stepping in the interior domain), that in this case is the main part of the error.

In the second set of plots that we can see in figure 6.(b) we have refined the mesh by a factor two, that is, h =√

2/12,∆t/h = 0.3, p = 10 and k = 2. The results obtained in this case are better than those obtained with the previousdiscretization parameters; the method is converging. However, the high-frequency spurious phenomena on the DGsolution is still present (even if considerably reduced) and seems to be preponderant in the error.

To exhibit this phenomenon more clearly we have performed a discrete Fourier transform of the recorded RP and DGsignals at point P that we define by (we remind that p = 10 in this case)

Pe(ω) = ∆t∑

tn+12 ∈[0,5]

πh,∆te (tn+1/2) e−iω(n+1/2)∆t,

Pi(ω) =∆t

p

∑

t2k+12p ∈[0,5]

p2k+12p

i e−iω 2k+12p ∆t.

(115)

Notice that the first function Pe(ω), that is plotted in figure 7.(a), is 2π/∆t periodic; any other frequency not beingon the interval [−π/∆t, π/∆t] will be equivalent to a frequency on this interval. In figure 7.(b) we represent the

function Pi(ω) on the same interval. We cannot see almost any difference between both plots which means that thelow frequencies of both signals are almost identical. However, this second function is 2pπ/∆t periodic; in the interiorregion we are using a time step that is p times smaller. In figure 7.(c) we represent this function on the whole interval

27

0 1 2 3 4 5−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Time

Pre

ssu

re

Pressure at point (1,0,0).

DGRPExact

0 1 2 3 4 5−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Time

v ⋅

n

Normal velocity at point (1,0,0).

DGExact

0.5 1 1.5 2 2.5 3 3.5 4 4.5

−0.1

−0.05

0

0.05

0.1

Time

Pre

ssu

re

Pressure at point (2/3,0,0).

DGExact

0.5 1 1.5 2 2.5 3 3.5 4 4.5

−0.05

0

0.05

0.1

Time

v x

First component of the velocity at point (2/3,0,0).

DGExact

(a) h =√

2/6. ∆t/h = 0.3. p = 10. k = 2.

0 1 2 3 4 5−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Time

Pre

ssu

re

Pressure at point (1,0,0).

DGRPExact

0 1 2 3 4 5−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Time

v ⋅

n

Normal velocity at point (1,0,0).

DGExact

0.5 1 1.5 2 2.5 3 3.5 4 4.5

−0.1

−0.05

0

0.05

0.1

Time

Pre

ssu

re

Pressure at point (2/3,0,0).

DGExact

0.5 1 1.5 2 2.5 3 3.5 4 4.5

−0.05

0

0.05

0.1

Time

v x

First component of the velocity at point (2/3,0,0).

DGExact

(b) h =√

2/12. ∆t/h = 0.3. p = 10. k = 2.

Figure 6: Energy preserving coupling. In both sets of figures from left to right, from top to bottom (the evolution intime of): pressure at P , normal velocity at P , pressure at Q, first component of the velocity at Q.

28

−1 −0.5 0 0.5 1−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

× π/∆ t

FFT of the pressure at (1,0,0). RP unknown

ReIm

(a) Plot of the complex function bPe(ω), ω ∈ (−π/∆t, π/∆t)defined in (115).

−1 −0.5 0 0.5 1−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

× π/∆ t

FFT of the pressure at (1,0,0). DG unknown

ReIm

(b) Plot of the complex function bPi(ω), ω ∈ (−π/∆t, π/∆t)defined in (115).

−10 −8 −6 −4 −2 0 2 4 6 8 10−0.04

−0.02

0

0.02

0.04

× π/∆ t

FFT of the pressure at (1,0,0). DG unknown

ReIm

(c) Plot of the complex function bPi(ω), ω ∈ (−10π/∆t, 10π/∆t) defined in (115).

Figure 7: Discrete Fourier transform of the RP and DG signals in figure 6.(b) on the top on the left. For frequencies in[−π/∆t, π/∆t] (that are all the frequencies discretized by the RP method) both plots are almost identical (representedin (a) and (b)). When we represent the spectrum of the DG signal in [−10π/∆t, 10π/∆t] the aliasing phenomenamakes appear some spurious high frequency component of the solution (see the oscillations for frequencies around±2πk/∆t, k ∈ 1, . . . , 5 in plot (c).

[−10π/∆t, 10π/∆t] (the region in between the dashed lines coincides with figure 7.(b)) where we observe the highfrequency components on that signal that are located in a neighborhood of ω = ±2πk/∆t, k ∈ 1, . . . , 5 and that areclearly unphysical. This is the so-called aliasing phenomenon.

Remark 4.2 These spurious oscillations could be partially reduced considering a smaller ratio ∆t/h (that is, reducingthe time step on the boundary for a fixed mesh) since one could select a smaller p. However, the retarded potentialmethod becomes too expensive in terms of memory and CPU time when the ratio ∆t/h is too small.

5 Reduction of the aliasing phenomenon: Post-processing in time

As shown in sections 3.1 and 4.2, the coupling scheme is very robust (since it is stable by construction under theusual CFL condition) and provides a good approximation for the exterior unknowns. However, the interior unknownspresent a high frequency component that pollutes the approximation. This error can be seen as resulting from thealiasing phenomenon due to the local time stepping. One should notice that most of the coupling terms for the interiorequations (given by (63)) are off-centered and only first order consistent. Inspired by previous works on space-timemesh refinement techniques [28], we propose a post-processing in time of the interior solution in order to obtain a newcoupling algorithm with the same stability properties as the previous one but with a second order truncation errorin time on the coupling terms. This post-processing will act as filter reducing the amplitude of the high frequencycomponents of the interior solution related to the aliasing.

29

−10 −8 −6 −4 −2 0 2 4 6 8 10−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

× π/∆ t(a) Plot of the complex function H(ω), ω ∈ (−10π/∆t, 10π/∆t) defined in (117).

−10 −8 −6 −4 −2 0 2 4 6 8 10−0.04

−0.02

0

0.02

0.04

× π/∆ t

FFT of the pressure at (1,0,0). DG unknown

ReIm

(b) Result of multiplying the complex function bPi(ω) (defined in (115)) by H(ω).

Figure 8: The post-processing in (116) can be interpreted as a filtering process. When one applies the post-processingto the signal analyzed in the previous section (see figure 7) one gets a signal whose discrete Fourier transform isrepresented in (b). The high frequency phenomena related to the aliasing is removed while the accuracy for the lowfrequencies is not perturbed.

5.1 A post-processing by average in time. A new coupling algorithm

Let us introduce the post-processed unknowns as the following particular mean value

Un+ 1

2 + kp

i :=1

p

p∑

j=1

Un+ 2(k+j)−1

2p

i . (116)

This value has not been chosen randomly. First of all, it is important to perform the mean value over p time steps(see remark 5.1 for more insight). Moreover, note that this quantity directly appears in the second equation of (72).

Remark 5.1 Let us come back to the DG signal analyzed in the previous section (see figure 7). This post-processingwill reduce the aliasing phenomena since the relation between the discrete Fourier transform (see (115)) of a signalobtained by the conservative scheme and its equivalent when we perform the post-processing is a multiplication by thefunction

H(ω) =1

10

10∑

j=1

eiωπ2j−10−1

20 =sin(ωπ/2)

10 sin(ωπ/20), (117)

that we represent in figure 8.(a). In figure 8.(b) we represent the product of the functions Pi(ω) (defined in (115)) andH(ω). One can observe that the high frequency spurious phenomena due to the aliasing is reduced. We refer to thereader to [28] for more details on this topic.

30

Our goal is to deduce a set of equations directly providing these post-processed unknowns. Performing similar averagesto the interior equations in (107) one gets that the new unknowns are the solution of

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

Find Un− 1

2+ kp

i and Πn+1e such that for k ∈ 1, . . . , p,

MUn− 1

2+ k+1p

i − Un− 1

2 + k−1p

i

2∆tp+ A U

n− 12+ k

p

i + C

[k

p

Πn+1e − Πn

e

∆t+

(1 − k

p

)Πne − Πn−1

e

∆t

]= F

n− 12+ k

p ,

n+1∑

m=1

Bn−mT (∆t) Πme − C∗ U

n+ 12 + 1

p

i + Un+ 1

2−1p

i

2∆t = 0,

(118)

where Fn+ 1

2+ kp := 1/p

∑pj=1 F n+ 2(k+j)−1

2p . Note that all the coupling terms are, in this case, second order consistentin time (see figures 9 and 10) and we can thus expect this method to be more accurate. The new coupling algorithmis also stable under the usual CFL condition since it can be reinterpreted as a post-processing of the previous one. Adirect stability proof does not seem trivial.

Ωe

Ωi

t0 t13 t

23 t1 . . . tn−1 tn−

23 tn−

13 tn tn+ 1

3 tn+ 23 tn+1

Figure 9: Schematic representation of the coupling on the interior equations proposed by the post-processed scheme(see the first p equations in (107)). The values of the exterior trace appearing on the interior equations in the time

interval (tn−12 , tn+ 1

2 ) are second order consistent convex linear combinations of Un− 1

2e and U

n+ 12

e . In the figure, p = 3.

Ωe

Ωi

t0 t13 t

23 t1 . . . tn−1 tn−

23 tn−

13 tn tn+ 1

3 tn+ 23 tn+1

Figure 10: Schematic representation of the coupling on the exterior equation proposed by the post-processed scheme(see the last equation in (107)). The value of the interior trace in the interior equation at some instants on the interval(tn, tn+1) is given by centered linear combination of two interior traces. In the figure we have the case, p = 3.

5.2 Implementation of the new algorithm

The computations are very similar to those performed in section 4.1.

Assume that we have already computed

Um− 1

2+ k+1p

i , (m, k) ∈ 1, . . . , n− 1 × 1, . . . , p, Πm+1e , m ∈ 0, . . . , n− 1,

we sought

Un− 1

2 + k+1p

i = Un− 1

2+ k+1p

i,d + Un− 1

2+ k+1p

i,c , k ∈ 1, . . . , p, Πn+1e ,

31

where once again we have split the interior unknowns into a decoupled term and a corrective term. We proceed asfollows

Prediction The decoupled unknowns Un− 1

2+ k+1p

i,d , k ∈ 1, . . . , p, solution of

MUn− 1

2 + k+1p

i,d − Un− 1

2+ k−1p

i,d

2∆tp+ A U

n− 12+ k

p

i,d +

(1 − k

p

)C

Πne − Πn−1

e

∆t− k

pC

Πne

∆t= F

n− 12+ k

p ,

for k ∈ 1, . . . , p, with initial contitions given by

Un− 1

2+ kp

i,d := Un− 1

2+ kp

i , k ∈ 0, 1,

are computed first.

Exterior solution The unknown Πn+1e is the solution of the following linear system

Mp(∆tp) Πn+1e = C∗

Un+ 1

2+ 1p

i,d + Un+ 1

2−1p

i,d

2∆tp −

n∑

m=1

BTn−m(∆t) Πme .

Correction The last step is to compute the corrective term Un− 1

2+ k+1p

i,c , k ∈ 1, . . . , p, solution of

MUn− 1

2+ k+1p

i,c − Un− 1

2+ k−1p

i,c

2∆tp+ A U

n− 12+ k

p

i,c +k

pC

Πn+1e

∆t= 0, k ∈ 1, . . . , p,

with zero initial conditions.

6 Numerical experiments for the post-processed algorithm

6.1 Results with Ωi = (−1, 1)3

In this section we compare the energy preserving coupling algorithm with the new post-processed scheme. To do sowe repeat the same experiments presented in section 4.2 using the new algorithm. We refer the reader to this lastsection for the settings of the experiments.

We show the results in figure 11. On the left we represent the maximum of the energy norm of the error in Ωiversus the space discretization step when ∆t/h = 0.3 and we use first (blue line) and second (green dashed line) orderpolynomials in the interior. Once again in both cases the global discretization procedure provides first order precisionand the post-processing does not seem to improve the global accuracy, due to the fact that we are using a low orderdiscretization for retarded potentials. However, when observing the approximate solution at a given point we cansee some improvement. In figure 12 we present the same outputs as those presented in figure 6 (when we were usingthe conservative scheme) when the post-processed scheme is used. We remind that the discretization parameters are∆t/h = 0.3 and k = 2. When h =

√2/6 we already obtain good results and the high frequency spurious oscillations we

had with the conservative scheme are significantly removed (see figure 12.(a)). The results for h =√

2/12 are also verysatisfactory (see 12.(b)). If we compare them to those in figure 6.(b) we remark that the high frequency phenomenahas almost disappeared and that the solution for large values of time is very close to zero with very few oscillations.

Moreover, when we represent the energy decay in Ωi one can see a significant improvement (of several orders ofmagnitud) clearly provided by the post-processing (compare the right pictures in figures 11 and 5).

6.2 Results with a L shaped domain

We finally present a numerical experiment where the interior domain is not convex. The physical characteristics areidentical to those in the previous section except that the initial condition is centered at xc = (2, 0, 0) and T = 10. Theinterior domain is Ωi = (−1, 3)× (−1, 1)× (−1, 3) \ [1, 3]× [−1, 1]× [1, 3] (see figure 13.(a)). Note that this simulation

32

10−0.9

10−0.7

10−0.5

10−2

h

Max

of

the

erro

r in

en

erg

y n

orm

Max of the error in energy norm. ∆ t/h = 0.3.

∆ t/h = 0.3, P1.

∆ t/h = 0.3, P2.

h

0 1 2 3 4 510

−8

10−6

10−4

10−2

Time

En

erg

y in

Ωi

Energy decay in Ωi. ∆ t/h = 0.3. P

2 polynomials.

h = 1/3 * 21/2

h = 1/6 * 21/2

h = 1/12 * 21/2

Figure 11: Post-processed scheme. On the left we present the maximum of the energy norm of the error in Ωi for thethree meshes in figure 4 for k = 1 and k = 2 and ∆t/h = 0.3 when we discretize using the post-processed scheme.Both configurations provide first order convergence rate. On the right we present the energy decay in Ωi for the threemeshes in figure 4 when k = 2.

could not be done with absorbing boundary conditions or PML techniques.

For this simulation we have considered the mesh in 13.(a) with ∆t = 0.054 and second order polynomials for theinterior discretization. We choose p = 10 in order to satisfy the CLF condition in the interior domain. In figure 13.(b)we show the energy decay in Ωi that behaves as expected (we remind that for time large enough, the energy of thecontinuous problem is identically equal to zero).

In figure 14 we show some snapshots of the solution at times t = 1.80562, t = 2.70562 and t = 3.60562. Onthe first row we have the interior pressure phi , on the second row we have the pressure at the interface πh,∆te , on thethird row we have the first component of the velocity field (vhi )x and on the fourth row we have the normal velocityat the interface vh,∆te . There are no visible spurious reflexion generated by the artificial interface obtaining a goodapproximation of the solution.

7 Conclusion and future work

We have introduced a hybrid discretization for transient wave propagation problems on unbounded domains allowingto use a discontinuous Galerkin method on the region containing heterogeneities (called the interior domain and that isassumed to be bounded) and the retarded potential method to account for the unbounded nature of the computationaldomain. The coupling formula enforcing a discrete energy identity ensures the stability of the method when theusual CFL condition in the interior domain is satisfied. The algorithm permits non-conforming meshes in space andtime allowing to use both coupled methods with quasi-optimal discretization parameters. The spurious phenomenaintroduced by the non-conformity in time is removed by a post-processing by averaging in time of the interior unknownsyielding to a stable and accurate method. We have also shown the feasibility of the whole discretization procedurethrough some numerical experiments on a model problem. The mid-term perspectives are the extension of thismethod to more general equations such as hyperbolic symmetric Friedrichs systems (that would include linearizedEuler equations) and the complete convergence theory of the method. Another interesting aspect is the extension ofthis work to the use of dissipative DG methods, i.e. non centered fluxes. One could think that this would be helpful toimprove the accuracy of the method and to attenuate the spurious oscillations observed with the conservative method.However, this extension is not straightforward and the time-stepping strategy must be adequately modified. This workis in progress.

A Matrix formulation for the bilinear form bT ((ϕh,∆t, ψh,∆t), (ϕh,∆t, ψh,∆t))

This appendix is devoted to a technical proof of a lemma concerning the relation between the bilinear form bT (·, ·)and the matrices BkT (∆t) appearing in (70).

33

0 1 2 3 4 5−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Time

Pre

ssu

re

Pressure at point (1,0,0).

DGRPExact

0 1 2 3 4 5−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Time

v ⋅

n

Normal velocity at point (1,0,0).

DGExact

0.5 1 1.5 2 2.5 3 3.5 4 4.5

−0.1

−0.05

0

0.05

0.1

Time

Pre

ssu

re

Pressure at point (2/3,0,0).

DGExact

0.5 1 1.5 2 2.5 3 3.5 4 4.5

−0.05

0

0.05

0.1

Time

v x

First component of the velocity at point (2/3,0,0).

DGExact

(a) h =√

2/6. ∆t/h = 0.3. p = 10. k = 2.

0 1 2 3 4 5−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Time

Pre

ssu

re

Pressure at point (1,0,0).

DGRPExact

0 1 2 3 4 5−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Time

v ⋅

n

Normal velocity at point (1,0,0).

DGExact

0.5 1 1.5 2 2.5 3 3.5 4 4.5

−0.1

−0.05

0

0.05

0.1

Time

Pre

ssu

re

Pressure at point (2/3,0,0).

DGExact

0.5 1 1.5 2 2.5 3 3.5 4 4.5

−0.05

0

0.05

0.1

Time

v x

First component of the velocity at point (2/3,0,0).

DGExact

(b) h =√

2/12. ∆t/h = 0.3. p = 10. k = 2.

Figure 12: Energy preserving coupling. In both sets of figures from left to right, from top to bottom (the evolution intime of): pressure at P , normal velocity at P , pressure at Q, first component of the velocity at Q.

34

(a) L-Coarse: # RP Nodes = 1615, # RP Triangles= 3226, # DG Nodes = 3972, # DG Tetrahedra= 18460.

0 2 4 6 8 1010

−6

10−5

10−4

10−3

10−2

10−1

TimeE

ner

gy

in Ω

i

Energy decay in Ωi. ∆ t = 0.054. P

2 polynomials.

(b) Energy decay in Ωi.

Figure 13: On the left, mesh of the computational domain Ωi = (−1, 3)× (−1, 1)× (−1, 3) \ [1, 3]× [−1, 1]× [1, 3]. Onthe right, energy decay in Ωi when k = 2 and ∆t = 0.054.

Lemma A.1 For any (ϕh,∆t, ψh,∆t) ∈ Xh,∆te (resp. (ϕh,∆t, ψh,∆t) ∈ Xh,∆t

e ) with vectors of unknowns given by

(Un− 1

2e ) (resp. (U

n− 12

e )), n ∈ 1, . . . , N we have

bT ((ϕh,∆t, ψh,∆t), (ϕh,∆t, ψh,∆t)) =N∑

m=1

m∑

n=1

Bm−nT (∆t) Un

e · Um

e , (119)

where the matrix BkT (∆t) is defined by (71).

Proof: From equation (14) we deduce that the bilinear form bT (·, ·) is the sum of four contributions. Each of thesecontributions is nothing but the bilinear form associated to each integral operator whose analytical expression is givenin proposition 2.3. We will formally proof that the terms associated to the matrices BkT,Z(∆t) correspond to thebilinear form associated to the operator ZΓ. For a rigorous proof of this result one could adapt lemma 3 in [15].

Let (ϕh,∆t, ϕh,∆t) ∈[P0

∆t(R+;P0

h(Γ))]2

. We have thus

ϕh,∆t(x, t) =

N−1∑

n=1

Nϕ∑

j=1

Φn− 1

2

j ϕj(x) ηn− 12(t), ϕh,∆t(x, t) =

N−1∑

m=1

Nψ∑

i=1

Φm− 1

2

i ϕi(x) ηm− 12(t).

Using the first equation in (33) we have

∫ T

0

∫

Γ

ZΓ ϕh,∆t · ϕh,∆t dγ dt = ρ0

Nϕ∑

j=1

N∑

n=1

Nϕ∑

i=1

N∑

m=1

∫ T

0

∫

Γ

∫

Γ

•

ηn− 1

2(t− τ) ϕj(y) ηm− 1

2(t) ϕi(x)

4π|x − y| dγx dγy dt Φn− 1

2

j Φm− 1

2

i

=

N∑

n=1

N∑

m=1

Nn,mT,Z (∆t) Φn− 1

2 · Φm− 12 ,

where

Nn,mT,Z (∆t) := − ρ0

∫

Γ

∫

Γ

Φj(y) Φi(x)

4π|x − y| J1n,m(τ) dγx dγy, (120)

and

J1n,m(τ) =

∫ T

0

•

ηn− 1

2(t− τ) ηm− 1

2(t) dt = ηm−n− 1

2(τ) − ηm−n+ 1

2(τ) = χ[tm−n−1,tm−n](τ) − χ[tm−n,tm−n+1](τ).

(121)

35

Figure 14: Propagation on the L shape domain. Each column corresponds to (the increasing) times t ∈1.80562, 2.70562, 3.60562. On the first row: phi the pressure in the interior. On the second row: πh,∆te the pressureon the interface. On the third row: (vhi )x the first component of the velocity field in the interior. On the fourth row:vh,∆te the normal trace of the velocity on the interface.

36

Note that the time integral in the previous expression should be considered in a formal sense (we refer to the lemma3 in [15] for a rigorous proof of this result) since ηn− 1

2(·) is the characteristic function of the interval [tn−1, tn], thus

its time derivative•

ηn− 1

2(·) = δ(t− tn−1) − δ(t− tn),

where δ(·) is the Dirac delta. Plugging (121) into (120) one deduces that the dependence on m and n on the matrixNn,mT (∆t) is only through m− n and N

n,mT,Z (∆t) = Bm−n

T,Z (∆t).

Similar computations can be done for the bilinear forms YΓ, WΓ and W∗Γ.

Acknowledgments

This work was supported by AIRBUS through the contract ADNUMO. The third author (J. Rodrıguez) is alsosupported by the MCINN (Spanish Ministry of Science and Innovation) contract RYC-2008-03543.

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