Optimal Polynomial Decay to Coupled Wave Equations and Its Numerical Properties

Abstract

In this work we consider a coupled system of two weakly dissipative wave equations. We show that the solution of this system decays polynomially and the decay rate is optimal. Computational experiments are conducted in the one-dimensional case in order to show that the energies properties are preserved. In particular, we use finite differences and also spectral methods.

1. Introduction

This paper is concerned with the polynomial stability and the optimal rate of decay of the 𝐶₀-semigroup of the coupled system of two wave equations: 𝑢_𝑡𝑡−Δ𝑢+𝑢_𝑡+𝛼𝑣=0inΩ×]0,∞[,𝑣_𝑡𝑡−Δ𝑣+𝛼𝑢=0inΩ×]0,∞[,𝑢=𝑣=0onΓ×]0,∞[,(𝑢(𝑥,0),𝑣(𝑥,0))=𝑢₀,𝑣₀,𝑢_𝑡(𝑥,0),𝑣_𝑡(𝑥,0)=𝑢₁(𝑥),𝑣₁(𝑥),inΩ, (1)(2)(3)(4) where Ω is an open bounded set of ℝ^𝑛 with smooth boundary Γ. Here 𝛼 is a small real positive constant.

This problem is motivated by an analogous problem in ordinary differential equations for coupled oscillators and has potential application in isolation of objects from outside disturbances. As an example in engineering, rubber-like materials are used to absorb vibration or shield structures from vibration. Modeling of structures such as beam or plates sandwiched with rubber or similar materials will lead to equations similar to those (1)-(2).

The exponential stability of coupled system of wave equations has been studied by Najafi et al. [1] in the case of two linear boundary feedback and by Komornik Rao [2] in the case of two nonlinear boundary feedback.

We should mention that this initial boundary value problem (1)–(4) has been studied by many authors. For example, in [3] Alabau Boussoira proved that the energy of associated coupled weakly dissipative system decays polynomially as 𝑡⁻¹ and this decay rate can improve since the initial data are more regular. In [4] Alabau Boussoira et al. showed the decay rates for an abstract coupled weakly dissipative system of using the same ideas introduced in [3]. For the case of the system (1)–(4) we can also cite the work of Santos et al. [5] where the authors proved 𝑡⁻¹ polynomial decay and this decay rate can improve since the initial data are more regular. The question we are interested in is to study optimal polynomial decay rates to the system (1)–(4). That is, we prove that the associated semigroup decays at a rate 𝑡^−1/2 and it is optimal. This work improves the results in [3–5]. Our result on the polynomial stability is based on Theorem 2.4 in [6].

Naturally, the system (1)–(4) is dissipative. Indeed, the energy of the solutions defined by 𝐸(𝑡)1∶=2_Ω𝑢²_𝑡1𝑑𝑥+2_Ω|∇𝑢|²1𝑑𝑥+2_Ω𝑣²_𝑡+1𝑑𝑥2_Ω|∇𝑣|²𝑑𝑥+𝛼_Ω𝑢𝑣𝑑𝑥 (5) is decreasing in 𝑡∈(0,∞), since 𝑑𝐸(𝑡)𝑑𝑡=−_Ω𝑢²_𝑡𝑑𝑥≤0. (6)

To verify the asymptotic behavior of systems like (1)–(4), only the work of Najafi [7] treated numerically the problem of lack of exponential decay to a system of two coupled wave equations. Here, we use two numerical approaches: one is in finite differences and in the other one we use the spectral methods. In particular, we explore the properties of numerical energy associated with a particular discretization in finite differences.

We conclude this introduction with an outline of the paper. In Section 2, we show the existence of solutions for the system (1)–(4). In the Section 3 we showed that the system (1)–(4) is polynomially stable with optimal decay rate. Section 4 is dedicated with the numerical aspects. We derive a numerical energy and we show that it preserves the conservation and dissipation laws. We use also spectral methods to see the behavior of the numerical eigenvalues when the damping is weak. Finally, in Section 5, we finished this work with a conclusion.

2. The Semigroup Setting

In this section we will study the existence and uniqueness of strong and global solutions for the system (1)–(4) using the semigroup techniques.

Let us denote by ℋ=𝐻¹₀(Ω)×𝐿²(Ω)×𝐻¹₀(Ω)×𝐿²(Ω) the Hilbert space with internal product given by ⟨𝑈,𝑉⟩_ℋ=_Ω∇𝑢₁⋅∇𝑣₁+𝑢₂𝑣₂+∇𝑢₃⋅∇𝑣₃+𝑢₄𝑣₄+𝛼𝑢₁𝑣₃+𝑢₃𝑣₁𝑑𝑥, (7) where 𝑈=(𝑢₁,𝑢₂,𝑢₃,𝑢₄)^𝑇 and 𝑉=(𝑣₁,𝑣₂,𝑣₃,𝑣₄)^𝑇.

Let us consider the unbounded operator 𝒜 in the energy space ℋ with 𝐷(𝒜)=𝐻²(Ω)∩𝐻¹₀(Ω)×𝐻¹₀(Ω)×𝐻²(Ω)∩𝐻¹₀(Ω)×𝐻¹₀(Ω), (8) defined by 𝒜=.0𝐼00Δ−𝐼−𝛼𝐼0000𝐼−𝛼𝐼0Δ0 (9)

Putting 𝜑=𝑢_𝑡 and 𝜓=𝑣_𝑡, (1) and (2) can be written as the following initial value problem: 𝑑𝑈𝑈(0)𝑑𝑡=𝒜𝑈,=𝑈₀, (10) with 𝑈=(𝑢,𝜑,𝑣,𝜓)^𝑇 and 𝑈₀=(𝑢₀,𝑢₁,𝑣₀,𝑣₁)^𝑇. Using the internal product (7), we obtain ℛ𝑒⟨𝒜𝑈,𝑈⟩_ℋ=−_Ω𝜑²𝑑𝑥≤0. (11) Thus, 𝒜 is a dissipative operator and therefore, using the Lumer-Phillips Theorem (see [8], Theorem 4.3), the operator 𝒜 generates a 𝐶₀-semigroup of contractions 𝑆(𝑡)=𝑒^𝒜𝑡 on ℋ.

Thus, we have the following result.

Theorem 1. Let one assume that 𝑈₀∈𝐷(𝒜); then there exists a unique solution 𝑈=(𝑢,𝜑,𝑣,𝜓) to system (1)–(4) satisfying 𝑈∈𝐶ℝ⁺;𝐷(𝒜)∩𝐶¹ℝ⁺;ℋ. (12)

3. Polynomial Decay and Optimal Result

To study the asymptotic behavior of the semigroup associated with (1)–(4), let us consider the spectral problem: −Δ𝑤_𝜈=𝜆_𝜈𝑤_𝜈𝑤inΩ,_𝜈=0onΓ, (13) where lim_𝜈→∞𝜆_𝜈=+∞. (14)

Theorem 2. Let 𝑆(𝑡) be 𝐶₀-semigroup of contractions generated by 𝒜. Then it follows that 𝑆(𝑡) is not exponentially stable.

Proof. Our main tool is to use Prüss’ result [9], which states that a semigroup 𝑒^𝒜𝑡 is exponentially stable if and only if the following conditions hold: 𝑖ℝ⊂𝜌(𝒜)(resolventset),∃𝐾>0,∀𝑈∈𝐷(𝒜),∀𝜆∈ℝ∶(𝑖𝜆𝐼−𝒜)⁻¹_ℋ≤𝐾. (15)(16) This way, let us consider the resolvent equation (𝑖𝜆𝐼−𝒜)𝑈=𝐹,with𝜆∈ℝ,𝐹∈ℋ; (17) that is, 𝑖𝜆𝑢−𝜑=𝑓¹,𝑖𝜆𝜑−Δ𝑢+𝛼𝑣+𝜑=𝑓²,𝑖𝜆𝑣−𝜓=𝑓³,𝑖𝜆𝜓−Δ𝑣+𝛼𝑢=𝑓⁴. (18)(19)(20)(21)
Let us take 𝑓¹=𝑓³=0 and 𝑓²=𝑓⁴=𝑤_𝜈. We look for solution of the form 𝑢=𝑎𝑤_𝜈, 𝑣=𝑏𝑤_𝜈, 𝜑=𝑐𝑤_𝜈, and 𝜓=𝑑𝑤_𝜈, with 𝑎,𝑏,𝑐,𝑑∈ℂ. From (19)–(21), we get 𝑎 and 𝑏 satisfying −𝜆²(𝑎+𝑏)𝑤_𝜈+(𝑎+𝑏)𝜆_𝜈𝑤_𝜈+𝛼(𝑎+𝑏)𝑤_𝜈+𝑐𝑤_𝜈=2𝑤_𝜈. (22)
Now, choosing 𝜆=𝜆_𝜈+𝛼, and using the above equation, we obtain 𝑐=2, so we have 𝑎=−2𝑖𝜆_𝜈+𝛼,𝑏=2𝑖𝜆_𝜈+1+𝛼𝛼,𝑑=−2𝛼+𝑖𝜆_𝜈+𝛼𝛼. (23)
Therefore, we have 𝑢=−2𝑖𝜆_𝜈𝑤+𝛼^𝜈,𝑣=2𝑖𝜆_𝜈+1+𝛼𝛼𝑤_𝜈,𝜑=2𝑤_𝜈,𝜓=−2𝛼+𝑖𝜆_𝜈+𝛼𝛼𝑤_𝜈. (24)
Now we claim that ‖𝑈‖_ℋ⟶+∞,as𝑣⟶+∞. (25)
In fact, using (7) and (24) and noting that lim_𝜈→∞𝜆_𝜈=∞, (26) we conclude that ‖𝑈‖²_ℋ≥_Ω−2𝛼+𝑖𝜆_𝜈+𝛼𝛼𝑤_𝜈²𝑑𝑥⟶∞. (27)
Recalling that 𝑖𝜆𝑈−𝒜𝑈=𝐹⟺𝑈=(𝑖𝜆𝐼−𝒜)⁻¹𝐹, (28) it follows from (27) and (16) that 𝑆(𝑡) is not exponentially stable.

In order to show the polynomial decay from semigroup associated with the system (1)–(4), first let us consider the product in ℋ of 𝑈=(𝑢,𝜑,𝑣,𝜓)^𝑇∈𝐷(𝒜) with the resolvent equation of 𝒜; that is, 𝑖𝜆‖𝑈‖²_ℋ−(𝒜𝑈,𝑈)_ℋ=(𝐹,𝑈)_ℋ. (29) Now, noting that _Ω𝜑²𝑑𝑥=−(𝒜𝑈,𝑈)^ℋ, (30) we get 𝑖𝜆‖𝑈‖²_ℋ+_Ω𝜑²𝑑𝑥=(𝐹,𝑈)_ℋ. (31) Then taking the real part, we obtain _Ω𝜑²𝑑𝑥≤‖𝑈‖_ℋ‖𝐹‖_ℋ. (32)

Lemma 3. The strong solution of the system (1)–(4) given by Theorem 1 satisfies _Ω|∇𝑢|²𝑑𝑥+_Ω|∇𝑣|²𝑑𝑥+𝛼_Ω(𝑢𝑣+𝑣𝑢)≤𝑑𝑥_Ω𝜓²𝑑𝑥+𝐾‖𝑈‖_ℋ‖𝐹‖_ℋ, (33) where 𝐾 is a positive constant.

Proof. Multiplying equalities (19) and (21) by 𝑢 and 𝑣, respectively, integrating on Ω, and summing up the result we get 𝑖𝜆_Ω𝜑𝑢𝑑𝑥_∶=𝐼1+_Ω|∇𝑢|²𝑑𝑥+𝛼_Ω𝑣𝑢𝑑𝑥+𝑖𝜆_Ω𝜓𝑣𝑑𝑥_∶=𝐼2+_Ω|∇𝑣|²𝑑𝑥+𝛼_Ω𝑢=𝑣𝑑𝑥_Ω𝑓²𝑢𝑑𝑥+_Ω𝑓⁴𝑣𝑑𝑥. (34)
Substituting 𝑖𝜆𝑢 given in (18) into 𝐼₁ and 𝑖𝜆𝑣 given in (20) into 𝐼₂, we have _Ω|∇𝑢|²𝑑𝑥+_Ω|∇𝑣|²𝑑𝑥+𝛼_Ω(𝑣𝑢+𝑢𝑣)=𝑑𝑥_Ω𝜑²𝑑𝑥+_Ω𝜑²+𝑑𝑥_Ω𝜑𝑓¹𝑑𝑥+_Ω𝜓𝑓³+𝑑𝑥_Ω𝑓²𝑢𝑑𝑥+_Ω𝑓⁴𝑣𝑑𝑥. (35) Using (32) our conclusion follows.

Lemma 4. The strong solution of the system (1)–(4) given by Theorem 1 satisfies 𝑐1−𝜆_Ω𝜓²𝑐𝑑𝑥≤|𝜆|_Ω|∇𝑣|²𝑑𝑥+_Ω|∇𝑢|²|𝜆|𝑑𝑥+𝐾²‖𝑈‖_ℋ‖𝐹‖_ℋ, (36) where 𝐾 is a positive constant and |𝜆|>1 large enough.

Proof. Multiplying equality (21) by 𝜓 and integrating on Ω, we get 𝑖𝜆_Ω𝜓²𝑑𝑥=_ΩΔ𝑣𝜓𝑑𝑥−𝛼_Ω𝑢+𝜓𝑑𝑥_Ω𝑓⁴𝜓𝑑𝑥. (37) Using equality (20) and Poincaré and Young inequalities, we have _Ω𝜓²𝑐𝑑𝑥≤|𝜆|_Ω|∇𝑣|²𝑑𝑥+_Ω|∇𝑢|²+𝑐𝑑𝑥|𝜆|_Ω𝜓²𝑑𝑥+𝐾‖𝑈‖_ℋ‖𝐹‖_ℋ, (38) where 𝑐 is a positive constant and |𝜆|>1 is large enough. The proof is now complete.

Now, we are in the position to prove the main result of this paper.

Theorem 5. The semigroup associated with the system (1)–(4) is polynomially stable and 𝑆(𝑡)𝑈_0ℋ≤𝐾√𝑡𝑈_0𝐷(𝒜). (39)
Moreover, this result is optimal.

Proof. From Lemmas 3 and 4 and for |𝜆|>1, we have ‖𝑈‖_ℋ≤𝐾|𝜆|²‖𝐹‖_ℋ, (40) which is equivalent to (𝜆𝐼−𝒜)⁻¹≤𝐾|𝜆|². (41)
Then using Theorem 2.4 in [6], we obtain 𝑆(𝑡)𝒜⁻¹=𝒪𝑡^−1/2⟹𝑆(𝑡)𝒜⁻¹𝐹_ℋ≤𝐾√𝑡‖𝐹‖_ℋ. (42)
Since 0∈𝜌(𝒜), it follows that 𝒜 is onto over ℋ; then taking 𝒜𝑈₀=𝐹, we get 𝑆(𝑡)𝑈_0ℋ≤𝐾√𝑡𝑈_0𝐷(𝒜). (43) Therefore the solution decays polynomially.
To prove that the rate of decay is optimal, we will argue by contradiction. Suppose that the rate 𝑡^−1/2 can be improved; for example, the rate is 𝑡^{−1/(2−𝜖)} for some 0<𝜖<2. From Theorem 5.3 in [10], the operator |𝜆|^{−2+(𝜖/2)}(𝜆𝐼−𝒜)⁻¹_ℒ(ℋ) (44) should be limited, but this does not happen. For this, let us suppose that there exists a sequence (𝜆_𝜇)⊂ℝ with lim_𝜇→∞|𝜆_𝜇|=∞ and (𝑈_𝜇)⊂𝐷(𝒜) for (𝐹_𝜇)⊂ℋ such that 𝑖𝜆_𝜇𝐼−𝒜𝑈_𝜇=𝐹_𝜇 (45) is bounded in ℋ and lim_𝜇→∞𝜆_𝜇^{−2+(𝜖/2)}𝑈_𝜇ℋ=∞. (46) Then, we can consider, for each 𝜇∈ℕ, 𝐹_𝜇=(0,𝑤_𝜇,0,𝑤_𝜇)^𝑇 and 𝑈_𝜇=(𝑢_𝜇,𝜑_𝜇,𝑣_𝜇,𝜓_𝜇)^𝑇, where, due to the boundary conditions, 𝑈_𝜇 are in the form 𝑢_𝜇=𝑎𝑤_𝜇, 𝑣_𝜇=𝑏𝑤_𝜇, 𝜑_𝜇=𝑐𝑤_𝜇, and 𝜓_𝜇=𝑑𝑤_𝜇.
Then following the same steps as in the proof of Theorem 2 we can conclude that 𝜆_𝜇^{−2+(𝜖/2)}𝑈_𝜇ℋ≥𝑂𝜇^𝜖/2⟶∞,as𝜇⟶∞. (47) Therefore the rate cannot be improved. The proof is now complete.

4. Numerical Approaches

In this section, we give a number of numerical experiments to certify the analytical results (see Figure 15). We use two numerical approaches to this aim: the spectral method to compute the numerical eigenvalues and the explicit time integration method to compute the numerical solutions and to derive the associated numerical energy. In particular, the numerical energy preserves the conservation/dissipation of energy. Moreover, by means of numerical simulations we can note a lack of numerical exponential decay considering only damping (see Figure 17).

Our numerical simulations are realized in the one-dimensional case. Then we consider the following one-dimensional system: 𝑢_𝑡𝑡−𝑢_𝑥𝑥+𝛼𝑣+𝑢_𝑡=0,in(0,𝐿)×(0,𝑇),𝑣_𝑡𝑡−𝑣_𝑥𝑥+𝛼𝑢=0,in(0,𝐿)×(0,𝑇),𝑢(0,𝑡)=𝑢(𝐿,𝑡)=0,𝑣(0,𝑡)=𝑣(𝐿,𝑡)=0,0<𝑡<𝑇,𝑢(⋅,0)=𝑢₀(⋅),𝑢_𝑡(⋅,0)=𝑢₁(⋅),𝑣(⋅,0)=𝑣₀(⋅),𝑣_𝑡(⋅,0)=𝑣₁(⋅).∀𝑥∈(0,𝐿). (48)(49)(50)(51)

4.1. Explicit Time Integration Method

For our purpose we define a computational mesh defining Δ𝑥=𝐿/(𝐽+1) and Δ𝑡=𝑇/(𝑁+1) for 𝐽,𝑁∈ℕ and the nets given by 𝑥₀=0<𝑥₁=Δ𝑥<⋯<𝑥_𝐽=𝐽Δ𝑥<𝑥_𝐽+1𝑡=𝐿,₀=0<𝑡₁=Δ𝑡<⋯<𝑡_𝑁=𝑁Δ𝑡<𝑡_𝑁+1=𝑇, (52) where 𝑥_𝑗=𝑗Δ𝑥 and 𝑡_𝑛=𝑛Δ𝑡 for 𝑗=0,1,2,…,𝐽+1 and 𝑛=0,1,2,…,𝑁+1.

We consider the following finite-difference discretization of (48)-(49): 𝑢_𝑗^𝑛+1−2𝑢^𝑛_𝑗+𝑢_𝑗^𝑛−1Δ𝑡²−𝑢^𝑛_𝑗+1−2𝑢^𝑛_𝑗+𝑢^𝑛_𝑗−1Δ𝑥²+𝑢_𝑗^𝑛+1−𝑢_𝑗^𝑛−12Δ𝑡+𝛼𝑣^𝑛_𝑗=0,𝑣_𝑗^𝑛+1−2𝑣^𝑛_𝑗+𝑣_𝑗^𝑛−1Δ𝑡²−𝑣^𝑛_𝑗+1−2𝑣^𝑛_𝑗+𝑣^𝑛_𝑗−1Δ𝑥²+𝛼𝑢^𝑛_𝑗=0, (53)(54) for all 𝑗=1,2,…,𝐽, 𝑛=1,2,…,𝑁. Such scheme is in fact explicit and its implementation is straightforward. The boundary conditions are 𝑢^𝑛₀=𝑢^𝑛_𝐽+1=0,𝑣^𝑛₀=𝑣^𝑛_𝐽+1=0,∀𝑛=1,2,…,𝑁, (55) and the initial conditions are given by 𝑢⁰_𝑗=𝑢𝑥_𝑗,0,𝑢¹_𝑗=𝑢⁰_𝑗+Δ𝑡𝑢_𝑡𝑥_𝑗,0,∀𝑗=1,2,…,𝐽,𝑣⁰_𝑗=𝑣𝑥_𝑗,0,𝑣¹_𝑗=𝑣⁰_𝑗+Δ𝑡𝑣_𝑡𝑥_𝑗,0,∀𝑗=1,2,…,𝐽. (56)(57)

The discretizations above are consistent with truncation error of second order in Δ𝑥 and Δ𝑡. Therefore, it is immediate by Lax Lemma [11] that (53)-(54) converges. For convenience, we use the stability criterion (taking into account the conservative case) due to Courant-Friedrichs-Lewy; namely, Δ𝑡≤Δ𝑥.

4.2. Energy Conserving Scheme

In this section we will build the numerical energy to system (53)–(57). Following the techniques performed, for example, in [12, 13], this energy is given by 𝐸^𝑛∶=Δ𝑥2^𝐽_𝑗=0𝑢_𝑗^𝑛+1−𝑢^𝑛_𝑗Δ𝑡²+𝑢^𝑛+1_𝑗+1−𝑢_𝑗^𝑛+1𝑢Δ𝑥^𝑛𝑗+1−𝑢^𝑛_𝑗+𝑣Δ𝑥_𝑗𝑛+1−𝑣^𝑛_𝑗Δ𝑡²+𝑣^𝑛+1_𝑗+1−𝑣_𝑗^𝑛+1𝑣Δ𝑥^𝑛𝑗+1−𝑣^𝑛_𝑗𝑣Δ𝑥+𝛼^𝑛_𝑗𝑢_𝑗^𝑛+1+𝑢^𝑛_𝑗𝑣_𝑗^𝑛+1. (58)

We note that 𝐸^𝑛 is the discrete version of the continuous energy (5) in one-dimensional setting. Moreover, one can show that 𝐸^𝑛 decreases. Instead of computing the time derivative of the energy we can use the summation by parts. The discrete energy 𝐸^𝑛 is an important numerical instrument to certify our analytical results concerning the polynomial stabilization of system (53)–(57).

We have the following result.

Theorem 6 (discrete energy). Let (𝑢^𝑛_𝑗,𝑣^𝑛_𝑗) be a solution of the finite difference scheme (53)–(57). Then for all Δ𝑡 and Δ𝑥, the discrete rate of change of energy of the numerical scheme (53)–(57) at the 𝑡_𝑛 instant of time is given by 𝐸^𝑛−𝐸^𝑛−1Δ𝑡=−^𝐽_𝑗=1𝑢_𝑗^𝑛+1−𝑢_𝑗^𝑛−12Δ𝑡², (59) for all 𝑛=1,…,𝑁,𝑁+1.

Proof. We shortened the proof. It is identical to the proof of Proposition 1 in [12]. Then, we can write 𝐸^𝑛+Δ𝑡^𝐽_𝑗=1𝑢_𝑗^𝑛+1−𝑢_𝑗^𝑛−12Δ𝑡²=𝐸^𝑛−1, (60) from where we have 𝐸^𝑛≤𝐸⁰,∀𝑛=1,…,𝑁,𝑁+1. (61)
Considering the conservative case we get 𝐸^𝑛=𝐸⁰,∀𝑛=1,2,…,𝑁. (62)

4.3. Discussion on Numerical Results

In this section, we consider two numerical approaches to certify the analytical results established in this work: the pseudospectral method to compute numerically the eigenvalues and also using the numerical energy 𝐸^𝑛 in (58).

For spectral analysis we apply the spectral method according to Trefethen’s book [14]. Using this numerical technique, we can see the numerical behavior of eigenvalues of (48)-(49). To see how this system is affected by a lack of exponential decay by means of several numerical simulations, we can consider two damping 𝛽₁𝑢_𝑡 and 𝛽₂𝑣_𝑡 where 𝛽₁,𝛽₂≥0. Then, making 𝛽₂ go to zero, we can see a numerical sequence of eigenvalues converging to zero.

We consider 𝛽₁=1 and we take 𝛽₂ going to zero. Figures 1, 2, 3, and 4 show that when 𝛽_𝑖≠0, 𝑖=1,2, the real part of complex roots approaches negative asymptotes −𝛽_𝑖/2, 𝑖=1,2.

Figure 1 

𝛽1=1, 𝛽2=1.

Figure 2 

𝛽1=1, 𝛽2=0.8.

Figure 3 

𝛽1=1, 𝛽2=0.6.

Figure 4 

𝛽1=1, 𝛽2=0.4.

On the other hand, in Figures 5 and 6, we can see a sequence of eigenvalues converging to 0 as 𝛽₂→0.

Figure 5 

𝛽1=1, 𝛽2=0.04.

Figure 6 

𝛽1=1, 𝛽2=0.

Next, we show results of numerical simulations of scheme (53)–(57) and its energy 𝐸^𝑛. Here we use the stability criterion Δ𝑡≤Δ𝑥.

4.4. Undamped Case

Here, we consider 𝛼=0.25, 𝐿=1, 𝑇=2s, 𝑢(𝑥_𝑗,0)=𝑣(𝑥_𝑗,0)=2sin(3𝜋𝑥_𝑗/𝐿), 𝑢_𝑡(𝑥_𝑗,0)=0, and 𝑣_𝑡(𝑥_𝑗,0)=3sin(3𝜋𝑥_𝑗/𝐿). At continuous level we obtain that 𝐸(0) ≈ 180,4029.

Figures 7, 8, 9, and 10 show the conservative behavior of numerical energy 𝐸^𝑛 to undamped system. We can see the conservative behavior of the energy 𝐸^𝑛 in agreement with Proposition 4.1. Moreover, it is clear that the accuracy of the approximations is achieved by means of refinements of type 𝜇∶=Δ𝑡/Δ𝑥≤1.

Figure 7 

Δ𝑡/Δ𝑥=0.9600.

Figure 8 

Δ𝑡/Δ𝑥=0.8960.

Figure 9 

Δ𝑡/Δ𝑥=0.7680.

Figure 10 

Δ𝑡/Δ𝑥=0.7168.

A simple analysis, by using absolute error 𝜖∶=|𝐸(0)−𝐸⁰| for each simulation, shows us the precision of the numerical energy. Table 1 shows the evolution of this error for each simulation.

Table 1 

4.5. Full and Partial Damping

It is clear that considering two damping to the system (53)-(54) its numerical energy 𝐸^𝑛 decays exponentially. See Figures 11 and 13. Therefore, in this case the energy 𝐸^𝑛 is controllable by an exponential of type 𝑒^{−𝜔𝑡𝑛} for some 𝜔>0. On the other hand, taking into account damping only one dissipative mechanism we observe a slow decay from value ≈180 to value ≈90 (see Figures 12 and 14). We can note a lack of 50 by cent in terms of exponential decay.

Figure 11 

Δ𝑡/Δ𝑥=0.9600.

Figure 12 

Δ𝑡/Δ𝑥=0.960.

Figure 13 

Δ𝑡/Δ𝑥=0.8960.

Figure 14 

Δ𝑡/Δ𝑥=0.8960.

Figure 15 

Partial damping.

Note that the numerical solutions of the function 𝑣 do not go to the state of rest. See Figures 16 and 18.

Figure 16 

Partial damping.

Figure 17 

Partial damping.

Figure 18 

Partial damping.

5. Conclusions

In this work we considered a coupled system of two weakly dissipative wave equations. That is to say, the damping mechanism acts partially on the system and it is well known that this system is not exponentially stable. Our main contribution is twofold: the optimality of the polynomial decay in accordance with a recent theoretical contribution due to Borichev and Tomilov [6] and its numerical certification by using finite differences method. In particular, we derive a numerical energy and we show the lack of exponential decay by means of several numerical experiments.

The analytical and numerical techniques performed in this work can be applied to systems where there exists a lack of exponential decay. See, for example, the work of Santos et al. for examples of systems with the property of lack of exponential decay and polynomial decay.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to acknowledge the support of the following corporations: UFPA, PROPESP, and FADESP through PAPQ. The authors thank the anonymous referees for their valuable comments as well as reference suggestions that helped to improve the presentation of this paper.