Local absorbing boundary conditions for nonlinear

Local absorbing boundary conditions for nonlinear wave equation on unbounded domain Hongwei Li∗ and Xiaonan Wu† Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong, People’s Republic of China

Jiwei Zhang‡ Courant Institute of Mathematics Sciences, New York University, New York, NY 10012, USA (Dated: August 9, 2011) The numerical solution of nonlinear wave equation on unbounded spatial domain is considered. The artificial boundary method is introduced to reduce the nonlinear problem on unbounded spatial domain to an initial boundary value problem on a bounded domain. Using unified approach, which is based on operator splitting method, we construct the efficient nonlinear local absorbing boundary conditions for the nonlinear wave equation, and give the stability analysis of the resulting boundary conditions. Finally, several numerical examples are given to demonstrate the effectiveness of our method. I.

INTRODUCTION

We consider the numerical solution of nonlinear wave equation on the unbounded spatial domain utt = a2 ∆u + f (u), (x, y) ∈ R2 , 0 < t ≤ T, (1) 2 u|t=0 = φ0 (x, y), ut |t=0 = φ1 (x, y), (x, y) ∈ R , (2) where ∆ denotes the two-dimensional spatial Laplace operator, u(x, y, t) represents the wave displacement, a is the given reference wave speed and φ0 (x, y), φ1 (x, y) are the initial values. The nonlinear force term f (u) here can be endowed with different forms, for example, −f (u) = sin(u), sin(u) + sin(2u), sinh(u) + sinh(2u), u3 − u et al., and Eq. (1) is called sine-Gordon equation, double sine-Gordon equation, double sinh-Gordon equation and φ4 equation, respectively. This equation rises in various areas of physical applications [1–4], such as the propagation of high-intensity laser pulses in plasmas and fluxons in the Josephson junctions, the motion of rigid pendula attached a stretched wire, electromagnetics, meteorology, solid geophysics, and dislocations in crystals. On the bounded spatial domain, studies have focused attention on the numerical scheme of the nonlinear wave equation (1) with Dirichlet, Neumann or periodic boundary condition. Bratsos [5] proposes a three time-level fourth-order explicit finite difference scheme for sineGordon equation. Djidjeli et al. [6] give a two-step oneparameter leapfrog scheme to simulate the damped sineGordon equation. Q. Sheng et al. [7] obtain the numerical solution of two-dimensional sine-Gordon equation via a split cosine scheme. In the numerical treatment of the problem (1)-(2) on unbounded domain, the given model need to be restricted to an appropriate bounded computational do-

∗ † ‡

[email protected] [email protected] [email protected]

main. There exist some solutions to address the unboundedness issue, such as the perfectly matched layer method [8], infinite element or boundary method [9], and the artificial boundary method [10–12, 14]. Here we use the artificial boundary method to reduce the problem on unbounded domain to a bounded (computational) domain, thus appropriate boundary conditions have to be constructed. Ideally, they should not only be easy to implement, but also imitate the perfect absorption of waves traveling out of the computational domain through the artificial boundary. For linear problems on unbounded spatial domain, many mathematicians, engineers and physicists have well developed the artificial boundary method in last three decades, refer to [10, 12, 13, 15, 16] and the review papers [17–21]. For nonlinear problems, it is in general difficult to construct such ideal absorbing boundary conditions (ABCs). Resorting to a linearized method, the exact artificial boundary conditions are reported in [22, 23] for certain nonlinear PDEs. To obtain local nonlinear ABCs, a novel unified approach is proposed in [24, 25], it has been successfully applied to problems like the nonlinear Schrödinger equations and semilinear parabolic equations [26, 27]. The basic idea underlying the unified approach is expressed in Sec. II for a great detail. For the nonlinear wave equation on unbounded domain, Han and Yin [28] derive the exact ABCs of the multidimensional Klein-Gordon equation. Dehghan and Shokri [2] solve the nonlinear Klein-Gordon equation using thin plate splines radial basis functions. Zheng [29] constructs exact non-reflecting boundary conditions (also called nonlocal ABCs) using a new DtN mapping for onedimensional sine-Gordon equation. Lindquist et al. [30] introduce spectral elements and high-order non-reflecting boundary conditions to solve Klein-Gordon equation. Hagstrom et al. [31] propose high-order local ABCs for two-dimensional nonlinear wave equation based on a modification of the ABCs of Higdon. Han and Zhang [4] used the operator splitting method to get the split local ABCs for Klein-Gordon equation. This paper aims

2 to extend the unified approach to construct the nonlinear local ABCs for general nonlinear wave equation (1) for one-dimensional and two-dimensional cases. The organization of the paper is as follows. In Sec. II, we give a detailed overview of the novel unified approach, and recall the local ABCs of Higdon for linear wave equation, then construct the local ABCs for nonlinear wave equation by extending unified approach and obtain an initial boundary value problem (IBVP). In Sec. III, we analyze the stability of the reduced IBVP by constructing an energy functional. The finite difference scheme is presented for the reduced IBVP in Sec. IV. In Sec. V, numerical examples are given to demonstrate the effectiveness and tractability of the proposed method, and, finally, concluding remarks are given in Sec. VI.

˜ for First of all, we derive the approximation operator L the linear operator L by making wave outgoing of the computational domain, then replace the operator L in (4). By restricting to the artificial boundaries and letting τ tend to zero, we arrive at one-way equation ˜ + PU. Ut = LU (5)

II. DESIGN OF NONLINEAR LOCAL ABSORBING BOUNDARY CONDITIONS

utt = a2 ∆u,

Review of Unified Approach

(3)

where L is the linear operator and P represents the nonlinear operator that governs the effect of the nonlinearity. The operators are given by v 0 LU := and PU := . a2 (uxx + uyy ) f (u) The well-known time-splitting approach means that the wave propagation carries out the action of a kinetic energy step and a potential energy step separately for a small time size τ . Hence, in a time interval from t to t+τ for small τ , we use the operator splitting U (x, y, t + τ ) ≈ e(L+P)τ U (x, y, t), in analog of the well-known Strang splitting [32] U (x, y, t + τ ) ≈ eLτ /2 ePτ eLτ /2 U (x, y, t).

Local Absorbing Boundary Conditions for Linear Wave Equation

In this subsection we recall the construction of LABCs for linear wave equation of the following form: (6)

Ut ≡ (ut , vt )T = LU.

(7)

In [34, 35], Higdon proposed the following LABC on the east boundary p Y cos θl ∂t + a∂x u = 0, (8) l=1

Unified approach is based on the well-known timesplitting method. The idea underlying the unified approach is to distinguish between incoming and outgoing waves along the boundaries for the linear subproblem, and to approximate the corresponding linear operator by using a ‘one-way operator’ (to make the wave outgoing), then unite the approximation operator and the nonlinear subproblem to yield nonlinear boundary conditions. Now, we construct the LABCs of Eq. (1) by extending the unified approach. Setting v = ut , U = [u, v]T , Eq. (1) can be equivalently written in the operator form Ut ≡ (ut , vt )T = LU + PU,

B.

which is equivalent to the operator form

In this section we are devoted to the derivation of the local absorbing boundary conditions (LABCs) for Eq. (1) by extending the unified approach described in [24, 25].

A.

Once the approximation operator is given, Eq. (5) will ˜ is discussed play the role of LABCs. The derivation of L in the following subsection.

(4)

where p is an integer and ±θl (l = 1, 2, · · · , p) are angles between the wave incident direction and the normal direction of the boundary (Here |θl | < π2 for all l). In the practical numerical simulation, it is expensive to calculate for very large p, instead, we simply select p = 2, on the east boundary we have cos θ1 cos θ2 utt + a(cos θ1 + cos θ2 )uxt + a2 uxx = 0. (9) Using the relation v = ut and setting 1 0 Q= , a(cos θ1 + cos θ2 )∂x cos θ1 cos θ2 Eq. (9) can be equivalently rewritten by v −1 ˜ Ut = Q ≡ LU. −a2 uxx

(10)

Comparing (7) with (10), we obtain the approximation ˜ for the operator L with operator L 0 1 −1 ˜ L ≈ L := Q . (11) −a2 ∂x2 0 By the same argument, the corresponding approxima˜ on the other artificial boundaries can be derived tion L from the corresponding west, north and south LABCs, which are respectively governed by cos θ1 cos θ2 utt − a(cos θ1 + cos θ2 )uxt + a2 uxx = 0, cos θ1 cos θ2 utt + a(cos θ1 + cos θ2 )uyt + a2 uyy = 0, cos θ1 cos θ2 utt − a(cos θ1 + cos θ2 )uyt + a2 uyy = 0.

For other strategies to design LABCs for linear wave equation, one refers to [4, 33] and references therein.

3 C.

Local Absorbing Boundary Conditions for Nonlinear Wave Equation

cos θ1 cos θ2 |f (u)| ≤ c|u| and 2(cos θ1 +cos θ2 ) ≤ a, then we have the following energy estimate:

We construct LABCs for nonlinear wave equation (1), and here discuss the derivation of LABC on the east boundary in detail. Recalling Eq. (5) in the procedure of the unified approach and the approximation operator ˜ of (11), we have the one-way equation L ˜ + PU. Ut = LU

(12)

Multiplying the operator Q to (12) and using v = ut , we thus arrive at the LABC of nonlinear wave equation on the east boundary: cos θ1 cos θ2 utt + a(cos θ1 + cos θ2 )uxt + a2 uxx − cos θ1 cos θ2 f (u) = 0. (13) Similarly, we obtain the corresponding LABCs on the west, north and south boundary: cos θ1 cos θ2 utt − a(cos θ1 + cos θ2 )uxt + a2 uxx − cos θ1 cos θ2 f (u) = 0, (14) cos θ1 cos θ2 utt + a(cos θ1 + cos θ2 )uyt + a2 uyy − cos θ1 cos θ2 f (u) = 0, (15) cos θ1 cos θ2 utt − a(cos θ1 + cos θ2 )uyt + a2 uyy − cos θ1 cos θ2 f (u) = 0. (16) Coupling those LABCs (13)-(16) with the nonlinear wave equations (1)-(2), we have the reduced IBVP on the bounded computational domain. III.

STABILITY ANALYSIS OF ABSORBING BOUNDARY CONDITIONS

Energy estimate is a well-known tool to prove the stability of partial differential equations. Ha-Duong and Joly [13] have showed strong well-posedness for hyperbolic IBVP by constructing an “energy” function which decays in time. The “energy” is not necessarily the physical one. In this section, we show that appropriate energy method leads to the stability result for nonlinear wave equation with our nonlinear LABCs. To prove the stability of the obtained IBVP, we only need to test the stability of the boundary condition on the east, since we can extend the idea to achieve the stability of boundary conditions on other boundaries. Throughout this section, the symbol c will denote a generic positive constant, not necessarily the same at different occurrences. We consider the nonlinear wave equation (1) and the east boundary condition (13) on the bounded domain. We have the following stability result: Theorem: Assume that the initial values satisfy (φ0 , φ1 ) ∈ H 2 (Ω) × H 1 (Ω), |F (ux )| ≤ M , |F (ut )| ≤ M ,

E(u; t) ≤ ect E(u; 0),

0 < t ≤ T,

(17)

where

E(u; t) =

Z

Ω

[cos θ1 cos θ2 u2tt + a2 cos θ1 cos θ2 (∇ut )2

+2 cos θ1 cos θ2 (M − F (ut )) + a2 u2xt + a4 (∇ux )2 +2a2 (M − F (ux )) + u2 + (∇u)2 + u2t ]dxdy. (18) M is a positive constant, (M − F (u))′ = f (u), Ω = {(x, y)|xw < x < xe , ys < y < yn }. Proof: Multiplying both sides of nonlinear wave equation (1) by ut , integrating on Ω, we have

Z 1 d ( [u2t + a2 (∇u)2 + 2(M − F (u))]dxdy) 2 dt Ω Z = ux ut dy, (19) ΓE

where ΓE = {(x, y, t)|x = xe , ys < y < yn , 0 ≤ t ≤ T }. Substituting u = ut into (19), we have

Z 1 d ( [u2tt + a2 (∇ut )2 + 2(M − F (ut ))]dxdy) 2 dt Ω Z = uxt utt dy, (20) ΓE

substituting u = ux into (19), we have

Z 1 d ( [u2xt + a2 (∇ux )2 + 2(M − F (ux ))]dxdy) 2 dt Ω Z = uxt uxx dy. (21) ΓE

Multiplying (20) by cos θ1 cos θ2 and (21) by a2 , then adding the two resulting equations, and using to (13) on

4 the east boundary, we get Z 1 d ( [cos θ1 cos θ2 u2tt + a2 cos θ1 cos θ2 (∇ut )2 2 dt Ω +2 cos θ1 cos θ2 (M − F (ut )) + a2 u2xt + a4 (∇ux )2 +2a2 (M − F (ux ))]dxdy) Z = (cos θ1 cos θ2 uxt utt + a2 uxt uxx )dy ΓE Z =− (a(cos θ1 + cos θ2 )uxt − cos θ1 cos θ2 f (u))uxt dy ZΓE =− a(cos θ1 + cos θ2 )u2xt dy ΓE Z 1 − cos θ1 cos θ2 ((uxt − f (u))2 − u2xt − f 2 (u))dy 2 ΓE Z 1 = (−a(cos θ1 + cos θ2 ) + cos θ1 cos θ2 )u2xt dy 2 ΓE Z 1 − cos θ1 cos θ2 (uxt − f (u))2 dy 2 ΓE Z 1 + cos θ1 cos θ2 f 2 (u)dy 2 ΓE Z Z ≤c f 2 (u)dy ≤ c u2 dy ΓE ΓE Z ≤ c (u2 + (∇u)2 )dxdy. (22) Ω

The first inequality of (22) is obtained when the paramcos θ1 cos θ2 eters θ1 and θ2 satisfy 2(cos θ1 +cos θ2 ) ≤ a, the last inequality of (22) is obtained by using the trace theorem. Using Z Z d 2 u dxdy ≤ (u2 + u2t )dxdy, (23) dt Ω Ω d dt

Z

Ω

(∇u)2 dxdy ≤

d dt

Z

Ω

u2t dxdy ≤

Z

((∇u)2 + (∇ut )2 )dxdy, (24)

Z

(u2t + u2tt )dxdy.

Ω

Ω

(25)

and combining (22)-(25) together, we have Z d ( [cos θ1 cos θ2 u2tt + a2 cos θ1 cos θ2 (∇ut )2 dt Ω +2 cos θ1 cos θ2 (M − F (ut )) + a2 u2xt + a4 (∇ux )2 +2a2 (M − F (ux )) + u2 + (∇u)2 + u2t ]dxdy) Z ≤ c (u2 + (∇u)2 + u2t + (∇ut )2 + u2tt )dxdy ZΩ ≤ c (u2 + (∇u)2 + u2t + cos θ1 cos θ2 u2tt

IV.

DISCRETIZATION

The finite difference method is given for the nonlinear wave equations (1)-(2) with nonlinear LABCs (13)-(16). We divide the bounded domain Ωi = [xw , xe ] × [ys , yn ] × [0, T ] by the uniform grids, and choose hx = (xw − xe )/J, hy = (ys − yn )/K and τ = T /N for the grid sizes in space and time, where J, K and N are three positive integers. unj,k represents the numerical approximation of the wave function u at the grid point (jhx , khy , nτ ). The grid points are given by Ωi = {(xj , yk , tn )|xj = xw + jh, yk = yn + kh, tn = nτ, j = 0, ..., J, k = 0, ..., K, n = 0, ..., N }. Denote the operators D·+ , D·− and D·0 by forward, backward and centered differences in x, y and t directions, respectively. S·0 represent centered sums, for example, n−1 St0 unj,k = (un+1 j,k + uj,k )/2. First of all, from time t = tn−1 to time t = tn+1 , where tn+1 = tn + τ , t0 = 0, we discterize the Eq. (1) at points (xj , yk , tn ) in the interior domain Ωi and obtain, Dt+ Dt− unj,k = a2 Dx+ Dx− St0 unj,k + Dy+ Dy− St0 unj,k + f (unj,k ),

where 1 ≤ j ≤ J − 1, 1 ≤ k ≤ K − 1. Then we discuss the discretized forms on the artificial boundaries and corners. For the sake of simplicity, the discretized difference schemes on the east boundary and east-north corner are only given. The LABC (13) on the east boundary can be discretized by,

a(cos θ1 + cos θ2 )Dx0 Dt0 unJ−1,k + cos θ1 cos θ2 Dt+ Dt− unJ−1,k +a2 Dx+ Dx− St0 unJ−1,k − cos θ1 cos θ2 f (unJ−1,k ) = 0,

(28)

where 1 ≤ k ≤ K − 1. For the corner point (J, K), and the corresponding discretized scheme has the form a(cos θ1 + cos θ2 )Dx0 Dt0 unJ−1,K + cos θ1 cos θ2 Dt+ Dt− unJ−1,K

Ω

+a2 cos θ1 cos θ2 (∇ut )2 + 2 cos θ1 cos θ2 (M − F (ut )) +a2 u2xt + a4 (∇ux )2 + 2a2 (M − F (ux )))dxdy, (26) i. e. dE(u; t) ≤ cE(u; t). dt

Thus Gronwall’s inequality leads to the conclusion (17). This completes the proof. We remark that if (φ0 , φ1 ) ∈ H 2 (Ω) × H 1 (Ω), E(u; t) can be uniformly estimated with the help of kφ0 k2H 2 (Ω) + kφ1 k2H 1 (Ω) . Furthermore, if the nonlinear wave equation is sine-Gordon equation, then the constant M = 1, F (u) = − cos(u), which satisfy the assumption in the theorem.

(27)

+a2 Dx+ Dx− St0 unJ−1,K + a(cos θ1 + cos θ2 )Dy0 Dt0 unJ,K−1 +a2 Dy+ Dy− St0 unJ,K−1 + cos θ1 cos θ2 Dt+ Dt− unJ,K−1 − cos θ1 cos θ2 f (unJ−1,K ) − cos θ1 cos θ2 f (unJ,K−1 ) = 0. (29) The discretization of other three boundary conditions can be obtained similarly.

5 TABLE I. The errors and orders for the one-dimensional sineGordon equation at time t = 5. L2 error 0.04013 0.01148 2.958e-3 7.595e-4

Order 1.81 1.96 1.96

L∞ error 0.2061 0.05116 0.01289 3.229e-3

Order 2.01 1.99 2.00

6 4 2

u(x,t)

Mesh h = 14 h = 18 1 h = 16 1 h = 32

8

0

t=5 −2 −4

V.

NUMERICAL RESULTS

t=0

−6

t=10 t=20 t=25 t=35

−8

From the derivation of LABCs for the two-dimensional nonlinear wave equation, we can easily obtain the corresponding LABCs by (13) and (14) for one-dimensional case. To demonstrate the performance of the proposed LABCs for different nonlinearities, sine-Gordon equation and φ4 equation are considered for one-dimensional case. For two-dimensional case, the sine-Gordon equation is discussed by using the various initial values. In the calculations, we set the reference wave speed a = 1.0, the examples used by Han and Zhang [4, 36], Bratsos [5] and Djidjeli et al. [6] are examined. A.

−10 −8

−6

−4

−2

0

2

4

6

8

x

FIG. 1. (Color online) The numerical solution of onedimensional sine-Gordon equation, which shows that the wave travels through the right artificial boundary, without causing dramatic reflection.

One-Dimensional Examples

4 2

As this soliton is associated with the fast variation of some quantity, it is often called “kink”. The theoretical solution is u(x, t) = 4 arctan(exp( √x−ct )). In this calcu1−c2 lation, the computational domain is set to be x ∈ [−8, 8], and the parameter c = 0.8. The errors in different norms and the corresponding orders of convergence are listed in Table I for different mesh sizes at time t = 5. From Table I one can see that the finite difference method has second order convergence rate in L2 -norm and L∞ -norm. This also shows that our boundary condition has a smaller effect on the numerical solution. Fig. 1 plots the evolution of numerical solution, no dramatic reflection is observed. Example 2. φ4 equation [3, 4] is simulated here, and the nonlinear force term f (u) = −γu3 + mu, where γ and m are two constant parameters. The initial values are given by: m mx φ0 (x) = √ tanh( p ), γ 2(1 − c2 ) (sech( √ x 2 ))2 m 2(1−c ) p φ1 (x) = − √ . γ 2(1 − c2 )

In this example, we use the fixed parameters m = 1,

0

u(x,t)

Example 1. We consider the sine-Gordon equation (in the form of f (u) = − sin(u)) with initial conditions [4]: p φ0 (x) = 4 arctan(exp(x/ 1 − c2 )), √ 4c exp(x/ 1 − c2 ) √ φ1 (x) = − √ . 1 − c2 (1 + exp(x/ 1 − c2 )2 )

−2

t=0

−4

t=5

−6

t=10

−8

t=20

−10

t=30

−12

t=40

−14

t=50

−16 −8

−6

−4

−2

0

2

4

6

8

x FIG. 2. (Color online) The numerical solution of φ4 equation, which shows that the reflection waves are negligible when the wave travel out the computational interval.

γ = π12 and c = 0.8, the truncated computational domain x ∈ [−8, 8]. Fig. 2 depicts the numerical solution of φ4 equation. There is no dramatic reflection when the wave travels through the right artificial boundary, which shows that the LABCs are nearly transparent for the wave propagation.

6 TABLE II. The errors and orders for the two-dimensional sine-Gordon equation at time t = 3. Mesh h = 12 h = 14 h = 18 1 h = 16

L1 error 0.0236 0.0067 0.0018 4.89e-4

B.

Order 1.82 1.90 1.88

L∞ error 0.3225 0.0921 0.0234 0.0059

Order 1.81 1.98 1.99

Two-Dimensional Examples

Example 3. We consider the sine-Gordon equation with the following initial conditions φ0 (x, y) = 4 arctan(exp(x + y − 7)), 4 exp(x + y − 7) , φ1 (x, y) = − 1 + exp(2x + 2y − 14) which has the exact solution u(x, y, t) = 4 arctan(exp(x+ y − t − 7)). Here the computational domain is chosen to be (x, y) ∈ [−7, 7] × [−7, 7]. Table II lists the corresponding errors and convergence orders in L1 -norm and L∞ -norm at time t = 3 for different mesh sizes. Table II shows that the convergence rates are close 2 in L∞ -norm when we vary the mesh size. Example 4. The superposition of two orthogonal line solitons [5, 6, 36] for the two-dimensional sine-Gordon equation is considered with the initial conditions: φ0 (x, y) = 4 arctan(exp(x)) + 4 arctan(exp(y)), φ1 (x, y) = 0. Soliton solution with the above initial conditions is called the superposition of two orthogonal line solitons in the literature [5, 6, 36]. The computational domain is (x, y) ∈ [−4, 4] × [−4, 4]. Fig. 3 presents the numerical solutions of superposition of two orthogonal line solitons at different times t = 0, 3, 6, 9.5. The numerical results depict the superposition of two orthogonal line solitons which move away from each other without disturbed. The break up of two orthogonal line solitons along the x direction and the y direction can be observed at time t = 0. Then the solitons move away from each other in the y = x direction. Finally, it is clear that the solitons move out the computational domain without dramatic reflection at the boundaries. The above procedure implies that the proposed LABCs are nearly transparent for the wave propagation. The energy of the solitons in the finite computational domain is plotted in Fig. 4, the energy decreases to zero, while the solitons move out the computational domain. The numerical solutions show an agreement with the published ones in literatures [5, 6, 36]. Remark: The computational domain is chosen to be (x, y) ∈ [−6, 6] × [−6, 6] in literatures [5, 6, 36]. The smaller computational domain in our method can be chosen, which illustrates that our proposed LABCs method is computationally much more efficient.

Example 5. Symmetric perturbation of a static line soliton [5–7, 36] is studied with the initial data: φ0 (x, y) = 4 arctan(exp(x + 1 − 2sech(y + 7) − 2sech(y − 7))), φ1 (x, y) = 0. The computational domain is (x, y) ∈ [−7, 7] × [−7, 7]. The numerical results of symmetric perturbation of a static line soliton are given in Fig. 5 in terms of sin( u2 ) at different times t = 0, 3, 5, 7, 9, 11. The wave displacement of the soliton is shown at initial time t = 0. Then the perturbation moves towards each other. The two symmetric dents move close to each other and collide around time t = 7 . The two symmetric dents continue to move and retain their shapes after the collision, which can be observed from the numerical solutions at time t = 3 and t = 11. Compare the results with that in [5–7, 36], we can see that the pictures are in agreement with the published ones. Example 6. Circular ring soliton [5–7, 36] We set the initial values of the circular ring soliton as follows: p φ0 (x, y) = 4 arctan(exp(3 − x2 + y 2 )), φ1 (x, y) = 0. The numerical solutions were sought over the bounded computational domain (x, y) ∈ [−7, 7] × [−7, 7] and are presented in Fig. 6 at several different times in terms of sin( u2 ). At time t = 0, the initial soliton seems to be two homocentric rings, then the soliton shrinks and appears like a single ring soliton at time t = 3. The soliton begins on the expansion from time t = 6. Radiations at the boundaries begin to form. The expansion is continued until at time t = 11, and the circular ring soliton is almost formed again. Finally, the shrinking process of the soliton appears to be again from t = 13. The center of the circular ring soliton does not move in the whole evolution process, and no observable reflection can be detected at all at the boundaries. These graphs are in complete agreement with the relevant example in [5–7, 36]. Example 7. Elliptical breather [5, 36] The elliptical breather is another important ring soliton solution to the two-dimensional nonlinear wave equation. The initial conditions of the elliptical breather are given by: r (x − y)2 (x + y)2 + )), φ0 (x, y) = 4 arctan(2sech(0.866 3 2 φ1 (x, y) = 0. In Fig. 7, the numerical solutions of the elliptical breather in terms of sin( u2 ) are shown at several different times in the computation domain (x, y) ∈ [−7, 7]×[−7, 7]. The same process as the circular ring soliton, the graphs show that the procedure of shrinking and expansion from time t = 2 to time t = 10. We can see that the center of elliptical breather does not move during the whole process, part of the soliton move out the bounded domain

7 t=3

t=0

12

12

10

10 u(x,y,t)

14

u(x,y,t)

14

8 6

8 6

4

4

2

2

0 4

2

0

−2

−4

y

−4

0

−2

2

0 4

4

2

0

−2

12

10

10 u(x,y,t)

12

u(x,y,t)

14

8 6

2

2 0 4 0

−2

−4 −4

y

4

x

6 4

−2

2

8

4

0

0

−2

t=9.5

14

2

−4

y

x t=6

0 4

−4

2

4

2

0

y

x

−2

−4 −4

0

−2

2

4

x

FIG. 3. (Color online) Superposition of two orthogonal line solitons at time t = 0, 3, 6, 9.5. We can see that the solitons move out the computational domain without dramatic reflection.

without observable reflection at the boundaries from time t = 10. The results agree well with the published ones in the literatures [5, 36].

120 100

Example 8. Collision of four expanding circular solitons [5–7, 36]

E(t)

80

Collision of four expanding circular solitons is an interesting phenomena of the solitary solution to the twodimensional sine-Gordon equation. Firstly, we introduce two definitions. Suppose

60 40 20 0 0

2

4

6

8

10

t FIG. 4. The energy of the superposition of two orthogonal line solitons. As the solitons move out the computational domain, the energy gradually decays to zero.

1 ϕ0 (x, y, x0 , y0 ) = 4 arctan(exp( (4 0.436 p − (x + x0 )2 + (y + y0 )2 ))), 1 ϕ1 (x, y, x0 , y0 ) = 4.13sech( (4 0.436 p − (x + x0 )2 + (y + y0 )2 )). Then the initial data of the juxtaposition of four circular

8

FIG. 5. (Color online) Symmetric perturbation of a static line soliton at time t=0, 3, 5, 7, 9, 11. When the solitons travel out the computational domain, the reflection waves are negligible.

FIG. 6. (Color online) The numerical solution of circular ring soliton at time t=0, 3, 6, 9, 11, 13.

ring solitons is set to: φ0 (x, y) = ϕ0 (x, y, 5, 5) + ϕ0 (x, y, 5, −5) + ϕ0 (x, y, −5, 5) + ϕ0 (x, y, −5, −5), φ1 (x, y) = ϕ1 (x, y, 5, 5) + ϕ1 (x, y, 5, −5) + ϕ1 (x, y, −5, 5) + ϕ1 (x, y, −5, −5).

In this calculation, the computational domain is (x, y) ∈ [−12, 12] × [−12, 12]. The numerical results of four circular solitons in terms of sin( u2 ) are depicted in Fig. 8 at times t = 0, 2, 4 and 14, respectively. Fig. 8 demonstrates precisely the collision between four expanding circular ring solitons in which the four smaller ring

solitons bounding an annular region emerge into a large ring soliton, and then move out the bounded domain with negligible reflection at the boundaries and corners.

VI.

CONCLUSION

Based on unified approach and the ABCs of Higdon, we have obtained the LABCs for the nonlinear wave equation in this paper. We use the energy estimate to show that the resulting LABCs are stable by constructing energy functional. The proposed LABCs involve no high

9

FIG. 7. (Color online) Elliptical breather at time t=0, 2, 5, 7, 9, 10.

FIG. 8. (Color online) The collision of four expanding circular solitons at time t=0, 2, 4, 14.

derivatives, and are thus amenable for standard finite difference method. Furthermore, the performance of the numerical examples illustrates that the given method is feasible and effective.

ACKNOWLEDGMENTS

This research is supported by FRG of Hong Kong Baptist University, and RGC of Hong Kong. We thank the referees for their careful reading of the paper and for sug-

10 gestions that led to an improved version of the paper.

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