PHYSICAL REVIEW E 89, 053305 (2014)

Local absorbing boundary conditions for a linearized Korteweg–de Vries equation

Wei Zhang,1,* Hongwei Li,2,† and Xiaonan Wu1,‡1Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong, P.R. China

2School of Mathematical Sciences, Shandong Normal University, Jinan, 250014, P.R. China(Received 5 August 2013; revised manuscript received 22 April 2014; published 13 May 2014)

The aim of this paper is to construct highly accurate local absorbing boundary conditions for a linearizedKorteweg–de Vries equation on unbounded domain. The local absorbing boundary conditions are derived byPade approximation with high accuracy, and a sequence of auxiliary variables are utilized to avoid the high-orderderivatives in the absorbing boundary conditions. Then the original problem on unbounded domain is replaced byan equivalent initial boundary value problem defined on a finite domain. The finite difference method is appliedto solve the reduced problem on the finite computational domain. Finally, numerical results are presented todemonstrate the effectiveness and accuracy of the proposed method.

DOI: 10.1103/PhysRevE.89.053305 PACS number(s): 02.70.Bf, 03.75.Lm, 42.25.Gy

I. INTRODUCTION

The Korteweg–de Vries (KdV) equation is a nonlinearpartial differential equation (PDE), which was originallydescribed as the propagation of a solitary wave on the watersurface. Due to the presence of soliton solutions, this equationhas attracted attention of mathematicians and physicists. TheKdV equation arises in various areas of physical applications,such as magneto hydrodynamics waves in warm plasma andacoustic waves in an inharmonic crystal. The linearized KdVequation has been used to describe physical models thatpossess soliton structures in the shallow water for fairly longwaves [1–3]. This paper is devoted to studying the numericalsolution of a linearized KdV equation on unbounded domain.

The initial value problem of a linearized KdV equation isgiven by the following form:

ut + uxxx = h(x,t), x ∈ R, t ∈ [0,T ], (1)

u(x,0) = u0(x), x ∈ R, (2)

u → 0, |x| → +∞, (3)

where h(x,t) and u0(x) denote source term and initial conditionwith compact supports; i.e., there is a finite interval [a,b] suchthat supp{h(x,t) ⊂ [a,b] × [0,T ]}, and supp{u0(x) ⊂ [a,b]}.

The numerical solution and analysis of this kind of PDEon unbounded domains has attracted attention worldwide.To cope with the difficulty brought by unboundedness of thedefined domain, one of the most popular approaches is to cutoff a finite computational domain from the unbounded domainand apply the so-called absorbing boundary conditions(ABCs) on the artificial boundaries. The appropriate ABCsshould not only be easy to implement but also imitate theperfect absorption of waves leaving the computational domainthrough the artificial boundaries.

In general, the absorbing boundary conditions can begrouped into two subcategories: global ABCs (also calledexact ABCs or nonreflecting boundary conditions) and localABCs [4]. In the past three decades, many mathematicians,

*wzhanghkbu@gmail.com†lhwqust@gmail.com‡xwu@hkbu.edu.hk

engineers, and physicists have made great contributions onthis subject. Zheng et al. [3] constructed the global (exact)absorbing boundary conditions for a linearized KdV equation.The authors separated the unbounded domain of an initialvalue problem into three subproblems and performed Laplacetransformation on two new unbounded domain subproblems.After solving three subproblems in the frequency domain,the global ABCs for linearized KdV equation are derived.Since the global ABCs try to simulate the effect of theexterior in an exact sense and are fully coupled in spaceand time, they are very expensive for practical computations.Zheng et al. developed a fast evaluation method to reduce thecomputational cost of convolution operations involved in theglobal ABCs. A kind of nonreflecting boundary conditionsthat are suitable for numerical purposes are presented inRef. [5], and the discretization of the nonreflecting boundaryconditions is studied in detail. While the local ABCs are localin both space and time, which are computationally efficientand tractable. The local ABCs were proposed by Lindman [6]and made famous by Engquist and Majda [7]. They proposedthe perfectly local absorbing boundary conditions for generalclasses of linear wave equations. However, it is limited bythe difficulties posed in applying the high-order derivatives inthe practical computations. In recent years, there has beennew progress on the local ABCs for PDEs on unboundeddomain. Hagstrom et al. [8–12] constructed high-order localabsorbing boundary conditions for time-dependent waves byintroducing special auxiliary variables. Guddati [13] designeda new arbitrary high-order ABCs based on continued fractionapproximation. Wu and Zhang [14] devised the high-orderlocal ABCs for heat equation and proved the stability ofthe coupled system. Zhang et al. [15,16] obtained the localABCs for the nonlinear Schrodinger equation on unboundeddomain by applying the operator splitting method. Usingthe designed local ABCs, Brunner et al. [17,18] studiedthe numerical solution of blow-up problems for semilinearparabolic equation on unbounded domain. Arnold et al. [19]proposed the discrete transparent boundary conditions for thetime-dependent Schrodinger equation on a circular domain.Daalen and coworkers [20,21] derived the radiation boundaryconditions for wave equations by variational principles andconservation laws. The radiation boundary conditions areapplicable to nonlinear and dispersive systems, and the reduced

1539-3755/2014/89(5)/053305(8) 053305-1 ©2014 American Physical Society

WEI ZHANG, HONGWEI LI, AND XIAONAN WU PHYSICAL REVIEW E 89, 053305 (2014)

problem by the conditions is well-posed. For more worksabout the local ABCs and high-order local ABCs of PDEson unbounded domains, we refer to Refs. [22–26] and thereview Ref. [27].

In this paper, we construct local absorbing boundaryconditions for the linearized KdV equation, which are derivedby Pade approximation with high accuracy. Then the originalproblem on unbounded domain is reduced to an equivalentinitial boundary value problem (IBVP) defined on a boundedcomputational domain. A finite difference scheme is presentedto solve the reduced IBVP, and some numerical results aregiven to demonstrate the convergence and accuracy of theproposed method.

II. DESIGN OF HIGHLY ACCURATE LOCALABSORBING BOUNDARY CONDITIONS

In this section we are devoted to the derivation of highlyaccurate local ABCs for the linearized KdV Eqs. (1)–(3)by applying Pade approximation. The studies concerningthe absorbing boundary conditions by rational approximationcan be found in the works Bruneau and coworkers [28],Szeftel [29,30], and others [31].

Following Ref. [3], we introduce the following artificialboundaries:

�a = {(x,t)|x = a,0 � t � T },�b = {(x,t)|x = b,0 � t � T }.

The unbounded domainR1 × [0,T ] are divided into three partsby the artificial boundaries �a and �b, namely, the boundeddomain Di = [a,b] × [0,T ], and the unbounded parts DL =(−∞,a] × [0,T ] and DR = [b, +∞) × [0,T ]. In order tostudy the numerical solution of linearized KdV equation, wemust find the boundary conditions on the artificial boundaries�a and �b, to reduce the original problem into an initialboundary value problem on the finite computational domain.

Next, we consider the following two problems:

ut + uxxx = 0, x > b, 0 � t � T , (4)

u(x,t)|x=b = u(b,t), 0 � t � T , (5)

u(x,0) = 0, x > b, (6)

u → 0, x → +∞, (7)

and

ut + uxxx = 0, x < a, 0 � t � T , (8)

u(x,t)|x=a = u(a,t), 0 � t � T , (9)

u(x,0) = 0, x < a, (10)

u → 0, x → −∞. (11)

Since u(b,t) and u(a,t) are not known, both problems are notcomplete, and cannot be solved. We can solve Eqs. (4)–(7)and (8)–(11), if the functions u(b,t) and u(a,t) are given.

Applying the Laplace transformation to Eqs. (4)–(7), weobtain

su + uxxx = 0, (12)

where

u(x,s) =∫ +∞

0e−stu(x,t)dt,s ∈ C,Re(s) > 0.

The general solution of Eq. (12) can be found as

u(x,s) = c1(s)eλ1(s)x + c2(s)eλ2(s)x + c3(s)eλ3(s)x, (13)

where

λ1(s) = − 3√

s, λ2(s) = − 3√

sω, λ3(s) = − 3√

sω2, ω = e2πi

3 .

Notice that

Re{λ1(s)} < 0, Re{λ2(s)} > 0,

Re{λ3(s)} > 0, ∀Re{s} > 0.

Therefore, we have c2(s) = 0 and c3(s) = 0 in Eq. (13), sinceu(x,s) vanishes as x → +∞. Similarly, we conclude c1(s) =0 from Eqs. (8)–(11). Thus, we have

u(b,s) − 1

λ21(s)

uxx(b,s) = 0, (14)

ux(b,s) − 1

λ1(s)uxx(b,s) = 0, (15)

u(a,s) + 1

λ1(s)ux(a,s) + 1

λ21(s)

uxx(a,s) = 0. (16)

Applying inverse Laplace transform to Eqs. (14)–(16),then we get the global absorbing boundary conditions. Un-fortunately, it is difficult to implement the inverse Laplacetransformation, since there involves expensive convolutionoperations. In order to overcome this disadvantage and obtainmanageable boundary conditions, we apply the effective Padeapproximate method. For calculation simplicity, we define thenew approximation as the following:

f (z) = (1 − z)α = 1 −∞∑i=1

biz

1 − aiz.

Then, we consider Pade approximation of λ1(s) = − 3√

s,

− 3√

s = −[1 − (1 − s)]13 = −1 +

∞∑i=1

biz

1 − aiz, (17)

where z = 1 − s. By comparing coefficients of nth-orderderivatives between Maclaurin series and Pade series, thecoefficients ai and bi (i = 1,2, . . . ,p) are determined uniquelyby the following equation:

2p∑i=1

Cizi = −

p∑i=1

biz

1 − aiz. (18)

Take the derivative on both sides of Eq. (18) continuously andset z = 0 each time, ai and bi are computed as

C1 = −p∑

i=1

bi, 2C2 = −p∑

i=1

aibi,

...

(2p − 1)!C2p = −(2p − 1)!p∑

i=1

a2p−1i bi .

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LOCAL ABSORBING BOUNDARY CONDITIONS FOR A . . . PHYSICAL REVIEW E 89, 053305 (2014)

0 2 4 6 8 100

0.5

1

1.5

2

2.5

s

3√s

p=5

Approximate solution

Exact solution

FIG. 1. (Color online) Pade approximation against the exactsolution of 3

√s with p = 5.

Denote di = ap

i bi , we get

b1 + b2 + · · · + bp = −C1,

a1b1 + a2b2 + · · · + apbp = −C2,

...

ap−11 b1 + a

p−12 b2 + · · · + ap−1

p bp = −Cp,

d1 + d2 + · · · + dp = −Cp+1,

TABLE I. L1 error in approximating 3√

s and 3√

s2 in the finiteinterval s = [0.2,9.9].

p = 3 p = 5 p = 7 p = 9

3√

s 8.356e-3 4.588e-4 2.657e-5 1.598e-63√

s2 1.970e-2 1.034e-3 5.887e-5 3.510e-6

a1d1 + a2d2 + · · · + apdp = −Cp+2,

...

ap−11 d1 + a

p−12 d2 + · · · + ap−1

p dp = −C2p.

Solving the above system, we obtain the coefficients ai andbi (i = 1,2, . . . ,p). Thus, λ1(s) = − 3

√s can be estimated

easily by truncated p terms, namely, λ1(s) ≈ λp(s) = −1 +∑p

i=1bi (1−s)

1−ai (1−s) .

Figure 1 shows the Pade approximation of 3√

s when p = 5.Comparing the approximate solution with the correspondingexact solution in this figure, one can observe that they arealmost indistinguishable.

Figure 2 plots the error of exact solution and the approxi-mate solution of 3

√s with different p. Table I lists the L1 error

in approximating 3√

s and 3√

s2 in the interval s = [0.2,9.9],which is defined by L1 = 1

K

∑Kk=1 |λm

1 (sk) − λmp (sk)|,(m =

1,2). We can see that the error decreases as the parameterp grows. Numerical results indicate that when p is greater

0 2 4 6 8 100

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Err

orin

appr

oxim

atin

g3√s

s

p=3

0 2 4 6 8 100

0.5

1

1.5

2

2.5x 10

−3

Err

orin

appr

oxim

atin

g3√s

s

p=5

0 2 4 6 8 100

0.5

1

1.5

x 10−4

Err

orin

appr

oxim

atin

g3√s

s

p=7

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

−5

Err

orin

appr

oxim

atin

g3√s

s

p=9

FIG. 2. (Color online) Error of Pade approximation to 3√

s with different p = 3, 5, 7, 9.

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WEI ZHANG, HONGWEI LI, AND XIAONAN WU PHYSICAL REVIEW E 89, 053305 (2014)

than 5, 3√

s can be approximated by the Pade approximationvery well.

Similarly, λ21(s) = ( 3

√s)2 can be estimated by using

the same method, with the corresponding parameters Ai

and Bi (i = 1,2, . . . ,p). Using Pade approximation toλ2

1(s) in Eq. (14), we have λ21(s) = 3

√s2 ≈ λ2

p(s) = 1 −∑p

i=1Bi (1−s)

1−Ai (1−s) . Let gr = 1λ2

p(s)uxx(b,t), a system of equations

is given as following

u − gr = 0, (19)

gr −p∑

i=1

Biwi = uxx, (20)

1 − s

1 − Ai + Aisgr = wi, i = 1,2, . . . ,p. (21)

Applying inverse Laplace transform to Eqs. (19)–(21), weobtain

u − gr = 0, (22)

gr −p∑

i=1

Biwi = uxx, (23)

(1 − Ai)wi + Ai

dwi

dt= gr − dgr

dt, i = 1,2, . . . ,p. (24)

Similarly, assume fr = − 1λp(s)

uxx(b,t). Using Pade approx-

imation to λ1(s) in Eq. (15), we get

ux + fr = 0,

fr −p∑

i=1

bi vi = uxx,

1 − s

1 − ai + aisfr = vi , i = 1,2, . . . ,p.

After performing inverse Laplace transformation on the abovesystem,

ux + fr = 0, (25)

fr −p∑

i=1

bivi = uxx, (26)

(1 − ai)vi + ai

dvi

dt= fr − dfr

dt, i = 1,2, . . . ,p. (27)

For Eq. (16), let fl = − 1λp(s)

ux(a,t) and gl = 1λ2

p(s)uxx(a,t).

Ai and ai (i = 1,2, . . . ,p) are defined as before. Applyinginverse Laplace transformation, we have

u − fl + gl = 0, (28)

fl −p∑

i=1

bi vi = ux, (29)

gl −p∑

i=1

Biwi = uxx, (30)

(1 − ai)vi + ai

dvi

dt= fl − dfl

dt, (31)

(1 − Ai)wi + Ai

dwi

dt= gl − dgl

dt, (32)

where i = 1,2, . . . ,p.Coupling the ABCs (22)–(32) with the linearized KdV

equation, we have the reduced IBVP on the bounded com-putational domain.

III. DISCRETIZATION

In this section, we discuss the discretized forms forthe reduced IBVP on the bounded computational domain[a,b] × [0,T ]. We divide the computational domain by a set oflines parallel to the x axis and t axis to form a grid. For giventwo positive integers J and N , we get the spatial and temporalmesh sizes by defining �x = b−a

Jand �t = T

N. Thus, the

uniform grid points can be written as xj = a + j�x,(j =0,1,2, . . . ,J ) and tn = n�t,(n = 0,1, . . . ,N ). un

j presents theapproximation of u(xj ,tn). We define the finite differenceoperators as

fn+1/2j = f n+1

j + f nj

2, δ+

t f nj = f n+1

j − f nj

�t.

We adopt the Crank-Nicolson finite difference method onlinearized KdV equation in the interior domain:

δ+t un

j + δ3xu

n+1/2j = h

n+1/2j ,

with

δ3xu

n1 = −un

4 − 6un3 + 12un

2 − 10un1 + 3un

0

2�x3,

δ3xu

nj = un

j+2 − 2unj+1 + 2un

j−1 − unj−2

2�x3,

j = 2,3, . . . ,J − 2.

Now we focus on the approximation on the artificial bound-aries. The discretized forms of absorbing boundary conditionon �b are given by

un+1/2J − gn+1/2

r = 0,

δ+x u

n+1/2J + f n+1/2

r = 0,

gn+1/2r −

p∑i=1

Biwn+1/2i = δ2

xun+1/2J ,

f n+1/2r −

p∑i=1

bivn+1/2i = δ2

xun+1/2J ,

(1 − ai)vn+1/2i + aiδ

+t vn

i = f n+1/2r − δ+

t f nr ,

(1 − Ai)wn+1/2i + Aiδ

+t wn

i = gn+1/2r − δ+

t gnr ,

TABLE II. L2 error and convergence order for the time space atT = 2.

�t = 116 Order �t = 1

32 Order �t = 164 Order

p = 5 1.870e-2 – 4.118e-3 2.18 8.797e-4 2.23p = 7 1.884e-2 – 4.680e-3 2.01 1.227e-3 1.93p = 9 1.866e-2 – 4.587e-3 2.02 1.135e-3 2.01

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LOCAL ABSORBING BOUNDARY CONDITIONS FOR A . . . PHYSICAL REVIEW E 89, 053305 (2014)

TABLE III. L2 error and convergence order for the spatial spaceat T = 2.

�x = 116 Order �x = 1

32 Order �x = 164 Order

p = 3 4.101e-3 – 1.770e-3 1.21 1.207e-3 0.55p = 5 3.044e-3 – 7.048e-4 2.11 1.283e-4 2.46p = 7 3.120e-3 – 7.796e-4 2.00 2.007e-4 1.96p = 8 3.107e-3 – 7.669e-4 2.02 1.881e-4 2.03p = 9 3.110e-3 – 7.694e-4 2.02 1.906e-4 2.01

where

δ+x un

J = unJ−2 − 4un

J−1 + 3unJ

2�x,

δ2xu

nJ = 2un

J − 5unJ−1 + 4un

J−2 − unJ−3

�x2,

i = 1,2, . . . ,p.

By the same way, the discretizations can be achieved for thelocal absorbing boundary condition on �a as follows:

un+1/20 − f

n+1/2l + g

n+1/2l = 0,

fn+1/2l −

p∑i=1

bivn+1/2i = δ+

x un+1/20 ,

gn+1/2l −

p∑i=1

Biwn+1/2i = δ2

xun+1/20 ,

TABLE IV. L2 error and convergence order for the spatial spaceat T = 4.

�x = 116 Order �x = 1

32 Order �x = 164 Order

p = 3 2.067e-3 – 9.024e-4 1.20 6.216e-4 0.54p = 5 1.542e-3 – 3.649e-4 2.08 7.221e-5 2.34p = 7 1.568e-3 – 3.904e-4 2.01 9.720e-5 2.01p = 8 1.567e-3 – 3.900e-4 2.01 9.682e-5 2.01p = 9 1.568e-3 – 3.907e-4 2.00 9.758e-5 2.00

(1 − ai)vn+1/2i + aiδ

+t vn

i = fn+1/2l − δ+

t f nl ,

(1 − Ai)wn+1/2i + Aiδ

+t wn

i = gn+1/2l − δ+

t gnl ,

where

δ+x un

0 = −un2 − 4un

1 + 3un0

2�x,

δ2xu

n0 = 2un

0 − 5un1 + 4un

2 − un3

�x2,

i = 1,2, . . . ,p.

The scheme is second order in �t and �x.

IV. NUMERICAL RESULTS

In this section, we report the numerical results of linearizedKdV equation with the highly accurate local absorbing

−6 −4 −2 0 2 4 6−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x

u(x,

t)

t=0.1

Numerical solutionExact solution

x

u(x,

t)

t=0.5

−6 −4 −2 0 2 4 6−0.4

−0.2

0

0.2

0.4

0.6

0.8

1Numerical solutionExact solution

−6 −4 −2 0 2 4 6−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

x

u(x,

t)

t=4

Numerical solutionExact solution

−6 −4 −2 0 2 4 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

u(x,

t)

t=7

Numerical solutionExact solution

FIG. 3. (Color online) Numerical solutions compared with exact solutions at different times as the parameter p = 7.

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WEI ZHANG, HONGWEI LI, AND XIAONAN WU PHYSICAL REVIEW E 89, 053305 (2014)

TABLE V. L2 error and convergence order for the time space atT = 7.

�t = 116 Order �t = 1

32 Order �t = 164 Order

p = 6 5.922e-4 – 1.374e-4 2.11 2.770e-5 2.31p = 7 6.061e-4 – 1.503e-4 2.01 3.711e-5 2.02p = 9 6.065e-4 – 1.505e-4 2.01 3.732e-5 2.01

boundary conditions to illustrate the effectiveness and accu-racy of our proposed method.

Example 1. We consider the following problem [3]:

ut + uxxx = 0, x ∈ R,

u(x,0) = e−x2, x ∈ R,

u → 0, x → ∞.

The fundamental solution of this linearized KdV equationis

u(x,t) = E(x,t) ∗ e−x2,

where E(x,t) = 13√3t

Ai( x3√3t

), Ai(·) is the Airy function, and ∗denotes the convolution operator on the whole real axis. Sincethe Gaussian functions decay very fast, the initial conditioncan be taken to have a compact support. In this example, wechoose the computational domain as x ∈ [−6,6].

In order to illustrate the performance of the proposed localABCs, L2 error and L∞ error are tested, which are defined by

L2(t) =⎛⎝ 1

J + 1

J∑j=0

|uex(xj ,t) − unu(xj ,t)|2⎞⎠

12

,

and

L∞(t) = maxj=0,1,...,J

{|uex(xj ,t) − unu(xj ,t)|},

where uex and unu denote the exact solution and numericalsolution of u, respectively.

For temporal error analysis, we choose �x = 1211 such that

the error in spatial discretization can be ignored. Table IIlists the L2 error and convergence order for time space with

0 1 2 3 4 5 6 7−10

−8

−6

−4

−2

t

Log 10

(L2 e

rror

)

p=3p=5p=7p=9

FIG. 4. (Color online) Numerical errors in L2 norm with differ-ent p for time T = 7.

0 1 2 3 4 5 6 7

−12

−10

−8

−6

−4

t

Log 10

(L∞ e

rror

)

p=3p=5p=7p=9

FIG. 5. (Color online) Numerical errors in L∞ norm with differ-ent p for time T = 7.

different values of parameter p. This table demonstrates thatthe convergence order is almost to 2 for time space.

For spatial error analysis, we choose time step �t = 10−3,such that the temporal error can be neglected. The errors andconvergence order for spatial space with different p at timeT = 2 and T = 4 are presented in Tables III and IV, respec-tively. We can conclude that the second-order convergence ofthe spatial error is clear when p � 5. Moreover, Fig. 3 plots thenumerical solution and exact solution at different times withthe mesh sizes �t = 0.02 and �x = 0.02 as the parameterp = 7. This figure shows that the numerical solution matchesthe exact solution very well. This figure also illustrates thatthe reflection waves are negligible at the artificial boundaries,when the waves travel out the computational domain.

In Table V, we list the L2 error and convergence order forthe time space at T = 7. From this table, one can see thatthe convergence order for the time space is nearly 2, and noinstability was observed for long time calculations.

Figures 4 and 5 depict the L2 and L∞ error with differentparameter p for time T = 7. Both L2 and L∞ error decay overtime, since the wave is propagating away from the area weare monitoring. We define the energy of the linearized KdVequation on the bounded domain by the expression E(t) =∫ b

au2(x,t)dx, which can be calculated numerically through

0 1 2 3 4 5 6 70.5

0.6

0.7

0.8

0.9

1

1.1

1.2

E(t

)

t

p=7

FIG. 6. (Color online) The energy of the wave remaining in thebounded domain with parameter p = 7.

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LOCAL ABSORBING BOUNDARY CONDITIONS FOR A . . . PHYSICAL REVIEW E 89, 053305 (2014)

−6 −4 −2 0 2 4 6−1

−0.5

0

0.5

1

x

u(x,

t)

t=0

−6 −4 −2 0 2 4 6−1

−0.5

0

0.5

1

x

u(x,

t)

t=0.1

Numerical solutionExact solution

−6 −4 −2 0 2 4 6−1

−0.5

0

0.5

1

x

u(x,

t)

t=0.15

Numerical solutionExact solution

−6 −4 −2 0 2 4 6−1

−0.5

0

0.5

1

x

u(x,

t)

t=0.2

Numerical solutionExact solution

FIG. 7. (Color online) Numerical solutions compared with exact solutions at different times as the parameter p = 9.

the numerical quadrature. In Fig. 6, we plot the energy of thewave remaining in the bounded domain with the mesh sizes�t = 0.02 and �x = 0.02. As the wave moves away fromthe bounded domain, the energy decays accordingly. From thefigures, one can observe that long time calculations with localABCs are stable.

Example 2. We choose the initial condition as

u(x,0) = sin(5x)e−x2/4.

The computational domain is x ∈ [−6,6].

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

1.2

1.4

E(t

)

t

FIG. 8. (Color online) The energy of the wave remaining in thebounded computational domain with p = 9.

The numerical solutions and exact solutions at differenttimes are presented in Fig. 7 as the mesh sizes �t = 5e-4and �x = 4e-2. From this figure, we can observe thatthe numerical solutions show an agreement with the exactsolutions, and there is no dramatic reflection when the wavetravels through the left artificial boundary.

Figure 8 plots the energy of the wave in the finitecomputational domain with the parameter p = 9. We cansee that the energy gradually decreases to zero, while thewave moves out of the computational domain. This figureimplies that the local absorbing boundary conditions are nearlytransparent for the wave propagation.

V. CONCLUSION

In our work, highly accurate local absorbing boundaryconditions for a linearized KdV equation are proposed. Ap-plying the absorbing boundary conditions yields an equivalentinitial boundary value problem defined on a finite domain.In order to obtain the numerical solution of the initialboundary value problem, the finite difference method isapplied. The numerical results are presented to demonstratethe accuracy of the proposed method. Future research di-rections would be to design the local ABCs for the fullynonlinear and multidimensional KdV equation on unboundeddomain.

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WEI ZHANG, HONGWEI LI, AND XIAONAN WU PHYSICAL REVIEW E 89, 053305 (2014)

ACKNOWLEDGMENTS

The authors gratefully acknowledge the anonymousreferees for their careful reading and many construc-tive suggestions, which lead to a great improvement of

the paper. This research is supported by FRG of HongKong Baptist University, RGC of Hong Kong, andNational Natural Science Foundation of China (GrantNo. 11326227).

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