Numerical stability analysis of steady solutions for the forced KdV equation based on the polynomial chaos expansion

European Journal of Mechanics B/Fluids 39 (2013) 71–86

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Numerical stability analysis of steady solutions for the forced KdV equation based on the polynomial chaos expansion Hongjoong Kim ∗ , Hye Jin Park, Daeki Yoon Department of Mathematics, Korea University, Seoul 136-701, Republic of Korea

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Article history: Received 20 April 2012 Received in revised form 18 October 2012 Accepted 30 October 2012 Available online 16 November 2012 Keywords: Forced KdV Solitary waves Stability Polynomial chaos

abstract Two-dimensional gravity–capillary waves can be modeled by the forced Korteweg–de Vries (fKdV) equation in subcritical flows when the Bond number is greater than one third. Four steady symmetric depression wave solutions and two elevation wave solutions for the fKdV equation have been found and time evolutions of their magnitude or spatial perturbations have been observed. We approach the fKdV equation as a stochastic equation by modeling the perturbation as a random variable and examine the stabilities of the steady solutions based on the polynomial chaos expansion framework. Polynomial chaos also provides surfaces, which encompass random fluctuations of unstable waves. The effects of several parameters on the stabilities and the surfaces have been also considered. © 2012 Elsevier Masson SAS. All rights reserved.

1. Introduction Interfacial waves of fluids and their many interesting features have received much attention over the decades. In particular, many researchers have been interested in two-dimensional surface waves generated by a localized pressure distribution and the Korteweg–de Vries (KdV) equation to model them in subcritical flows. There have been numerous theoretical, mathematical, and computational approaches for KdV-type equations. See [1–6] and the references therein. A rigorous justification of the formal asymptotic method for interfacial solitary waves was given by Shen and Sun [7]. Shen et al. derived asymptotically a forced Korteweg–de Vries (fKdV) equation and found two types of symmetric steady-state solitary-wave-like solutions of fKdV equations in [8]. Choi et al. obtained in [9] a forced modified Korteweg–de Vries (fmKdV) equation when KdV theory fails and found numerically its symmetric steady-state solutions. Larkin [10] proved the existence and uniqueness of strong and weak global solutions for the fmKdV equation in a bounded domain. Besides the derivation of various KdV-type equations, there has been great interest in the stability of their solutions. Camassa and Wu [11,12] performed a stability analysis of steady solitary-wave solutions when the forcing term is given as a sech4 (x)-profile, and confirmed their analytical findings with accurate numerical simulation. Shen et al. [8] studied the stability of solitary-wave-like

∗

Corresponding author. E-mail address: [email protected] (H. Kim).

0997-7546/$ – see front matter © 2012 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechflu.2012.10.007

solutions of fKdV equations in time and Bona et al. [13] performed theoretical analysis of the stability of a damped KdV equation in a quarter plane. Pava and Natali [14] studied periodic traveling wave solutions for the critical KdV equation analytically, while Johnson performed the research on the orbital stability for a fourparameter family of periodic stationary traveling wave solutions to the generalized KdV equation in [15]. The two-dimensional stability of the localized solitary waves from the steady forced KdV equation was examined numerically by Grimshaw et al. [16] and their results are in good agreement with the results of Camassa and Wu [11]. Recently Chardard et al. derived solutions of the stationary forced KdV equation and studied their stability analytically and numerically in [17], and Donahue and Shen [18] performed numerical stability analysis on the hydraulic fall and cnoidal-wave solution of the forced KdV equation. Maleewong et al. [19] considered two-dimensional steady gravity-capillary waves generated by a localized pressure distribution with a constant speed U in water of finite depth h, and derived the forced Korteweg–de Vries equation, 2ηt + (F − 1)ηx − 2

3 2

η

2



 x

+ τ−

1 3



ηxxx = F 2 px (x),

√

(1)

when the Froude number F = U / gh < 1 and the Bond number τ = T /ρ gh2 > 1/3. T is the coefficient of surface tension and ρ is the water density. p(x) represents the localized pressure distribution. They then investigated the stability of the waves in [16]. In this article, we extend the work of Grimshaw et al. [16] and Kim et al. [20]. Grimshaw et al. [16] used τ = 0.4 and F = 0.9 and the same values are used in this study. p(x) is

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Fig. 1. (Left) Four depression wave solutions for τ = 0.4, ϵ = ±0.01 and F = 0.9. (Right) The values of η(0) of the depression wave solutions in F for τ = 0.4 and ϵ = ±0.01.

set to p(x) = ϵ sech2 (x), where ϵ is the magnitude of the applied pressure. The stability is checked by observing the propagation of waves using perturbed steady solutions as the initial condition. We found 6 time-independent symmetric solitary-wave-like solutions for ϵ = ±0.01, 4 depression wave solutions and 2 elevation wave solutions, to the forced KdV equation and simulation results show that one depression wave and one elevation wave are stable. It has been observed that the perturbation affects the behaviors of waves and characteristics of waves are different depending on the magnitudes and signs of the perturbation. Thus, the Eq. (1) is interpreted as a stochastic equation in this study modeling the perturbation as a random variable as in [21,22]. The solution η is then a function of the spatial–temporal variable (x, t ) and the random variable ξ . There have been many studies on such type of problems, especially those with the random variable ξ = ˙ (t ) for a Brownian motion W (t ). Cameron and Martin [23] W proved that such a solution can be separated into deterministic and random variables by a Fourier series with respect to the Hermite polynomials called Wiener Chaos expansion (WCE) and that the WCE for a second-order random process converges in the mean square sense. Mikulevicius and Rozovskii performed theoretical analysis and estimated error bounds for the WCE in [24, 25]. Ghanem and Spanos extended the Hermite polynomial chaos using the finite element method to study uncertainty problems in solid mechanics in [26,27]. When the random variable ξ is not Gaussian, Hermite polynomial may not work well and alternative polynomial basis needs to be used. Askey and Wilson classified the hypergeometric orthogonal polynomials and presented their properties in [28]. For instance, orthogonal polynomials in the classification satisfy certain differential equations, and they are orthogonal if the inner product is defined using the probability density function of some appropriate random distribution as the weight function. For example, Jacobi polynomials are orthogonal when the weight function from the Beta distribution is used. Xiu and Karniadakis [29] applied various polynomial chaos into stochastic differential equations. They then modeled uncertainty in diffusion problems in [30]. Later Hou et al. [31,32] extended the problem to stochastic partial differential equations from fluid mechanics. Lin et al. considered the stochastic KdV equation with random forcing in [33]. But their studies were mostly focused on a randomness driven by Brownian motion and the effect of Hermite polynomial chaos. This paper is organized as follows: In Section 2, we observe that evolutions of symmetric solitary-wave-like solutions are different with respect to the magnitudes and signs of the perturbation. A numerical scheme based on the polynomial chaos is constructed for the forced KdV equation with a random initial condition in Section 3. Simulation results in Section 4 show that only one depression wave and one elevation wave are stable among six

steady symmetric solitary-wave-like solutions. Polynomial chaos used in the computational simulation also provides surfaces, which give at each time curves encompassing random fluctuations of unstable waves. These curves may be used for the prediction of propagations of waves. Preliminary results of this study have been presented at a conference [34]. 2. Two-dimensional steady gravity-capillary waves 2.1. Time-independent symmetric solitary-wave-like solutions The time-independent symmetric solitary-wave-like solutions of the steady forced KdV equation

(F 2 − 1)ηx −

3 2

η2



  1 + τ − ηxxx = F 2 px (x), x 3

(2)

can be obtained when the Eq. (2) is transformed using the spectral method and then a shooting method with a matching process is performed as Choi et al. did in [9]. We consider the pressure distribution p(x) = ϵ sech2 (x) similarly to Grimshaw et al. [16], where ϵ is the magnitude of the applied pressure. Fig. 1(Left) shows four depression wave solutions for τ = 0.4 and F = 0.9, waves with labels 1 and 2 for ϵ = 0.01 and waves with labels 3 and 4 for ϵ = −0.01. The solutions have been found for various F values and Fig. 1(Right) shows these F values and the values of the corresponding depression wave solutions at x = 0. Fig. 2(Left) shows two elevation wave solutions labeled 5 and 6 for τ = 0.4, ϵ = −0.01 and F = 0.9. Fig. 2(Right) shows the values of η(0) of the elevation wave solutions in F for τ = 0.4 and ϵ = −0.01. Grimshaw et al. in [16] and Kim and Moon in [34] could find only 5 solutions with labels 1–5 and could not find one elevation wave solution with label 6. In order to explicitly state the correspondence between the solution waves in the current study and the 5 corresponding solutions of Grimshaw et al., the same nomenclature as those in [16] have been used. 2.2. Numerical stability of steady solutions In order to identify the stable solution, the equation is solved using the perturbed time-independent solutions as the initial condition as Grimshaw et al. performed in [16] or Kim did in [20]. Two types of perturbations are considered in this study. One is the magnitude perturbation, where the initial data is given in the form of

η0 (x) = (1 + ξ )η∗ (x),

(3)

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Fig. 2. (Left) Two elevation wave solutions for τ = 0.4, ϵ = −0.01 and F = 0.9. (Right) The values of η(0) of the elevation wave solutions in F for τ = 0.4 and ϵ = −0.01.

Fig. 3. Time evolutions of the steady waves perturbed in magnitude (3) by ξ = 5%. (Top) The depression waves with labels 1 and 2 for ϵ = 0.01, (Middle) the depression waves with labels 3 and 4 for ϵ = −0.01, (Bottom) the elevation waves with labels 5 and 6 for ϵ = −0.01.

where η∗ (x) represents 6 steady solutions in Figs. 1(Left) and 2(Left) and ξ is a perturbation. The other type is the spatial perturbation, where the initial condition is given by

η0 (x) = η∗ (x − ξ ).

(4)

Fig. 3 compares the evolutions in time when all 6 steady solution waves are perturbed in magnitude (3) by ξ = 5%. Among 4

depression waves, the depression wave labeled 1 (Top Left) seems to be the only stable wave. The depression wave labeled 2 (Top Right) generates a traveling wave. The depression wave labeled 3 (Middle Left) is oscillatory in the beginning, then a traveling wave is generated after a long time. The depression wave labeled 4 (Middle Right) shows that the wave is split into an oscillatory wave and a traveling wave moving to the right. After a long time (not

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Fig. 4. Perturbed waves in magnitude (3) by ξ = 5% after a sufficiently long time. (Left) The unperturbed wave with label 1 (at t = 0) and the depression waves with labels 1 (at t = 200), 2 (at t = 200), 3 (at t = 1000), 4 (at t = 500). (Right) The unperturbed wave with label 5 (at t = 0) and the elevation waves with labels 5 (at t = 200) and 6 (at t = 200).

Fig. 5. Time evolutions of the steady waves perturbed in magnitude (3) by ξ = −5%. (Top) The depression waves with labels 1 and 2 for ϵ = 0.01, (Middle) the depression waves with labels 3 and 4 for ϵ = −0.01, (Bottom) the elevation waves with labels 5 and 6 for ϵ = −0.01.

shown), the oscillatory wave is split again into stable and unstable waves, which implies that the depression wave labeled 4 is not stable. This observation is consistent with findings of Grimshaw et al. [16]. In case of the elevation waves, the elevation wave labeled

5 (Bottom Left) is stable while the elevation wave labeled 6 (Bottom Right) is also split into several waves. Fig. 4(Left) compares the unperturbed depression wave labeled 1 at t = 0 (solid line) with these solution waves from (1) and

H. Kim et al. / European Journal of Mechanics B/Fluids 39 (2013) 71–86

75

Fig. 6. Time evolutions of the depression wave with label 4 for ϵ = −0.01 perturbed in magnitude (3) by (Top) −10% and −5%, (Bottom) 5% and 10%.

Fig. 7. Time evolutions of the depression wave with label 2 for ϵ = 0.01 perturbed in magnitude (3) by (Left) ξ = −10% and (Right) ξ = −25%.

Fig. 8. (Left) The mean and (Right) the variance of the depression wave with label 1 for (3) with ξ ∈ [−0.1, 0.1].

(3) with ξ = 5% after a sufficiently long time. It is shown that the depression wave labeled 2 for ϵ = 0.01 converges to the stable depression wave labeled 1 after a sufficiently long time. Fig. 4(Right) compares the unperturbed elevation wave labeled 5 at t = 0 (solid line) with the elevation waves labeled 5 and 6 at t = 200 and shows that the elevation wave labeled 6 also converges to the stable elevation wave labeled 5 after a long time.

For comparison purposes, the forced Korteweg–de Vries equation (1) has been solved with (3) for several values of ξ . Fig. 5 shows the time evolutions of six waves with ξ = −5% of the magnitude perturbation instead of 5% in Fig. 3. The figure confirms again that the depression wave solution labeled 1 for ϵ = 0.01 and the elevation wave solution labeled 5 for ϵ = −0.01 are stable. When Figs. 3 and 5 are compared, we can additionally notice

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Fig. 9. (Left) The mean and (Right) the variance of the elevation wave with label 5 for (3) with ξ ∈ [−0.1, 0.1].

Fig. 10. Means for (3) with ξ ∈ [−0.1, 0.1]. (Top) the depression waves with labels 2 (ϵ = 0.01) and 3 (ϵ = −0.01), (Bottom) the depression wave with label 4 (ϵ = −0.01), and the elevation wave with label 6 (ϵ = −0.01).

that the behaviors of waves are changed when the perturbation is changed from 5% to −5%, which attracts our attention. For instance, the traveling wave corresponding to the depression wave labeled 4 in Fig. 5 moves to the left while that in Fig. 3 moves to the right. Fig. 6 compares evolutions of the depression wave labeled 4 when perturbed in magnitude by 4 different values, −10%, −5%, 5%, 10%. Figs. 3, 5 and 6 clearly show that the depression wave labeled 4 is unstable. In addition, more importantly, the figures show that the characteristics of the evolution of the depression wave labeled 4 are affected by the value of ξ . While the depression wave labeled 2 for ξ = 0.5 in Fig. 3(Top Right) generates a traveling wave, the wave labeled 2 for ξ = −0.5 in Fig. 5(Top Right) does not generate a traveling wave but its amplitude decreases to that of the wave labeled 1 after a sufficiently long time. Fig. 7 shows that evolutions of the depression wave labeled 2 do not generate a traveling wave for other negative values of ξ as well and only the amplitude change in time is observed. We perform in-depth analysis of the effect of the perturbation in the next sections. 3. Polynomial chaos Numerical results in Section 2.2 show that the depression wave labeled 1 and the elevation wave labeled 5 are stable given a

specific value of ξ = −5% or 5%. As seen in Section 2.2, however, the characteristics of the evolution changes as ξ changes, and thus the analysis of stability based on a few sample values of ξ above may not be sufficient. In order to confirm that the waves with labels 1 and 5 are stable, let us interpret the situation by assuming that ξ may take any value within a certain interval, i.e., assuming that ξ is a random variable following, for instance a uniform distribution on [−10%, 10%] = [−0.1, 0.1]. Then the solution η can be considered as a random function, and the forced KdV equation (1) can be regarded as a stochastic differential equation with a random initial condition. The Monte Carlo (MC) methods is one way to solve a stochastic differential equation but the computational cost of MC is expensive. As Kim showed [35], the polynomial chaos expansion is an efficient tool to find statistical moments of the solution for the stochastic differential equations. Thus, let us apply this polynomial chaos expansion to this stochastic fKdV equation to calculate the mean and the variance of the waves. 3.1. Jacobi polynomial chaos (α ∗ ,β ∗ )

Let Jn

(x) represent the Jacobi polynomial of degree n with (α ∗ ,β ∗ ) (x) is defined

two parameters α ∗ and β ∗ . Jacobi polynomial Jn

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Fig. 11. Variances for (3) with ξ ∈ [−0.1, 0.1]. (Top) the depression waves with labels 2 (ϵ = 0.01) and 3 (ϵ = −0.01), (Bottom) the depression wave with label 4 (ϵ = −0.01), and the elevation wave with label 6 (ϵ = −0.01).

by

(α ∗ + 1)n ∗ ∗ Jn(α ,β ) (x) = n! 

× 2 F1 −n, n + α + β + 1; α + 1; ∗

∗

∗

1−x



2

,

where the Pochhammer symbol (a)n is defined by

(a)n =



if n = 0 if n = 1, 2, . . . ,

1 a(a + 1) · · · (a + n − 1)

and the generalized hypergeometric series r Fs is defined by r Fs (a1 , . . . , ar ; b1 , . . . , bs ; z ) =

∞  (a1 )k · · · (ar )k z k k=0

(b1 )k · · · (bs )k k!

. (α ∗ ,β ∗ )

See [36,37] for more information. Jacobi polynomials Jn be also derived from the Rodriguez formula ∗ ∗ Jn(α ,β ) (x) =

(−1)

(x) can

(α ∗ ,β ∗ )

2n n!(1 − x)α (1 + x)β  dn  ∗ ∗ × n (1 − x)n+α (1 + x)n+β dx ∗

(α ∗ ,β ∗ )

(α ∗ ,β ∗ )

(x) = 1. Setting α ∗ = 0, β ∗ = 0 results in the (0,0) Legendre polynomials and for notational simplicity Jn (x) will be denoted by Jn (x). First few Legendre polynomials Jn (x) are J1 (x) = x, (α ∗ ,β ∗ )

{Jn

J2 (x) =

2

,

J3 (x) =

10x3 − 6x 4

+

,...

∗ ,β ∗ )

, Jn(α

∗ ,β ∗ )



1

⟩≡

∗ ∗ (α ∗ ,β ∗ ) Jm (x)Jn(α ,β ) (x)dµ,

−1

∗

(2n +

α∗

+

β ∗ )(2n

+

α∗

+

(α ∗ ,β ∗ )

β∗

+ 1)

J n −1

for n = 1, 2, 3, . . .. Let us normalize {Jn

(x),

(6)

(x)} with respect (α ∗ ,β ∗ )

to the inner product (5) and use the same notation {Jn (x)} to represent the normalized Jacobi polynomial chaos below for (α ∗ ,β ∗ )

(5)

where the measure µ is expressed by dµ(x) = w(x)dx, using the probability density function of the Beta distribution

Γ (α ∗ + β ∗ + 2)(1 − x)α (1 + x)β w(x) = ∗ ∗ 2α +β +1 Γ (α ∗ + 1)Γ (β ∗ + 1)

2(n + α ∗ )(n + β ∗ )

(α ∗ ,β ∗ )

(x)} are orthogonal when the inner product is defined

by

⟨Jm(α

(x)} satisfy a recurrence relation, 2(n + 1)(n + α ∗ + β ∗ + 1) (α ∗ ,β ∗ ) Jn+1 (x) xJn (x) = ∗ ∗ ∗ ∗ (2n + α + β + 1)(2n + α + β + 2) (β ∗ )2 − (α ∗ )2 ∗ ∗ + Jn(α ,β ) (x) ∗ ∗ ∗ ∗ (2n + α + β )(2n + α + β + 2) polynomials {Jn

with J0

3x2 − 1

∈

as the weight function. The Beta distribution with α ∗ = 0 and β ∗ = 0 becomes the uniform distribution. Note also that Jacobi

n

∗

Fig. 12. Fluctuations of the depression wave with label 4 for (3) with ξ [−0.1, 0.1] at t = 100 for various sample values of ξ .

(α ∗ ,β ∗ )

notational simplicity so that ⟨Jm , Jn ⟩ = δmn . See [36,38] for more properties of various orthogonal polynomials. Let I be the set of multi-indices with finitely many non-zero components,



∗

I=

α = (α1 , α2 , . . .)|αi ∈ {0, 1, 2, . . .}, |α| ≡

∞  i =1

 αi < ∞ .

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H. Kim et al. / European Journal of Mechanics B/Fluids 39 (2013) 71–86

Fig. 13. Fluctuations of the depression wave with label 4 for (3) with ξ ∈ [−0.1, 0.1] at t = 100 with the curves (12) for σ = 1, 2, 3, 4.

Fig. 14. Fluctuations of the depression wave with label 4 for (3) with ξ ∈ [−0.1, 0.1] at t = 50, 100, 150, 200 with the curves (12) for σ = 3.

For each multi-index α = (α1 , α2 , . . .), a multi-variate Jacobi polynomial of ξ = (ξ1 , ξ2 , . . .) is defined by Jα (ξ ) =

∞ 

Jαi (ξi )

in this study, we represent the stochastic solution η of the Eq. (1) with random initial condition (3) or (4) by the Fourier series with respect to the Jacobi polynomials with both parameters (α ∗ , β ∗ ) set to zero

i=1

where Jαi (ξi ) is the normalized Jacobi polynomial of order αi . Since the random variable is assumed to follow the uniform distribution

η(x, t , ξ ) =

 α∈I

ηα (x, t )Jα (ξ ),

H. Kim et al. / European Journal of Mechanics B/Fluids 39 (2013) 71–86

79

Fig. 15. Fluctuations of (Top) the depression waves with labels 2 and 3 and (Bottom) the elevation wave with label 6 for (3) with ξ ∈ [−0.1, 0.1] at t = 100 with the curves (12) for σ = 3.

Fig. 16. The means and the variances for (3) with ξ ∈ [−0.3, 0.3]. (Top) The depression wave solution with label 1 for ϵ = 0.01 and (Bottom) the elevation wave solution with label 5 for ϵ = −0.01.

where the Fourier coefficients ηα = E [η(x, t , ξ )Jα ]. This η(x, t , ξ ) can be simply written as

η(x, t , ξ ) = ηˆ 0 J0 +

∞ 

ηˆ i J1 (ξi ) +

i=1

+

j ∞  i   i=1 j=1 k=1

∞  i 

ηˆ ij J2 (ξi , ξj )

i =1 j =1

ηˆ ijk J3 (ξi , ξj , ξk ) + · · · ,

(7)

where Jn (ξ1 , ξ2 , . . . , ξn ) denotes the polynomial chaos of order n in the n independent and identically distributed random variables (ξ1 , ξ2 , . . . , ξn ). For notational simplicity, we follow the notation of [29] so that (7) can be rewritten as

η(x, t , ξ ) =

∞  α=0

ηα Jα (ξ ),

(8)

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H. Kim et al. / European Journal of Mechanics B/Fluids 39 (2013) 71–86

Fig. 17. Means for (3) with ξ ∈ [−0.3, 0.3]. (Top) The depression waves with labels 2 and 3, (Bottom) the depression wave with label 4, and the elevation wave with label 6.

Fig. 18. Variances for (3) with ξ ∈ [−0.3, 0.3]. (Top) The depression waves with labels 2 and 3, (Bottom) the depression wave with label 4, and the elevation wave with label 6.

where there is a one-to-one correspondence between the functions Jn (ξ1 , ξ2 , . . . , ξn ) in (7) and Jα (ξ ) in (8) and also between ηˆ 1,2,...,n and ηα . Then (1) can be written as

 2

∞ 

 η α Jα

+ (F 2 − 1)

α=0

 −3



η β Jβ

 ∞ 

β=0

 +

τ−

3

  ∞ α=0

∞ 

ηα Jα

 +

x

ηγ Jγ 

 =F xxx

∞ 

(ηα )x Jα − 3

α=0

τ−

1 3

 ∞ α=0

(ηα )xxx Jα = F 2

∞  β,γ =0

∞ 

  ηβ ηγ x Jβ Jγ

(pα )x Jα .

α=0

Since Jα ’s are an orthonormal basis, Jβ Jγ can be written in terms of Jα ’s,

x

η α Jα

p(ξ )Jα (ξ )w(ξ )dξ . Since Jα ’s are not dependent on x

(ηα )t Jα + (F 2 − 1)



γ =0

1

2



α=0



α=0

t

∞ 

∞ 

where pα = or t,

2

∞  α=0

 ,

pα Jα x

Jβ Jγ =

∞  α=0

eαβγ Jα ,

H. Kim et al. / European Journal of Mechanics B/Fluids 39 (2013) 71–86

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Fig. 19. Fluctuations of (Top) the depression waves with labels 2 and 3 and (Bottom) the depression wave with label 4 and the elevation wave with label 6 for (3) with ξ ∈ [−0.3, 0.3] at t = 100 with the curves (12) for σ = 3.

Fig. 20. Means for (4) with ξ ∈ [−1.5, 1.5]. (Top) The depression waves with labels 1 and 2 for ϵ = 0.01, (Middle) the depression waves with labels 3 and 4 for ϵ = −0.01, (Bottom) the elevation waves with labels 5 and 6 for ϵ = −0.01.

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H. Kim et al. / European Journal of Mechanics B/Fluids 39 (2013) 71–86

Fig. 21. Variances for (4) with ξ ∈ [−1.5, 1.5]. (Top) The depression waves with labels 1 and 2 for ϵ = 0.01, (Middle) the depression waves with labels 3 and 4 for ϵ = −0.01, (Bottom) the elevation waves with labels 5 and 6 for ϵ = −0.01.

Fig. 22. Fluctuations of (Top) the depression waves with labels 2 and 3 and (Bottom) the depression wave with label 4 and the elevation wave with label 6 for (4) with ξ ∈ [−1.5, 1.5] at t = 100 with the curves (12) for σ = 3.

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83

Fig. 23. Means for (4) with ξ ∈ [−4.0, 4.0]. (Top) The depression waves with labels 1 and 2 for ϵ = 0.01, (Middle) the depression waves with labels 3 and 4 for ϵ = −0.01, (Bottom) the elevation waves with labels 5 and 6 for ϵ = −0.01.

where eαβγ = written as ∞ 



Jα (x)Jβ (x)Jγ (x)w(x)dx. Then the equation can be



α=0

∞ 

2 (ηα )τ + (F 2 − 1) (ηα )x − 3

 +

τ−

1



3

η(x, t , ξ ) = eαβγ ηβ ηγ



β,γ =0

x

(ηα )xxx − F 2 (pα )x Jα = 0.

Since Jα ’s are orthonormal, we derive an infinite system of ηα ’s, ∞ 



β,γ =0

3

 +

τ−

1 3



(10)

It should be noted that the polynomial chaos expansion satisfies a system of deterministic equations, whose solution determines statistical moments of the solution. Since the resultant system is deterministic, it needs be solved only once, and thus the computational loads will be reduced.

 x

∀α = 0, 1, 2, . . . .

When this infinite system is truncated to a system of finite dimension P, this becomes P 

ηα Jα (ξ ).

4. Results eαβγ ηβ ηγ

  1 + τ− (ηα )xxx − F 2 (pα )x = 0,

2 (ηα )τ + (F 2 − 1) (ηα )x − 3

P 

α=0





2 (ηα )τ + (F 2 − 1) (ηα )x − 3

and we obtain the algorithm for the finite order polynomial chaos,

eαβγ ηβ ηγ

β,γ =0

(ηα )xxx − F 2 (pα )x = 0,



Two types of perturbations have been considered in this study. Section 4.1 considers the stabilities of the time-independent symmetric solitary-wave-like solutions for the forced Korteweg–de Vries equation when they are perturbed in magnitude as in (3) and Section 4.2 considers the spatial perturbations (4). The spectral method [39] is used for the spatial differentiation and the fourth order Runge Kutta method is used for temporal discretization. The polynomial chaos (10) with P = 4 is used in this study.

 x

∀α = 0, 1, 2, . . . , P

4.1. Magnitude perturbations (9)

In this section, evolutions of the steady symmetric depression and elevation solitary-wave-like solutions have been considered

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Fig. 24. Variances for (4) with ξ ∈ [−4.0, 4.0]. (Top) The depression waves with labels 1 and 2 for ϵ = 0.01, (Middle) the depression waves with labels 3 and 4 for ϵ = −0.01, (Bottom) the elevation waves with labels 5 and 6 for ϵ = −0.01.

when they are perturbed in magnitude. That is, we consider

 

2ηt + (F 2 − 1)ηx −

3

η2

2  η(x, 0) = (1 + ξ )η∗ (x),

 x



+ τ−

1 3



ηxxx = F 2 px (x)

(11)

where η∗ (x) represents steady solutions in Figs. 1 and 2 and the perturbation ξ is a random variable following a uniform distribution. A small perturbation in magnitude is considered in Section 4.1.1 and a moderate perturbation in Section 4.1.2. 4.1.1. Magnitude perturbations by a small amount Let us consider the stabilities of the depression and elevation waves labeled 1–6 when they are initially perturbed up to 10% of their amplitudes. That is, the fKdV equation is solved with the initial condition (3) when ξ is a uniform random variable over the interval [−0.1, 0.1]. Polynomial chaos provides Fourier coefficients, which can be used to obtain statistical moments of the solution. In order to make sure that the depression wave labeled 1 and the elevation wave labeled 5 are stable, let us examine their means and variances in time. Fig. 8 shows the mean and the variance of the depression wave labeled 1 for ϵ = 0.01. Fig. 8(Right) shows that the variance for the depression wave labeled 1 is zero in time, which means that the depression wave labeled 1 at each time is identical to the mean

at that time. Fig. 8(Left) shows its mean, which indicates that the mean of the depression wave labeled 1 does not change in time. Based on these figures, we may conclude that the depression wave labeled 1 for ϵ = 0.01 does not change in time and eventually that it is stable in time. Fig. 9 shows the mean and the variance of the elevation wave labeled 5 for ϵ = −0.01. Similarly to Fig. 8 for the stable depression wave labeled 1, the variance is zero in time and the mean does not change in time. Thus, the figure confirms the claim that the elevation wave labeled 5 for ϵ = −0.01 is also stable. Figs. 10 and 11 show the means and the variances of the depression waves labeled 2, 3, 4 and the elevation wave labeled 6, respectively. Contrary to the stable waves labeled 1 and 5, the means in Fig. 10 change in time, which imply that the waves are expected to change in time. Fig. 11 for the variances show that the waves are not stable. Based on Figs. 8–11 above, we may conclude that the polynomial chaos allow us to identify stable wave solutions among time-independent solutions. As explained in Section 2.2, the characteristics of the unstable solitary-wave-like solutions such as waves labeled 2, 3, 4, and 6 are different depending on the perturbation. Fig. 12 shows the fluctuations of the depression wave labeled 4 at t = 100 for various sample values of the perturbation ξ . Since the variance is a measure how random waves are spread about the mean, we may expect that

H. Kim et al. / European Journal of Mechanics B/Fluids 39 (2013) 71–86

85

Fig. 25. Fluctuations of (Top) the depression waves with labels 2 and 3 and (Bottom) the depression wave with label 4 and the elevation wave with label 6 for (4) with ξ ∈ [−4.0, 4.0] at t = 100 with the curves (12) for σ = 3.

4.2. Spatial perturbations

a certain interval around the mean,

[(mean) − κ(sdv), (mean) + κ(sdv)]

(12)

will encompass most fluctuations of the solution, where (mean) and (sdv) are the mean and the standard deviation of the random solution at each space and time. Fig. 13 shows the fluctuations in Fig. 12 with these curves (thick lines) defined by (12) for several σ values. The figure clearly shows that the curves (12) with σ = 3 or 4 appropriately bound the random fluctuations in Fig. 12. Fig. 14 shows the curves (12) and the fluctuations of the depression wave 4 at various times, t = 50, 100, 150, 200. Curves (12) are efficient for other unstable waves as well. For instance, Fig. 15 shows the results for the unstable depression waves labeled 2 and 3 and the unstable elevation wave labeled 6. Similarly to the depression wave labeled 4 in Figs. 13 and 14, fluctuations of the unstable waves are well bounded by the curves (12). 4.1.2. Magnitude perturbations by a moderate amount Let us consider the stabilities of the depression and elevation waves 1–6 when the magnitude is perturbed up to 30% of their amplitudes. That is, the fKdV equation is solved with the initial condition (3) when ξ is a uniform random variable over the interval [−0.3, 0.3]. Fig. 16 shows the means and the variances of the depression wave labeled 1 for ϵ = 0.01 and the elevation wave labeled 5 for ϵ = −0.01, which show that they are still stable when they are perturbed in magnitude by the amount above. Figs. 17 and 18 show the means and the variances of the depression wave labeled 2 for ϵ = 0.01, the depression waves labeled 3 and 4 for ϵ = −0.01 and the elevation wave labeled 6 for ϵ = −0.01, which imply that they are unstable. The curves (12) bounds fluctuations of the unstable depression waves labeled 2, 3, 4 and the unstable elevation wave labeled 6 sufficiently well even for the magnitude perturbation by a moderate amount. Fig. 19 shows the curves (12) for σ = 3 and fluctuations of those unstable waves at t = 100.

In this section, evolutions of the steady symmetric depression and elevation solitary-wave-like solutions have been considered when they are spatially perturbed. That is, we consider

  

2ηt + (F − 1)ηx − 2

3 2

η(x, 0) = η∗ (x − ξ ).

η

2

 x



+ τ−

1 3



ηxxx = F 2 px (x)

(13)

The spatial perturbation by a small amount is considered in Section 4.2.1 and by a moderate amount in Section 4.2.2. 4.2.1. Spatial perturbations by a small amount Let us consider the stabilities of the depression and elevation waves labeled 1–6 when they are spatially perturbed by a small amount. The fKdV equation is solved with the initial condition (4) when ξ is a uniform random variable over the interval [−1.5, 1.5]. Figs. 20 and 21 show the means and the variances of the depression waves labeled 1–4 and the elevation waves labeled 5 and 6, from which we can conclude that the depression wave labeled 1 and the elevation wave labeled 5 are stable for the spatial perturbation by a small amount and the other 4 waves are unstable. The curves (12) bounds fluctuations of the unstable depression waves labeled 2, 3, 4 and the unstable elevation wave labeled 6 sufficiently well for the spatial perturbation by a small amount. Fig. 22 shows the curves (12) for σ = 3 and fluctuations of those unstable waves at t = 100. 4.2.2. Spatial perturbations by a moderate amount Let us consider the stabilities of the depression and elevation waves labeled 1–6 when the waves are spatially perturbed by a moderate amount. The fKdV equation is solved with the initial condition (4) when ξ is a uniform random variable over the interval [−4.0, 4.0]. Figs. 23 and 24 show the means and the variances of the waves. The figures show that the depression wave 1 and the elevation wave 5 are still stable when they are spatially perturbed by a

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moderate amount, and that the depression wave labeled 2, 3 and 4 and the elevation wave labeled 6 are unstable. The curves (12) bounds fluctuations of the unstable depression waves labeled 2, 3, 4 and the unstable elevation wave labeled 6 sufficiently well even for the spatial perturbation by a moderate amount. Fig. 25 shows the curves (12) for σ = 3 and fluctuations of those unstable waves at t = 100. 5. Conclusions In this paper, the polynomial chaos expansion has been extended to the analysis of the stability of the stochastic Korteweg–de Vries equation. Four steady symmetric depression wave solutions and two elevation wave solutions have been found and time evolutions of their magnitude or spatial perturbations have been observed. Polynomial chaos allows us to identify stable solutions and also to construct surfaces, which encompass random fluctuations of unstable solutions. Effects of several parameters on stabilities and the surfaces have been also examined. When the random pressure distribution is introduced, physical characteristics of the solutions for the forced KdV equation are affected. Its analytical study including the error analysis will be the directions of our future research. Acknowledgments The authors are grateful to the anonymous referees for their valuable comments and suggestions. This research of Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (20120003004). References [1] H. Washimi, T. Taniuti, Propagation of ion-acoustic solitary waves of small amplitude, Phys. Rev. Lett. 17 (1966) 996–998. [2] T.R. Akylas, On the excitation of long nonlinear water waves by moving pressure distribution, J. Fluid Mech. 141 (1984) 455–466. [3] R. Grimshaw, N. Smyth, Resonant flow of a stratified fluid over topography, J. Fluid Mech. 169 (1986) 429–464. [4] T.Y. Wu, Generation of upstream advancing solitons by moving disturbances, J. Fluid Mech. 184 (1987) 75–99. [5] L.K. Forbes, Critical free surface flow over a semicircular obstruction, J. Eng. Math. 22 (1988) 3–13. [6] H. Sha, J.M. Vanden Broeck, Internal solitary waves with stratification in density, J. Aust. Math. Soc. B 38 (1997) 563–580. [7] M.C. Shen, S.M. Sun, Asymptotic method for interfacial solitary waves in a compressible fluid, Methods Appl. Anal. 3 (1) (1996) 135–156. [8] S.P. Shen, R.P. Monohar, L. Gong, Stability of the lower cusped solitary waves, J. Phys. Fluid. 7 (1995) 2507–2509. [9] J.W. Choi, S.M. Sun, M.C. Shen, Steady capillary-gravity waves on the interface of two-layer fluid over an obstruction-forced modified K-dV equation, J. Eng. Math. 28 (1994) 193–210. [10] N.A. Larkin, Modified kdv equation with a source term in a bounded domain, Math. Methods Appl. Sci. 29 (7) (2006) 751–765. [11] R. Camassa, T.Y-T. Wu, Stability of some stationary solutions for the forced KdV equation, Physica D 51 (1–3) (1991) 295–307.

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