Numerical solution of two-dimensional nonlinear sine-Gordon equation using localized method of approximate particular solutions

Engineering Analysis with Boundary Elements 108 (2019) 95–107

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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound

Numerical solution of two-dimensional nonlinear sine-Gordon equation using localized method of approximate particular solutions LingDe Su Institute of Mathematics and Information Science, North-Eastern Federal University, Russia

a r t i c l e

i n f o

Keywords: Sine-Gordon equation Generalized thin plate splines (GTPS) Radial basis functions (RBFs) Localized method of approximate particular solutions (LMAPS) Numerical solution

a b s t r a c t Sine-Gordon equation is one of the most famous nonlinear hyperbolic partial differential equations, it arises in many science and engineering fields. In the present work, we consider the numerical solution of two dimensional sine-Gordon equation using localized method of approximate particular solutions (LMAPS), in this technique, the method of approximate particular solutions (MAPS) occurs on some local domains, that greatly reduces the size of the collection matrix, and by combining the conditional positive radial basis function (RBF) generalized thin plate splines (GTPS) with additional low-order polynomial basis to avoid selecting shape parameters during localization. This method is effective compared with other existing methods and since this method is really meshless, it can be used to solve the nonlinear model with complicated computational domains. Several numerical examples are given to demonstrate the ability and accuracy of the present approach for solving nonlinear sineGordon equation.

1. Introduction The sine-Gordon equation is an important dynamical model in nonlinear science, and also plays a central role in many scientific fields, such as in differential geometry [1], transmission of ferromagnetic waves [2], motion of dislocations in crystals [3], DNA-soliton dynamics, nonlinear optics [4], etc. Since sine-Gordon equation has many kinds of soliton solutions, it has attracted widespread attention. Actually, soliton solution does not undergo any deformation when it propagates through the medium, even after interacting with other solitons [5]. Many soliton solutions have been identified in various wave and particle systems in nature. The solitons occupy the core position in the theory of nonlinear differential equations and have a wide range of physical applications, including condensed-matter physics [6], shallow-water waves, plasma physics [7], fluid dynamics and fibre optics, etc. As an important nonlinear partial differential equation with many kinds of soliton solutions, sine-Gordon equation has been investigated by many scientists. A lot of researchers have been spending a great deal of effort to solve sine-Gordon equation exactly. Wazwaz obtained several exact solutions of one dimensional sine-Gordon equation by using tanh method [8]. In [9], S. Johnson with his collaborators offered three new exact solutions for sine-Gordon equation in two dimensions. Chen and Gao [10] obtained hybrid solutions of two dimensional sine-Gordon equation based on a constructed Wronskian form expansion method. More works about exact solutions of sine-Gordon equation can be seen [11–14] and the references therein.

Although more and more new exact solutions of sine-Gordon equation are found, there are still many soliton solutions that have not yet been solved, so the numerical methods are mentioned. About the numerical solution of one dimensional sine-Gordon equation, we refer readers to [15,16] and the references therein. In this paper, we focus on higher dimensional case, finite element method (FEM) was considered in [17] to solve two dimensional sine-Gordon equation, in [18] a damped sine-Gordon equation in two space variables was solved using explicit method, [19] proposed a split cosine scheme for simulating solitary solutions of the sine-Gordon equation in two dimensions, the method of lines [20], radial basis functions collection method (RBFCM) [21] and the meshless local Petrov–Galerkin method (MLPGM) [22] were also discussed for solving 2D sine-Gordon equation. In recent years, Dehghan and Ghesmati in their work [23] adopted a local radial point interpolation method (LRPIM) to simulate the two-dimensional nonlinear sine-Gordon equation, Jiwari and his collaborators considered the 2D sine-Gordon equation based on polynomial differential quadrature method (PDQM) in [24], a new numerical scheme which is obtained by the Fourier pseudo-spectral method and the fourth order average vector field method (AVF) was proposed in [25] for one and two dimensional sine-Gordon equation, also high order compact alternating direction implicit scheme was given in [26], space-time spectral method was proposed in [27] and a well-posed moving least squares approximation was considered in [28], etc. In this paper we consider the nonlinear sine-Gordon by using localized method of approximate particular solutions (LMAPS), the LMAPS

E-mail address: [email protected] https://doi.org/10.1016/j.enganabound.2019.08.018 Received 16 February 2019; Received in revised form 2 July 2019; Accepted 9 August 2019 Available online 5 September 2019 0955-7997/© 2019 Elsevier Ltd. All rights reserved.

L. Su

Engineering Analysis with Boundary Elements 108 (2019) 95–107

Table 1 Some commonly used types of RBFs.

was first introduced in 2010 in Yao’s works [29,30] and has been used in many problems in science and engineering [31–35]. The LMAPS based on RBFs approximation is a true meshless method, it calculates on some scatter points, that avoids grid generation, compared with the traditional mesh method, such as finite element method (FEM), finite volume method, the meshless method saves a lot of times. Since the value of RBFs at any point in space domain depends only on the distance from the fixed point, the LMAPS can be extended to high dimension problems. Also no numerical integration is required in LMAPS so that it can easily handle irregular calculation areas. Compared with the global version of the method of approximate particular solutions (GMAPS), the localized method produces a sparse matrix and reduces the ill-conditional problem of the collection matrix, which makes it useful for solving large-scale problems, in [30] about one million points were used to consider two dimensional problem and obtained good accuracy within a reasonable times. In LMAPS, the collection by MAPS is considered on overlapping local domains, and to approximate differential operator using a weighted combination of the function values on every corresponding local domain, by padding the weight vector with zeros based on a map, which mapping from elements in a local domain to scatters in the computational area. Then the collection matrix is a sparse one, that drastically reduces the size of collection matrix, but, many small matrices need to be solved. When using LMAPS, RBFs interpolation is required, many commonly used positive RBFs, such as inverse multiquadrics (IMQ) and Gaussian, have shape parameters. The selection of shape parameters is still a prominent subject, inappropriate shape parameters greatly affect the results. In our paper, we take the conditional positive RBF generalized thin plate splines (GTPS) with additional low-order polynomial basis, avoiding the choice of shape parameters while guaranteeing the positive definiteness of the collection matrix on every local domains. The layout of the article is as follows : In section 2, we introduce and discretize the sine-Gordon equation. Section 3 gives an explanation of LMAPS and applies it on the nonlinear sine-Gordon equation. The results of numerical experiments are presented in section 4. Section 5 is dedicated to a brief conclusion. Some references are introduced at the end.

x∈Ω x∈Ω x ∈ 𝜕Ω, 𝑡 > 0,

Mat𝑒́ rn

Poisson

K𝜈 ( r ) r𝜈

𝐉𝜈 (𝑟)𝑟−𝜈

RBFs

Linear

Cubic

PS

GTPS

𝜙(r)

r

r3

𝑟2𝜅−1

r2𝜅 ln r

1-D space 2,3-D space

=

(1 − 𝑟)+ (1 − 𝑟)2+

2 (1 − (1 −

4 𝑟)3+ (3𝑟 𝑟)4+ (4𝑟

+ 1) + 1)

(1 − 𝑟)5+ (8𝑟2 + 5𝑟 + 1) (1 − 𝑟)6+ (35𝑟2 + 18𝑟 + 3)

1 𝑙+1 1 1 Δ𝑢 (x) + Δ𝑢𝑙 (x) + Δ𝑢𝑙−1 (x) + 𝜚(x) sin 𝑢𝑙 (x), 4 2 4

(2.6)

rearranging Eq. (2.6) we can get, ( ) ( ) 𝛽𝜏 𝑙+1 𝛽𝜏 𝑙−1 1 1 1+ 𝑢 (x)− 𝜏 2 Δ𝑢𝑙+1 (x) = 2𝑢𝑙 (x)+ 𝜏 2 Δ𝑢𝑙 (x) − 1 − 𝑢 ( x) 2 4 2 2 1 + 𝜏 2 Δ𝑢𝑙−1 (x) + 𝜏 2 𝜚(x) sin 𝑢𝑙 (x). (2.7) 4 When 𝑙 = 0, for approximation of 𝑢−1 , we use the initial velocity (2.3), the details will be given in the following section. Then, the original problem of nonlinear sine-Gordon Eqs. (2.1)–(2.3) with boundary conditions (2.4) or (2.5) is transformed into the discrete Eqs. (2.7) with the same initial and boundary conditions. 3. Solving sine-Gordon equation using LMAPS In this section we consider the discrete Eqs. (2.7) by applying LMAPS. Assuming there are n collocation points {x𝑗 }𝑛𝑗=1 in the domain Ω, including nI interior points and nB boundary points, where 𝑛𝐼 + 𝑛𝐵 = 𝑛. We create a local domain Ωi for any point x𝑖 by choosing m nearest neighbor points of x𝑖 , for some different points, the corresponding local domains can intersect. For convenience, we assume Ω𝑖 = {x𝑘𝑖 }𝑚 , easily see that 𝑘=1 every element x𝑘𝑖 ∈ Ω𝑖 corresponds to a unique point x𝑗 ∈ {x𝑗 }𝑛𝑗=1 . Fig. 1 shows the local domains with 𝑚 = 5, it can be seen intuitively that there is a correspondence between any local domains Ωi and the all collocation points {x𝑗 }𝑛𝑗=1 . To consider (2.7) with initial conditions (2.2)–(2.3) and boundary conditions (2.4) or (2.5) using LMAPS, we need to approximate the second order differential operator Δ. On any local domain Ωi , the method of approximate particular solutions (MAPS) based on RBFs interpolation is used to approximate the solution 𝑢(x) for point x ∈ Ω𝑖 ,

(2.1)

(2.2)

(2.3)

with Dirichlet boundary conditions, 𝑢(x, 𝑡) = ℎ(x, 𝑡),

IMQ √ 1∕ 𝑟2 + 𝑐 2

Wendland’s compactly supported-RBFs

and initial velocity, 𝑢𝑡 (x, 0) = 𝑔(x),

exp(−𝑐𝑟2 )

0

with initial condition, 𝑢(x, 0) = 𝑓 (x),

𝜙(r)

Conditionally Positive Definite RBFs

Consider the following sine-Gordon equation on a connected and bounded domain Ω ⊆ ℝ2 with continuous boundary 𝜕Ω, Ω = Ω ∪ 𝜕Ω, x ∈ Ω, 𝑡 ⩾ 0,

Strictly Positive Definite RBFs Gaussian

2. Discrete sine-Gordon equation

𝑢𝑡𝑡 + 𝛽𝑢𝑡 = Δ𝑢 + 𝜚(x) sin 𝑢,

RBFs

(2.4)

𝑢(x) ⋍

or the following boundary conditions, 𝜕𝑢 (x, 𝑡) = ℎ(x, 𝑡), x ∈ 𝜕Ω, 𝑡 > 0, (2.5) 𝜕x where Δ is the Laplace operator, x = (𝑥, 𝑦), 𝛽 ⩾ 0 is the damping coefficient, and 𝑢(x, 𝑡) is sufficiently differentiable solution of nonlinear sineGordon Eq. (2.1). In Eq. (2.1), the term with parameter 𝛽 is called dissipative term, when 𝛽 = 0, Eq. (2.1) is an undamped sine-Gordon equation, when 𝛽 > 0, it’s a damped one. Let 𝜏 = 𝑡𝑙+1 − 𝑡𝑙 be the time step size, notation 𝑡𝑙+1 represents the time value at 𝑙 + 1 steps and 𝑡0 = 0. For any 𝑡𝑙 ⩽ 𝑡 ⩽ 𝑡𝑙+1 , using the symbol 𝑢𝑙 (x) = 𝑢(x, 𝑡𝑙 ) we discretize Eq. (2.1) with the weighted scheme in the following,

𝑚 ∑ 𝑘=1

𝜆𝑘 Φ(𝑟𝑘 ) + Ψ(x),

(3.1)

where 𝑟𝑘 = ‖x𝑘𝑖 − x‖ and ΔΦ(𝑟) = 𝜙(𝑟), 𝜙(r) is a RBF. Eq. (3.1) can be written without the additional term Ψ(x), when the RBF is a positive definite one, such as IMQ and Gaussian RBFs. But for polyharmonic splines (PS) and generalized thin plate splines (GTPS) such conditionally positive definite RBFs, the additional term Ψ(x) is always needed. Some commonly used RBFs in meshless methods are given in Table 1, in Table 1 c > 0 is shape parameters, K𝜈 is a modified first kind Bessel function of order 𝜈, J𝜈 is Bessel function of the first kind of order 𝜈, 𝜅 ∈ ℕ+ and the cut-off function (𝑟)+ which is defined to be r if 0 ⩽ r ⩽ 1 and to be zero elsewhere. More information about RBFs can be seen in books [36–38]. Let P𝑞𝑑 be a space of d-variate polynomial of order not exceeding more than q and polynomials {𝑝1 , 𝑝2 , … , 𝑝𝑠 } be the basis of P𝑞𝑑 , thus,

𝑢𝑙+1 (x) − 𝑢𝑙−1 (x) 𝑢𝑙+1 (x) − 2𝑢𝑙 (x) + 𝑢𝑙−1 (x) +𝛽 2𝜏 𝜏2 96

L. Su

Engineering Analysis with Boundary Elements 108 (2019) 95–107

x4i i

x3i

xi = x1i

x5i

x2i

Fig. 1. Seven five-node local domains and local domain Ωi with five nearest points of x𝑖 .

a map 𝜑 ∶ x𝑘𝑖 → x𝑗 or (i, k) → j, where 𝑖 = 1, 2, … , 𝑛 and 𝑘 = 1, 2, … , 𝑚,

Ψ(x) can be written as follows [37], Ψ(x) =

𝑠 ∑ 𝑖=1

𝑚 ∑ 𝑘=1

𝜁𝑖 𝑝𝑖 (x), where 𝑠 =

𝜆𝑘 𝑝𝑗 (x𝑘𝑖 )

̃ 𝐔𝑛 , F 𝐔𝑛 = 𝐀

(𝑞 − 1 + 𝑑)! , 𝑑!(𝑞 − 1)!

= 0, 𝑗 = 1, 2, … , 𝑠.

̃ = [𝑎̃𝑖𝑗 ] is a n × n sparse matrix obtained by padding every row where 𝐀 vector with zero entries based on the map 𝜑 and combining all matrix on every local domain. For any interior points, take the operator F as Δ, for any boundary points, take the operator F as the corresponding boundary operator, the discrete Eqs. (2.7) with boundary conditions can be written as,

(3.2)

In this paper, we consider the nonlinear sine-Gordon equation on two dimensional space, where x = (𝑥, 𝑦) and 𝑠 = (𝑞 + 2)(𝑞 + 1)∕2, based on the works [35,39], the additional polynomial basis can be taken as the following forms and the degree of the augmented polynomial in the computation should be consistent with the order of GTPS radial basis function, 𝑞

{𝑝1 , 𝑝2 , … , 𝑝𝑠 } = {1, 𝑥, 𝑦, 𝑥 , 𝑥𝑦, 𝑦 , … , 𝑥 , 𝑥 2

2

𝑞−1

𝑦, … , 𝑥𝑦

𝑞−1

B𝐔𝑙+1 = C𝐔𝑙 + D𝐔𝑙−1 + 𝜏 2 [𝜚𝑆]𝑙 + [𝐻]𝑙+1 ,

F 𝑢(x) ⋍

B𝐔1 = C𝐔0 + D𝐔−1 + 𝜏 2 [𝜚𝑆]0 + [𝐻]1 ,

𝑞

, 𝑦 },

𝑘=1

𝜆𝑘 F Φ(𝑟𝑘 ) + F Ψ(x).

where F 𝐔𝑚 = F 𝚼𝑚+𝑠 𝚼−1 𝑚+𝑠 , from (3.6) can get,

⋯,

(B − D)𝐔1 = C𝐟 − 2𝜏D𝐠 + 𝜏 2 [𝜚𝑆]0 + [𝐻]1 , using the notation

𝑢𝑙𝑖

= 𝑢(x𝑖

, 𝑡𝑙 ),

(3.12)

where

𝐟 = [𝑓1 , 𝑓2 , … , 𝑓𝑛 ]𝑇 , 𝐔𝑙 = [𝑢𝑙1 , 𝑢𝑙2 , … , 𝑢𝑙𝑛 ]𝑇 , 𝑇 𝐔𝑙±1 = [𝑢𝑙±1 , 𝑢𝑙±1 , … , 𝑢𝑙±1 𝑛 ] , 1 2

𝐠 = [𝑔1 , 𝑔2 , … , 𝑔𝑛𝐼 , 0, … , 0]𝑇 , 𝑇 [𝐻]𝑙+1 = [0, … , 0, ℎ𝑙+1 , … , ℎ𝑙+1 𝑛 ] , 𝑛 +1 𝐼

(3.6) 𝑇 F 𝑢(x𝑚 𝑖 )] ,

[𝜚𝑆]𝑙 = [𝜚1 sin(𝑢𝑙1 ), 𝜚2 sin(𝑢𝑙2 ), … , 𝜚𝑛𝐼 sin(𝑢𝑙𝑛 ), 0, … , 0]𝑇 , 𝐼

let A𝑚+𝑠 =

F 𝐔𝑚 = A𝑚 𝐔𝑚 .

and the sparse matrices B = [𝑏𝑖𝑗 ], C = [𝑐𝑖𝑗 ] and D = [𝑑𝑖𝑗 ] are given as follows, ⎧− 1 𝜏 2 𝑎̃𝑖𝑗 , 𝑖 ≠ 𝑗, ⎪ 4 𝑏𝑖𝑗 = ⎨(1 + 𝛽𝜏 ) − 1 𝜏 2 𝑎̃𝑖𝑗 , 4 ⎪𝑎̃ , 2 ⎩ 𝑖𝑗

(3.7) x𝑘𝑖

(3.11)

substitute 𝑢−1 (x) = 𝑢1 (x) − 2𝜏𝑔(x) as vector form into Eq. (3.10) and combined with the initial condition (2.2), we obtain,

where 𝐙𝑠 = [𝜁1 , 𝜁2 , ⋯ , 𝜁𝑠 ]𝑇 , 𝚲𝑚 = [𝜆1 , 𝜆2 , ⋯ , 𝜆𝑚 ]𝑇 , 𝐔𝑚 = 𝑇 [𝑢(x1𝑖 ), 𝑢(x2𝑖 ), ⋯ , 𝑢(x𝑚 𝑖 )] . For convenience, the coefficient matrix in left hand of (3.4) is denoted as 𝚼𝑚+𝑠 , the invertibility of 𝚼𝑚+𝑠 was proved in Yao’s Ph.D. thesis [29]. Thus, the coefficients can be obtained, ( ) ( ) 𝚲𝑚 𝐔𝑚 = 𝚼−1 , (3.5) 𝑚+𝑠 𝟎 𝐙𝑠 𝑠

F 𝑢(x2𝑖 ),

𝑢−1 (x),

𝑢1 (x) − 𝑢−1 (x) = 𝑔(x), 2𝜏

Combining Eq. (3.1) with (3.2) we can obtain the following linear system on the local domain Ωi , ( )( ) ( ) 𝚽𝑚𝑚 𝐏𝑚𝑠 𝚲𝑚 𝐔𝑚 ≃ , (3.4) 𝐏𝑇𝑚𝑠 𝟎𝑠𝑠 𝐙𝑠 𝟎𝑠

[F 𝑢(x1𝑖 ),

(3.10)

we use the initial velocity (2.3) to approximate

(3.3)

Combine with Eq. (3.3), we have, ( ) ( ) ( ) F 𝐔𝑚 𝚲𝑚 𝐔𝑚 = F 𝚼𝑚+𝑠 = F 𝚼𝑚+𝑠 𝚼−1 , 𝑚 + 𝑠 𝟎𝑠 𝐙𝑠 𝟎𝑠

(3.9)

note that when 𝑙 = 0 Eq. (3.9) has the following form,

And, for any linear partial differential operator F , we have, 𝑚 ∑

(3.8)

{x𝑗 }𝑛𝑗=1 ,

Because every ∈ Ω𝑖 corresponds to a unique x𝑗 ∈ we can reformulate (3.7) in terms of global U𝑛 instead of the local Um based on 97

for 𝑖 = 1, 2, … , 𝑛𝐼 , 𝑖 = 𝑗,

for 𝑖 = 1, 2, … , 𝑛𝐼 , for 𝑖 = 𝑛𝐼 + 1, 𝑛𝐼 + 2, … , 𝑛,

L. Su

Engineering Analysis with Boundary Elements 108 (2019) 95–107

T=3

T=1

L T=3

L T=1 RMSE T = 1

−2

Errors

10

Errors

−2

10

−3

10

RMSE T = 3

−3

10

−4

10

−4

10 1/10

1/20

1/50

1/10

1/100

1/20

1/50

1/100

Fig. 2. The L∞ and RMSE of T = 1, 3 with different time steps 𝜏.

Fig. 3. The analytical solutions and absolute errors with 𝑇 = 7 using LMAPS.

⎧ 1 𝜏 2 𝑎̃𝑖𝑗 , 𝑖 ≠ 𝑗, ⎪2 𝑐𝑖𝑗 = ⎨2 + 1 𝜏 2 𝑎̃𝑖𝑗 , 𝑖 = 𝑗, ⎪0 , 2 ⎩ ⎧ 1 𝜏 2 𝑎̃𝑖𝑗 , 𝑖 ≠ 𝑗, ⎪4 𝑑𝑖𝑗 = ⎨−(1 + 𝛽𝜏 ) + 1 𝜏 2 𝑎̃𝑖𝑗 , 2 4 ⎪0 , ⎩

the same time, except for the first time step, the sparse matrices are the same at every time steps, so we only need to invert one sparse system. Additionally, solving sparse system at each time step is faster and more efficient than a full one. So, although many small matrices must be solved in LMAPS, reasonable calculation costs can be maintained [29].

for 𝑖 = 1, 2, … , 𝑛𝐼 , for 𝑖 = 1, 2, … , 𝑛𝐼 , for 𝑖 = 𝑛𝐼 + 1, 𝑛𝐼 + 2, … , 𝑛, for 𝑖 = 1, 2, … , 𝑛𝐼 , 𝑖 = 𝑗,

for 𝑖 = 1, 2, … , 𝑛𝐼 , for 𝑖 = 𝑛𝐼 + 1, 𝑛𝐼 + 2, … , 𝑛,

4. Numerical example

Then, we can easily and efficiently obtain all approximate solutions by using the existing sparse system solver to solve (3.9) and (3.12). At

We present several typical numerical examples of two dimensional sine-Gordon equation to demonstrate the effectiveness of the algorithm, 98

L. Su

Engineering Analysis with Boundary Elements 108 (2019) 95–107

T=3

0

T=5

10

0

10

L2

L2

L∞

L∞

RMSE

RMSE

−1

10

−1

Errors

Errors

10

−2

10

−2

10

−3

10

−3

10 1

2

3

4

1

κ

2

3

4

κ

Fig. 4. The L2 , L∞ and RMSE errors with different orders of GTPS RBF.

Exact solutions

The star−like domain 6

4

6 2

y

u

4 2

0

0 −2

6 4 2

−4

−6 −4

0

−2 −2

−6 −6

−4

−2

0 x

2

4

0 2

−4

6

x

−6

4 6

y

Fig. 5. The profiles of computational domain (left) and the exact solutions at 𝑇 = 2 (right).

we take the radial basis functions, GTPS ∶ 𝜙(𝑟) = 𝑟2𝜅 ln(𝑟),

For the error estimation and convergence analysis, the L2 , L∞ and RMSE are used, which are defined in the following,

𝜅 ∈ ℕ+ ,

and we can easily get, Φ(𝑟) =

1 1 𝑟2𝜅+2 ln 𝑟 − 𝑟2𝜅+2 , 4(𝜅 + 1)2 4(𝜅 + 1)3

√ √ 𝑛 √∑ 𝐿2 = √ (𝑢𝑖 − 𝑢̃ 𝑖 )2 ,

𝜅 ∈ ℕ+ ,

𝑖=1

where ΔΦ = 𝜙, and for the boundary condition (2.5) we use 𝜕 Φ(r)/𝜕 x and 𝜕 Φ(r)/𝜕 y, which are obtained from, 𝜕Φ(𝑟) 𝑥dΦ(𝑟) = , 𝜕𝑥 𝑟d𝑟

𝑦dΦ(𝑟) 𝜕Φ(𝑟) = . 𝜕𝑦 𝑟d𝑟

𝐿∞ = max |𝑢𝑖 − 𝑢̃ 𝑖 |, 1⩽𝑖⩽𝑛

√ √ 𝑛 √1 ∑ 𝑅𝑀𝑆𝐸 = √ (𝑢 − 𝑢̃ 𝑖 )2 , 𝑛 𝑖=1 𝑖

where n is the number of nodes, ui is the value of exact solution at the point x𝑖 , while 𝑢̃ 𝑖 is the numerical one. 99

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Engineering Analysis with Boundary Elements 108 (2019) 95–107

t=0

t=2

1 sin(u/2)

sin(u/2)

1

0.5

0

0.5

0 5

5

−6

0 0 −5

4

−4 0

2

6

x

−6

0

−4 −2

−5

t=5

1 sin(u/2)

sin(u/2)

y

t=7

1

0.5

0

0.5

0 5

5

−6

0 0 4

−4

−2

0

2

6

x

−6

0

−4 −5

−5

−2

2

6

x

y

4

t=9

y

t=11

1 sin(u/2)

1 sin(u/2)

2

6

x

y

4

−2

0.5

0

0.5

0 5

5

−6

0 0 −5 x

4 6

−6

0

−4

−4

−2

0

2

−5 x

y

Fig. 6. Symmetrically perturbed static line solitons at different times.

100

4 6

2 y

−2

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t=2

t=0

14

12

12

10

10

8

8

u

u

14

6

6

4

4

2

2 0

0 5

5

5 0

5 0

0 −5

0 −5

−5

y

−5

y

x

x

t=7

t=4

14

12

12

10

10

8

8

u

u

14

6

6

4

4

2

2

0

0 5

5

5 0

5 0

0 −5

0 −5

−5 y

x

y

−5 x

Fig. 7. Superposition of two orthogonal line solitons at different times using LMAPS. Table 2 The errors L2 , L∞ and RMSE using LMAPS with different T.

4.1. Test problem In this example we consider the following two dimensional sineGordon equation on the square domain [−7, 7] × [−7, 7] as test problem, 𝜕2 𝑢 𝜕2 𝑢 𝜕2 𝑢 = + − sin 𝑢, 𝜕𝑡2 𝜕𝑥2 𝜕𝑦2

T

HADIM [26] L2

𝑡 ∈ [0, 𝑇 ],

1 3 5 7

with the initial conditions, ( ) 𝑢(𝑥, 𝑦, 0) = 4 arctan exp(𝑥 + 𝑦) , 4 exp(𝑥 + 𝑦) 𝜕𝑢 (𝑥, 𝑦, 0) = − , 𝜕𝑡 1 + exp(2𝑥 + 2𝑦)

LMAPS L2

L∞

7.4 × 10 1.5 × 10−3 2.0 × 10−3 1.9 × 10−3 −4

2.35 × 10 5.15 × 10−4 5.72 × 10−4 6.98 × 10−4 −4

RMSE

L∞

3.1 × 10 3.9 × 10−3 5.1 × 10−3 5.7 × 10−3 −3

4.02 × 10 2.52 × 10−4 3.95 × 10−4 5.85 × 10−4 −4

2.18 × 10−5 2.82 × 10−5 3.62 × 10−5 4.01 × 10−5

We also discuss the impact of time steps 𝜏 on results, the L∞ and RMSE vary with time steps 𝜏 are shown in Fig. 2. The figures are obtained with 𝜅 = 4, 𝑚 = 35 and the collocation points are chosen by using 𝑑𝑥 = 𝑑𝑦 = 0.25. From the figures we can see that the errors decrease as the time step decrease. The results with different numbers of collocation points are also discussed, Table 3 gives the L∞ and RMSE error with different n for different T. The results in this table are got using 𝜅 = 4, 𝑚 = 35 and 𝜏 = 0.01. It is can be seen that, more nodes are used in the computation, smaller errors are gotten and for large number of nodes LMAPS is still very efficient and can produce higher accuracy, which is a good illustration of the ability of the LMAPS to solve large scale problems.

the theoretical solution of this problem is given as, ( ) 𝑢(𝑥, 𝑦, 𝑡) = 4 arctan exp(𝑥 + 𝑦 − 𝑡) , the boundary conditions are calculated from the theoretical solution. Firstly, we consider this problem with Dirichlet boundary conditions (2.4), Table 2 gives the L2 , L∞ norms and root-mean-square (RMS) of errors with different T, compared with the results in [26] obtained by high order compact alternating direction implicit method (HADIM). In this computation using LMAPS, we choose 𝜅 = 4, the number of nearest points 𝑚 = 35 and the same time step and space points with [26]. 101

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= 1.0

=0

15

10

10

u

u

15

5

5

0

0

5

5 −5

−5 0

0

0

0

−5

5

−5

5

y

y

x

x

= 0.5

= 1.5

15

10

10 u

u

15

5

5

0

0

5 −5

5 −5

0

0

0

0

−5

5

5

y

x

−5 y

x

Fig. 8. Superposition of two orthogonal line solitons with different 𝛽 when 𝑡 = 3. Table 3 The L∞ and RMSE errors with different numbers of collocation points n. n

T=1

T=3

L∞ 41 × 41 57 × 57 101 × 101 141 × 141

RMSE

2.01 × 10 5.04 × 10−4 5.59 × 10−4 4.01 × 10−4 −3

T=5

L∞

2.05 × 10 5.27 × 10−5 3.18 × 10−5 2.19 × 10−5 −4

RMSE

2.65 × 10 8.17 × 10−4 3.27 × 10−4 2.52 × 10−4 −3

L∞

3.38 × 10 1.01 × 10−4 1.69 × 10−4 2.82 × 10−5 −4

RMSE

2.38 × 10 7.88 × 10−4 5.88 × 10−4 3.95 × 10−4 −3

4.15 × 10−4 1.31 × 10−4 1.14 × 10−5 3.62 × 10−5

time step 𝜏 = 0.001, while in our method we use 𝜏 = 0.01. In addition, in the recent work of Li [41], the meshless singular boundary method (SBM) was considered, and from Fig. 2 in [41], it can be seen that the SBM yields almost the same results as the MK-MLPG with the same division of time and space as we used. These are good illustrations of that, LMAPS is more effective and capable. Fig. 4 shows the L2 , L∞ and RMSE errors with differen 𝜅 of GTPS RBF, from the figures we can see that, the higher order of polyharmonic splines is used, the smaller errors are obtained. Thirdly, to illustrate the ability of LMAPS for solving nonlinear sineGordon equation with complicated domains, we consider this test problem with Dirichlet boundary conditions on the star-shape area Ω, which is defined by the parametric equation as follows,

Secondly, this test problem with boundary conditions (2.5) is considered using LMAPS. The comparison with the results of polynomial differential quadrature method (PDQM) [24] and moving Kriging-based MLPG method (MK-MLPG) [40] is given in Table 4. In this computation we choose the same collocation points with [40] by 𝑑𝑥 = 𝑑𝑦 = 0.25, but the time step 𝜏 = 0.01, the number of nearest nodes 𝑚 = 75 for local domains and the order of GTPS 𝜅 = 4, the graphs of analytical solutions and absolute error with 𝑇 = 7 are presented in Fig. 3. Table 4 illustrates the good accuracy of LMAPS, it can be seen easily that, LMAPS obtains smaller errors than MK-MLPG method. And although the errors of PDQM looks the same with LMAPS, it should be note that, the results of MK-MLPG [40] and PDMQ [24] are got with 102

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Fig. 9. Two-periodic solitoff solutions at different times using LMAPS. Table 4 The comparison of numerical results using LMAPS with other methods. T

PDQM [24] L∞

1 3 5 7

MK-MLPG [40] RMSE

2.7 × 10 2.0 × 10−3 3.3 × 10−3 5.9 × 10−3 −3

L2

0.5 × 10 0.5 × 10−4 0.7 × 10−3 1.1 × 10−3 −3

LMAPS RMSE

L∞

4.80 × 10 1.20 × 10−1 1.78 × 10−1 2.06 × 10−1 −2

7.67 × 10 1.48 × 10−2 1.92 × 10−2 2.82 × 10−2 −3

Ω = {(𝑥, 𝑦)|𝑥 = 𝜌 cos 𝜃, 𝑦 = 𝜌 sin 𝜃, 𝜃 ∈ [0, 2𝜋]},

L2

8.42 × 10 2.11 × 10−3 3.12 × 10−3 3.62 × 10−3 −4

RMSE

L∞

3.42 × 10 4.69 × 10−2 5.57 × 10−2 6.43 × 10−2 −2

7.86 × 10 5.83 × 10−3 7.21 × 10−3 8.06 × 10−3 −3

6.01 × 10−4 8.22 × 10−4 9.77 × 10−4 1.12 × 10−3

conditions [20–22],

where 𝜌 = 3 + 3 cos2 (4𝜃), the profiles of the computational domain and the exact solution at 𝑇 = 2 are given in Fig. 5. In this computation, we use 𝜅 = 3, time step 𝜏 = 0.01 and the collocation points are selected uniformly and 𝑛𝐵 = 500. Tables 5 and 6 give the three errors at different time moment with different numbers of local domains m, the results in Table 5 are obtained using uniformly distributed points with 𝑛𝐼 = 3281, while Table 6 using 𝑛𝐼 = 11949.

( ( )) 𝑓 (x) = 4 tan−1 exp 𝑥 + 1 − 2sech(y + 7) − 2sech(y − 7) ,

g(x) = 0,

𝜕𝑢 𝜕𝑢 and the boundary conditions 𝜕𝑥 = 𝜕𝑦 = 0. Fig. 6 presents the perturbation of a single soliton in terms of sin (u/2) with different t. These graphs are obtained by using LMAPS with 𝑑𝑥 = 𝑑𝑦 = 0.25, the time step 𝜏 = 0.01 and 𝜅 = 3, 𝑚 = 25. The figures show that two moved symmetric dents interacted at time 𝑡 = 7 and retained their shape after the collision, these results are in good agreement with [20–22] and the pictures are also consistent with those given in [20–22].

4.2. Perturbation of a line soliton In this example perturbation of a line soliton solution of (2.1) is considered on [−7, 7] × [−7, 7] with 𝛽 = 0.05, 𝜚(x) = −1 and the initial 103

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Engineering Analysis with Boundary Elements 108 (2019) 95–107

Fig. 10. Two-sawtooth-solitoff solution at different times by LMAPS.

Table 5 The L2 , L∞ and RMSE errors with different m using 𝑛𝐼 = 3281 and 𝑛𝐵 = 500. m

45 55 65 75

T=1

T=2

T=4

L2

L∞

RMSE

L2

L∞

RMSE

L2

L∞

RMSE

6.94 × 10−3 3.27 × 10−3 3.67 × 10−3 3.68 × 10−3

1.02 × 10−3 6.51 × 10−4 6.73 × 10−4 6.77 × 10−4

1.12 × 10−4 6.05 × 10−5 5.97 × 10−5 5.99 × 10−5

9.43 × 10−3 4.62 × 10−3 4.64 × 10−3 4.29 × 10−3

1.31 × 10−3 1.16 × 10−3 9.93 × 10−4 8.64 × 10−4

1.53 × 10−4 7.51 × 10−5 7.55 × 10−5 6.97 × 10−5

9.61 × 10−3 5.95 × 10−3 6.11 × 10−3 5.81 × 10−3

1.27 × 10−3 1.49 × 10−3 1.51 × 10−3 1.41 × 10−3

1.56 × 10−4 9.67 × 10−5 9.92 × 10−5 9.44 × 10−5

Table 6 The L2 , L∞ and RMSE errors with different m using 𝑛𝐼 = 11949 and 𝑛𝐵 = 500. m

65 75 95

T=1

T=2

L2

L∞

RMSE

L2

L∞

RMSE

2.88 × 10−2 2.94 × 10−2 3.06 × 10−2

6.59 × 10−3 6.48 × 10−3 6.83 × 10−3

2.57 × 10−4 2.63 × 10−4 2.74 × 10−4

5.04 × 10−2 9.82 × 10−2 1.85 × 10−1

6.82 × 10−3 3.39 × 10−2 3.88 × 10−2

4.52 × 10−4 1.36 × 10−3 1.66 × 10−3

104

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Engineering Analysis with Boundary Elements 108 (2019) 95–107

t=0 t=0

10 5

sin(u/2)

1

0

0 −1

−5 y

10

−10

0

−15 −10

−20 10 −20

−25

0 −10

y −30

−30 −30

−20

−25

−20

−15

x

−30

−10 x

−5

0

5

10

−5

0

5

10

t=2 t=2

10 5

sin(u/2)

1

0

0 −1

−5 y

10

−10

0

−15 −10

−20 10 −20

−25

0 −10

y −30

−30 −30

−20

−25

−20

−15

x

−30

−10 x

t=4 t=4

10 5

sin(u/2)

1

0

0 −1

−5 y

10

−10

0

−15 −10

−20 10 −20

−25

0 −10

y −30

−30 −30

−20

−25

−20

−15

x

−30

−10 x

−5

0

5

10

t=7 t=7

10 5

sin(u/2)

1

0

0 −1

−5 y

10

−10

0

−15 −10

−20 10 −20

−25

0 −10

y −30

−30 −30

−20 −30

x

−25

−20

−15

−10 x

Fig. 11. Collision of four expanding ring solitons at different times using LMAPS.

105

−5

0

5

10

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4.3. Superposition of two orthogonal line solitons

4.6. Collision of four expanding ring solitons

In this example, the superposition of two line solitons is obtained with the initial condition [17,20–23],

The last example we consider the collision of four expanding circular ring solitons with 𝜚(x) = −1 and initial conditions as following [19–22] ( ( )) √ 4 − (𝑥 + 3)2 + (𝑦 + 3)2 −1 𝑓 (𝑥, 𝑦) = 4 tan exp , 0.436

𝑓 (𝑥, 𝑦) = 4 tan−1 (exp(𝑥)) + 4 tan−1 (exp(𝑦)), 𝜕𝑢 𝜕𝑥

𝑔(𝑥, 𝑦) = 0,

𝜕𝑢 𝜕𝑦

and the boundary conditions = = 0. The computational domain is [−6, 6] × [−6, 6] and coefficient of nonlinear term 𝜚(𝑥, 𝑦) = −1. Firstly, we consider the undamped problem with 𝛽 = 0. The solutions u(x, y, t) with different t are presented in Fig. 7 by using LMAPS. In the computation using 𝑑𝑥 = 𝑑𝑦 = 0.25, 𝜏 = 0.01 and 𝜅 = 2, 𝑚 = 25. These graphs show that, two orthogonal line solitons parallelled to the diagonal 𝑦 = −𝑥 are undisturbed moving from each other in the direction of 𝑦 = 𝑥, this conclusion is consistent with the result in [17,20–23] and the figures are the same as those, which are given in [17,20–23]. Secondly, we consider the problem with different dissipative. In general, the presence of the dissipative term delays the propagation of the solitons. To examine this point, the graphs at 𝑡 = 3 with different 𝛽 are shown in Fig. 8. From the figures, we can see the delay becomes obvious when the larger value of 𝛽 is chosen, this is agree with [20,21].

( 𝑔(𝑥, 𝑦) = 4.13∕ cosh

In this paper we extend the localized method of approximate particular solutions (LMAPS) to solve second-order time-dependent hyperbolic sine-Gordon equation. In the computation we use generalized thin plate splines (GTPS) radial basis function (RBF) with additional low-order polynomial basis to avoid selecting shape parameters during localization. Numerical examples involving line and ring solitons, as well as two kinds of new solitons, two-sawtooth-solitoff solution and two-sawtoothsolitoff solution, demonstrate the effectiveness of the algorithm. The results of numerical examples are in agreement with many published papers. It should be note that, the LMAPS can be used to solve nonlinear sineGordon problem with complicated computational domains, because of its meshless features. Moreover, LMAPS can handle large-scale problems very well, from the results of test problem, we can see that even collocation points are selected almost 20 thousands, LMAPS can also obtain high accuracy results. In addition, using LMAPS to solve nonlinear sineGordon equation can get numerical solutions closer to the exact ones, the comparison of LMAPS with some other existed methods illustrates accuracy and effectiveness. In the end, it is worth mentioning that in the calculation process, the selection of the nearest m neighbor points for each computational point is very important. In the case of using a large number of interpolation points, the consideration of an effective search algorithm is worthwhile. It has been proven that the Kd-tree algorithm is very effective among many search algorithms [29,30]. In the calculation of the numerical examples in the previous section, we adopted the Kd-tree search algorithm [42] to find the nearest m neighbor points.

(

1 [ (1.2𝑥 + 1.6𝑦 + 2𝑡) arctan(1.2𝑥 + 1.6𝑦 + 2𝑡) 1+𝜍 ] ]) + 𝑥 + 3𝑦 + 3𝑡 , 𝜍 sn

) (𝑥 + 3)2 + (𝑦 + 3)2 , 0.436

5. Conclusion

Two-periodic solitoff solution of two-dimensional sine-Gordon equation with 𝛽 = 0 and 𝜚(𝑥, 𝑦) = 1 is considered, we use the analytical solution, which was first given in [14], has the following form, [ √

√

and the boundary conditions (2.5) is zero. The figures of the estimated solutions in terms of sin (u/2) with different times for 𝛽 = 0.05 are given in Fig. 11. These graphs are obtained extended across 𝑥 = −10 and 𝑦 = −10 by symmetry relations from one-quarter of the domain. In this computation we use LMAPS with 𝑑𝑥 = 𝑑𝑦 = 0.5, 𝜏 = 0.01 and 𝜅 = 2, the number of nearest points 𝑚 = 25. The result shows a complex collision of four expanding ring solitons, which is in excellent agreement with corresponding results given in [19–22].

4.4. Two-periodic solitoff solution

𝑢(x, 𝑡) = 4 arctan

4−

where sn is Jacobi elliptic sine function and constant 𝜍 is the modulus of the Jacobi elliptic function. In the example, Dirichlet boundary conditions are obtained form the analytical solution, as well as the initial conditions and the computational domain is [−8, 4] × [−8, 4]. Fig. 9 presents the graphs of estimated solution of the two-periodic solitoff solution with different time t using LMAPS and 𝜍 = 0.9. In these computation, we use 𝑑𝑥 = 𝑑𝑦 = 0.24 and the time step 𝜏 = 0.01, 𝜅 = 3 and number of neighbor nodes 𝑚 = 35. From the figures we can see that two traveling waves propagate with different velocities in different directions construct the solitoff type solution, this is consistent with that described in [14]. 4.5. Two-sawtooth-solitoff solution In this example we consider a two-sawtooth-solitoff solution, which was given in W-X Chen’s work [14], with 𝜚(𝑥, 𝑦) = 1, 𝛽 = 0 and the computational domain is [−2, 6] × [−8, 0], the solution has the form as follows, ] [ √ ( 1 [ ]) 𝑢(x, 𝑡) = 4 arctan , 7sech 2 (3𝑥 + 4𝑦 + 5𝑡) + 𝑥 + 3𝑦 + 3𝑡 + 3 𝜍 sn 1+𝜍

Acknowledgments The author would like to thank Professor C. S. Chen for carefully reviewing our article and providing his valuable comments and positive suggestions. And also thanks Professor Yalchin Efendiev and Professor Vasily I. Vasil’ev for their valuable suggestions when the author revised the manuscript. The work was supported by the mega-grant of Russian Federation Government (14.Y26.31.0013) and RFBR (project 17-01-00689).

where sn and 𝜍 are the same as in example 5.4 and sech is hyperbolic secant function. In the example, we will use the Dirichlet boundary conditions and all known conditions are obtained form the analytical solution. Fig. 10 shows the graphs of estimated solution of the twosawtooth-solitoff solution with different time t using LMAPS and 𝜍 = 1. In this computation, we use 𝑑𝑥 = 𝑑𝑦 = 0.16 and the time step 𝜏 = 0.01, the degree of GTPS 𝜅 = 3 and number of neighbor nodes 𝑚 = 35. From the figures we can see that a kink soliton and an antikink soliton, with different travelling velocities, constructs the two toothed-splitoff, and during the propagation process, the angle between the two solitoff waves is constant, which is the same as described in [14].

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