1 Introduction
The sineGordon equation (SGE) is a nonlinear hyperbolic partial differential equation (PDE) involving the d’Alembert operator and the sine of the unknown function, and the SGE plays an important role in many mathematical physics applications. It was originally introduced by
Bour (1862) and rediscovered by Frenkel and Kontorova (1939). Further details about the background and applications of the SGE can be found in Aero et al. (2009); Drazin and Johnson (1989); Scott et al. (1973); Taleei and Dehghan (2014). In this paper, we focus on developing an effective means for the numerical solution of the SGE in an arbitrary number of dimensions. A dimensional SGE generally takes the form:(1)  
where is a positive integer, is the Laplacian operator in spatial dimensions, . The initial conditions associated with Eq.(1) are given by
(2)  
(3) 
and the Neumann boundary conditions are
(4) 
where denotes the (typically exterior) normal to the boundary of the domain, and is the boundary of , i.e . The real parameter weights the dissipative term. When , Eq.(1) reduces to an undamped SGE in spatial variables, while when , the damped SGE is obtained. The function can be interpreted as a Josephson current density, while and in Eqs.(2) and (3) represent wave modes or the kink and velocity, respectively.
The dimensional SGE first appeared in a strictly mathematical context in differential geometry regarding the theory of surfaces of constant curvature Lamb (1971). Moreover, it is well known that the dimensional SGE arises in many important systems, such as the Thirring model, the Coulomb gas system and the ferromagnetic model, etc. Minnhagen (1985, 1987); Ni et al. (1990). Because of its wide applications, the dimensional SGE has been studied with a variety of numerical methods, including finite difference methods (FDM) and finite element methods (FEM), etc. Argyris and Haase (1987). Recently, additional solution methods have been proposed including collocation Dehghan and Shokri (2008a); Lakestani and Dehghan (2010), the boundary integral approach Dehghan and Mirzaei (2008a); Dehghan and Ghesmati (2010a), and a combination of the finite difference with the the diagonally implicit RungeKuttaNyström (DIRKN) method Mohebbi and Dehghan (2010), etc.
There is recent interest in the SGE in higher dimensions. As Barone et al. (1971) pointed out, Eq.(1) also has been applied in many branches of physics for the and cases. The exact solutions for the undamped SGE in higher dimensions have been obtained by Hirota’s method Hirota (1973), Lamb’s method Zagrodzinsky (1979), the Bäcklund transformation Christiansen and Olsen (1979) and Painlevé transcendents Kaliappan and Lakshmanan (1979), etc. Moreover, numerical solutions for the dimensional undamped SGE have been proposed by Christiansen and Lomdahl (1981) using a generalized leapfrog method, Guo et al. (1986) using two finite difference schemes, Argyris et al. (1991) using finite elements. Xin (2000) studied the SGE as an asymptotic reduction of the two level dissipationless MaxwellBloch system, Sheng et al. (2005) presented a numerical method with a split cosine scheme, and Bratsos (2005) used a threetime level fourthorder explicit finite difference scheme to solve the undamped SGE. Following a similar approach, Bratsos (2007) transformed the SGE to a secondorder initial value problem with the help of the method of lines. Numerical approaches for the damped SGE were proposed by Nakajima et al. (1974) who considered dimensionless loss factors and unitless normalized bias, and Gorria et al. (2004) investigated nonlinear wave propagation in a planar wave guide consisting of two rectangular regions joined by a bent domain of constant curvature using as a model of the kink solution to the SGE. Additionally, Dehghan and Mirzaei (2008b) developed the dual reciprocity boundary element method for both the undamped and damped dimensional SGE, and Jiwari et al. (2012b) obtained a numerical scheme based on a polynomial differential quadrature method.
Although the SGE is nonintegrable except for , some properties and exact solutions for the dimensional SGE have been obtained by various methods. Kobayashi and Izutsu (1976) extensively studied the exact traveling wave solutions of the SGE in the field of theoretical physics. Many additional mathematical methods have been proposed for finding traveling wave solutions of the SGE. Feng (2004) applied the Painlevé analysis to the study of an approximate SGE and its traveling solitary wave solution in dimensional space. With the help of exact solutions to the cubic nonlinear KleinGordon fields, Lou et al. (2005) studied the exact solutions for the dimensional SGE. Adopting and , de la Hoz and Vadillo (2012) generalized the exact soliton solution for the dimensional SGE:
(5) 
where . By adopting the proper ansatz, more general solutions can be obtained for the multidimensional SGE, including the threedimensional case allowing for nonconstant Aero et al. (2009). Obtaining the exact solution for the general SGE would be ideal, but unfortunately it is very difficult for practical engineering problems that are usually complex in nature. Despite numerical methods commonly used in many types of linear and nonlinear PDEs, de la Hoz and Vadillo (2012) remarked that there were no references to the numerical treatment of the SGE for dimensions larger than three, which motivated them to propose a numerical method for the dimensional SGE based on using operational matrices.
Many standard numerical methods for solving PDEs are widely used in engineering, but they usually require the construction and update of a mesh, which is an inherent disadvantage. In order to overcome these difficulties, recently the meshless numerical method has attracted attention. This method can establish a system of algebraic equations over the entire problem domain without using a predefined mesh. Rather, a set of scattered nodes, called field nodes, are used within the problem domain as well as on the boundaries of the domain Liu and Gu (2005). The meshless method does not require a priori
information about the relationship between the nodes for the interpolation or approximation of unknown functions over the field of variables
Liu and Gu (2005). Employing the meshless method, Dehghan and Shokri (2008a)studied the onedimensional nonlinear SGE and used Thin Plate Spline Radial Basis Functions (TPSRBF) to approximate the solution, and they also applied the TPSRBF method to both the KleinGordon equation
Dehghan and Shokri (2009) and the twodimensional SGE Dehghan and Shokri (2008b). A series of meshless approaches have been presented Asgari and Hosseini (2013); Dehghan and Ghesmati (2010b); Jiang and Wang (2012); Jiwari et al. (2012a); Karamanli and Mugan (2013); Mirzaei and Dehghan (2009, 2010); Pekmen and TezerSezgin (2012); Shao and Wu (2014); Taleei and Dehghan (2014). Moreover, since the nodal distribution for most existing meshless methods is preassigned, Xu et al. (2015) proposed a numerical twostep meshless method for solitonlike structures based on the optimal sampling density of kernel interpolation.In dealing with highdimensional PDEs, obtaining good quality approximate solutions is a difficult problem because of the socalled ‘curse of dimensionality’. High dimensional model representation (HDMR)
Li et al. (2001); Rabitz and Alis (1999, 2000); Rabitz et al. (1998); Sobol (1993)provides a viable approach based on the fact that highdimensional functions often can be efficiently expressed as sums of lowdimensional functions. The HDMR decomposition is also well known in statistics as the ANOVA (analysis of variance) decomposition
Efron and Stein (1981); Fisher (1925); Griebel and Holtz (2010); Stone (1994). In recent years, the HDMR decomposition has been under rapid development becoming an important tool for understanding highdimensional functions Griebel and Holtz (2010); Griebel et al. (2010, 2013); Li et al. (2001); Luo et al. (2014a, b); Rabitz and Alis (1999, 2000); Rabitz et al. (1998); Sobol (1993). In this paper, we will use a HDMR decomposition in conjunction with the Fourier hyperbolic cross (HC) approximation Luo et al. (2017).The remainder of the paper is organized as follows: Section 2 presents a function approximation method using HDMRHC. Then a new meshless numerical scheme is proposed in Section 3 for solving the dimensional SGE using the HDMRHC approximation. In Section 4, we provide several examples with a comparative numerical error analysis. Section 5 summarizes the relevant results.
2 Function approximation using HDMRHC and the partition of unity
2.1 HDMRHC approximation
Let , and be a variate function which is periodic in each variable, where is the torus given by Luo et al. (2017)
and the function space is defined by
(6) 
with the norm
where denotes a dimensional multiindex with the norm
(7) 
and the th order mixed derivative is given by
(8) 
We consider the multivariate Fourier series of
(9) 
where the Fourier coefficients are defined by
For a nonempty set , let
and
Note that is the set consisting of the
dimensional zero vector. Then
can be decomposed into the following form(10) 
and we refer to this as a HDMR decomposition of ; then a multiple Fourier series can be decomposed with an HDMR structure
(11)  
If let
(12) 
that is,
(13) 
then we define the Fourier HDMRHC partial sum up to th order Luo et al. (2017)
(14) 
Suppose is the number of points , which depends on both and , and
then Eq.(14) can be rewritten as
(15) 
Moreover, it follows that, for any , and , the bound
holds pointwise almost everywhere, where the constant depends on and .
From the definition of , we find that is smaller as becomes larger. The convergence rate of the Fourier HDMRHC partial sums is closely related to the value of . In particular, for a given accuracy, when is large, generally is well approximated by a low order truncated HDMR, and this is the basic starting point of the Fourier HDMRHC approximation. For example, if there is a function of variables, and suppose and the desired accuracy is , then a th order Fourier HDMRHC partial sum is just a nd order truncated HDMR of the function when . Therefore, we expect that a low order truncated HDMRHC can be used to effectively capture the behavior of a highdimensional function and its derivatives.
2.2 Partition of unity
In this subsection, we will discuss how to approximate a function via HDMRHC. Suppose is a function defined on , where . First, we divide the domain into subdomains, denoted as , thus . In each subdomain , let be the centre of the region, and let denote all nodes in , i.e., . Further, suppose can be represented as , then we have
(16) 
where
(17) 
is the characteristic function of
satisfying(18) 
From Eq.(15), at any point can be approximated as
(19) 
where with the same for all nodes and is the th unknown coefficient. Utilizing the values of these nodes, there are equations with one for each node, then we have following matrix form
(20) 
where
is the vector of function values at the nodes, and
is the vector of undetermined coefficients with
where . Suppose exists (i.e., this condition can always be satisfied Powell (1992); Wendland (1998)), then can be obtained by solving Eq.(20), i.e.
(21) 
From Eq.(16), we have
(22) 
where . Then
(23) 
3 Solution for dimensional SGEs
Now, we present the numerical scheme for solving the dimensional SGE based on using the Fourier HDMRHC approximation. Suppose the approximated function of the field function , () is formally denoted as
(27) 
where is the number of field nodes used in the selected domain, and is the vector that collects the true nodal function values for these field nodes, and is the centre of this selected region. Further, the derivatives of at any point can be approximated as
(28) 
where denotes one element of .
In this paper, the time derivatives are approximated by the timestepping method and we have the following approximation:
(29) 
(30) 
where is the time step, and is the approximate value of at , . Moreover the CrankNicolson scheme is used to approximate at three respective times as
(31) 
To manage the nonlinearity, a Quasilinearization Method (QLM) is adopted. The QLM is very effective for dealing with the nonlinear aspects of the SGE and other PDEs. In this fashion the nonlinear term in Eq. (1) can be represented as
(32) 
Thus, Eq.(1) can be discretized as
(33)  
where , and .
Suppose field nodes are denoted as , where is determined by both and . Then from Eq.(26) we have an approximation of the field variable according to the HDMRHC approximation
(34) 
where depends on and . The derivatives of can be approximated as
(35) 
Therefore, for any point , the approximation in Eq.(33) can be written as
(36)  
Let , , and , , , , then Eq.(36) can be rewritten as
(37) 
For all field nodes we have following matrix form:
(38) 
where .
In using a meshless strong method to solve the PDE, the solution can be unstable if there is a derivative boundary condition, so the fictitious points method is used to impose derivative boundary conditions Liu and Gu (2005). Suppose there are nodes on the boundary, then along the derivative boundaries, another fictitious points are added outside of the domain. Two sets of equations are established at each derivative boundary node: one for the derivative boundary condition, and the other for the governing equation. With the
additional degrees of freedom,
, added into the system, then Eq.(36) can be rewritten as(39) 
and for a node at , that is, on the derivative boundary, the derivative boundary conditions have the form
(40) 
Assembling Eqs.(39) and (40) for the corresponding nodes, the discretized global system equation becomes
(41) 
where , , and are matrices, , . At the first time level, i.e. , we adopt the following:
(42) 
and
(43) 
where and are the initial conditions for all nodes introduced in Eqs.(2) and (3).
4 Numerical experiments
In this section, the proposed meshless numerical scheme is applied to several examples to show the efficiency and accuracy for the dimensional SGE. As mentioned in the previous section, to approximate the time derivatives we use a finite difference method, so an iterative scheme is employed to reach the final time . In order to test the performance of the numerical solution, we use the error and rootmeansquare (RMS) error norms defined as
(44) 
and
(45) 
where is the number of nodes, is the exact solution, and is the numerical solution. To assess both the stability and the solution accuracy, we compute the condition number of the system matrix, which is defined as
(46) 
where depends on the parameter and the number of nodes .
4.1 Test problem for a dimensional SGE
The test problem for a dimensional SGE has the following form Djidjeli et al. (1995); Dehghan and Shokri (2008b); Jiwari et al. (2012b)
(47) 
where , and the initial conditions are
(48) 
and the Neumann boundary condition is
(49) 
The analytic solution of this problem is:
(50) 
In this example, both the proposed HDMRHC meshless method and the radial basis function (RBF) method in Dehghan and Shokri (2008b) are used to numerically solve the equation. Since the field nodes of the RBF method in Dehghan and Shokri (2008b) is the Sobol sequence with , then we adopt the same total number of field nodes for the HDMRHC meshless method (). The time step is set to . The results of the two different measures of error are presented in Table 1. We see that with the same number of field nodes, the errors of the proposed HDMRHC meshless method are smaller than those of the RBF method. The condition numbers at particular times are also listed in Table 1.
error  RMSerror  

Time()  RBF  HDMRHC  RBF  HDMRHC  
1.0  0.0670  0.0326  0.0050  0.0043  
3.0  0.0834  0.0343  0.0103  0.0045  
5.0  0.1015  0.0355  0.0145  0.0045  
7.0  0.1516  0.0368  0.0187  0.0047 
Note: The results of RBF method come from Ref. Dehghan and Shokri (2008b)
4.2 Test problem for a dimensional SGE
To further test the proposed HDMRHC scheme, we choose a dimensional example, which involves all the implementation issues explained in the previous subsection. The exact solution has the form of Eq.(5)
where , , and
(51) 
The initial conditions are
(52)  
and the Neumann boundary conditions are
(53) 
Both the proposed HDMRHC method and the RBF method are used to solve the equation. The Sobol sequence is chosen as the field nodes with a total number of for the HDMRHC method ( and we use the HDMR approximation up to order ) and for the RBF method. In this case the time step is chosen as . Table 2 presents , RMS errors and the condition number at some selected times .
error  RMSerror  

Time()  RBF  HDMRHC  RBF  HDMRHC  
1.0  0.2071  0.1083  0.0130  0.0076  
3.0  0.1956  0.0910  0.0143  0.0079  
5.0  0.2132  0.1205  0.0165  0.0080  
7.0  0.2203  0.1124  0.0187  0.0081 
5 Conclusions
In this paper, we propose a new meshless solution method for highdimensional sineGordon equations. First, we present a function approximation using the HDMRHC decomposition. Then we divided the whole domain into several subdomains with the help of the partition of unity, and obtain a function approximation at any random point in each subdomain. Hence, we develop a numerical procedure for the highdimensional SGEs by a meshless strong solution method. The timestepping method is used to approximate the time derivatives of SGEs, and a quasilinearization scheme is performed to treat the nonlinearity of the equation. Finally, to demonstrate the accuracy of the proposed method with two numerical experiments. The examples suggest that the proposed procedure is attractive for solving highdimensional SGEs.
Acknowledgements.
The authors X.X. and X.L. acknowledge support from the National Science Foundation (Grant No. CHE1763198), and H.R. acknowledges support from the Templeton Foundation (Grant No. 52265).References
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