A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations

Abstract

A space-time fully decoupled wavelet integral collocation method (WICM) with high-order accuracy is proposed for the solution of a class of nonlinear wave equations. With this method, wave equations with various nonlinearities are first transformed into a system of ordinary differential equations (ODEs) with respect to the highest-order spatial derivative values at spatial nodes, in which all the matrices in the resulting nonlinear ODEs are constants over time. As a result, these matrices generated in the spatial discretization do not need to be updated in the time integration, such that a fully decoupling between spatial and temporal discretization can be achieved. A linear multi-step method based on the same wavelet approximation used in the spatial discretization is then employed to solve such a semi-discretization system. By numerically solving several widely considered benchmark problems, including the Klein/sine–Gordon equation and the generalized Benjamin–Bona–Mahony–Burgers equation, we demonstrate that the proposed wavelet algorithm possesses much better accuracy and a faster convergence rate than many existing numerical methods. Most interestingly, the space-associated convergence rate of the present WICM is always about order 6 for different equations with various nonlinearities, which is in the same order with direct approximation of a function in terms of the proposed wavelet approximation scheme. This fact implies that the accuracy of the proposed method is almost independent of the equation order and nonlinearity.

1. Introduction

In this study, we consider a numerical solution of the general nonlinear wave problem

$$\{\begin{array}{l}\alpha u_{tt}+\beta u_t-\gamma \Delta u_t-\lambda \Delta u+\rho \nabla u=g\left(u,\nabla u,\Delta u\right)+f, x\in \Omega , t\ge 0\hfill \\ u\left(x,0\right)=g_1\left(x\right), u_t\left(x,0\right)=g_2\left(x\right), u\left(x,t\right)=h\left(x,t\right) \mathrm{for} x\in \Gamma \hfill \end{array}$$

(1)

in which α, β, γ, and ρ are constants; $\Gamma$ is the boundary of the problem domain $\Omega ={[0,T]}^2$; u is the unknown function of time and space; g can be an arbitrary continuous function; f is the source term; g₁(x), g₂(x), and h(x, t) describe the initial and boundary conditions. The general form (1) covers several types of well-known wave problems arising in many scientific and engineering fields, such as nonlinear optics [1,2,3], solid state physics [4,5,6,7,8], and quantum field theory [9,10,11]. For instance, when $\alpha =1$, $\gamma =0$, $\beta =0$, $\lambda =1$, $\rho =0$, and $g(u)=au+b{u}^c$ with constants a, b, and c, Equation (1) degrades to the so-called nonlinear Klein–Gordon equation:

$$u_{tt}-\Delta u+au+b{u}^c=f$$

(2)

which is a vital mathematical model in field theory [9] and relativistic quantum mechanics [10,11]. Moreover, Equation (2) will become the sine–Gordon, sinh–Gordon, Liouville, Dodd–Bullough–Mikhailov, and Tzitzeica–Dodd–Bullough equations when replacing the nonlinear term $g(u)=au+b{u}^c$ by $\sin u$ [2], $\sinh u$ [3], $e^u$ [3], $e^u+e^{-2u}$ [8], and $e^{-u}+e^{-2u}$ [8], respectively.

Given the broad applications of nonlinear wave Equation (1) and the severe limitation in the application scope of various analytical and semi-analytical methods [4,5,6,7,8], a considerable number of studies have been conducted to develop numerical methods for solving the nonlinear wave problem (1) [3,9,10,11]. For example, Shao and Wu [9] proposed a Chebyshev tau meshless method to solve the nonlinear Klein–Gordon and sine–Gordon equations. Bulbul and Sezer [11] studied the same problems by using the Taylor polynomial collocation method. The differential quadrature method is also employed to solve the nonlinear Klein–Gordon and sine–Gordon equations [10]. In these numerical methods, the space-time domain is discretized simultaneously by handling the temporal dimension as an extra spatial dimension [9,10,11]. Therefore, such methods are usually referred to as space-time coupled formulations. Although the space-time coupled methods feature some attractive attributes, such as the absence of a time step restriction in stability and good capacity to capture sharp gradients, their computational cost is highly expensive, making them unacceptable for available computing resources, especially for three-dimensional problems [9,10,11,12,13]. Thus, the space-time decoupled formulations seem to be more popular in solving nonlinear wave equations, in which the initial boundary value problems are first transformed into a system of ordinary differential equations (ODEs) by spatial discretization and then further solved by using time integration methods [14,15,16,17,18,19,20,21,22]. For instance, Kuo and Luis [14] applied the Galerkin method to transform the Klein–Gordon equation into a system of nonlinear ODEs and then obtained its numerical solution by using the finite difference scheme. A similar space-time decoupled process is also employed to solve the sine–Gordon equation [15]. Many other space-time decoupled formulations for solving nonlinear wave problems, which combine various numerical techniques such as the finite element method [16], the collocation method [17], the finite difference method (FDM) [18,19,20,22], and the Runge–Kutta method [21], can be found in existing research.

The abovementioned space-time decoupled formulations normally entail costly computational resources compared with space-time coupled formulations [9,10,11,12,13,14,15]. However, in these space-time decoupled formulations, the matrices generated in the spatial discretization of nonlinear terms are normally dependent on the time-dependent unknown vector [14,15,21,22,23]. As a result, these matrixes have to be recalculated at each time step, thereby consuming a lot of computations. Since the recalculation of matrices representing the spatial discretization of nonlinear terms actually re-performs the spatial discretization at each time step, the decoupling between spatial and temporal discretization in these space-time decoupled methods is incomplete [23]. To remove this shortcoming, Liu et al. [24,25] proposed a space-time fully decoupled wavelet Galerkin method to solve nonlinear wave problems and the two-dimensional (2-D) Burgers’ equations. In this method, all the matrices generated in the spatial discretization are completely independent of time. Hence, these matrices never need to be recalculated in the time integration [24,25]. However, the convergence rate in space of this wavelet Galerkin method seems to be dependent on the nonlinearity and order of the equation. For example, its convergence rate can reach about order 7 for the sine–Gordon equation but less than order 4 for the Klein–Gordon equation with cubic nonlinearity [25]. When the wavelet Galerkin method is used to solve the second-order Burgers’ equation and fourth-order bending problem of thin rectangular plates, the former achieves fifth -order convergence [24], while the latter only features third-order convergence [26]. In fact, the situation that the convergence rate is sensitive to the form of nonlinearity and order of equation has also been encountered in many existing numerical methods [3,25,27,28,29,30].

Recently, a wavelet integral collocation method (WICM) was developed to solve nonlinear boundary value problems, and it shows high-order convergence for problems with various nonlinearities [31]. For this method, various derivatives of unknown function in the equation are denoted as new functions, and the equation is directly discretized by being satisfied at certain given points. Moreover, the new functions representing derivatives of the unknown function are correlated by integral approximation based on wavelet series expansion. Owing to the interesting property that the error order of the wavelet-based integral approximation scheme is independent of the integral order, the order of the error that is generated by the whole discretization process becomes consistent with that of the direct function approximation, so that the accuracy and convergence rate of the method does not depend on the order of the equation. Clearly, this unique property makes the WICM significantly different from most conventional methods.

Although nonlinearity, space-time coupling, and high-order derivatives pose major challenges to the accurate and efficient solution of nonlinear wave problems, most recent studies have demonstrated that these difficulties can be resolved very effectively when using the function expansion scheme based on Coiflet-type basis functions [31]. Therefore, for the solution of the nonlinear wave problem shown in Equation (1), we adopted the WICM method [31] in spatial discretization and a multi-step method [32] based on wavelet approximation in time discretization. This processing strategy was expected to allow the resulting algorithm possessing the following advantages: (1) spatial discretization and temporal discretization are completely decoupled; (2) both the spatial discretization and temporal discretization maintain high accuracy consistent with the direct approximation of a function, and almost independent of the order of the equation; (3) nonlinearity can be dealt with in a simple and unified way.

In order to effectively produce the establishment, analysis, and verification of such an algorithm, in Section 2, we discuss how we constructed a scheme to accurately approximate multiple-order integrals of functions in a bounded domain, which was used in developing the spatial discretization scheme; next, in Section 3, by using the WICM associated with the wavelet approximation method for multiple-order integrals, we first transformed the nonlinear initial-boundary-value wave problem into a system of ODEs. All the matrices in this semi-discretization system are constant and do not need to be updated in the subsequent time integration. In other words, a fully decoupling between spatial and temporal discretization was achieved. Next, a wavelet linear multi-step method [31] was employed to further solve these resulting nonlinear ODEs. Eventually, the solution method on nonlinear wave problems is established accordingly. In Section 4, the validation and convergence of the proposed wavelet method are examined through several widely used benchmark problems, including the Klein/sine–Gordon equation and the generalized Benjamin–Bona–Mahony–Burgers equation.

2. Wavelet Approximation of Multiple Integrals in a Bounded Domain

Following our previous works [24,25,31,32,33,34], a continuous function f(x) defined on the interval [0, T] can be approximated by the wavelet expansion

$$f(x)\approx P^{j}f(x)={\displaystyle \sum _{k=2^m}^{2^{m}T}f\left(k/2^m\right)}{\Phi }_{m,k}(x),\hspace{1em}x\in [0,T],$$

(3)

in which m is the resolution level controlling the accuracy, and the modified wavelet basis combining the boundary extension can be expressed as

$${\Phi }_{m,k}(x)=\{\begin{array}{l}\phi (2^{m}x+M_1-k)+{\displaystyle \sum _{j=-{\alpha }_2}^{-1}{T}_{L,k}(\frac{j}{2^m})\phi (2^{m}x+M_1-j)}, 0\le k\le {\alpha }_1\\ \phi (2^{m}x+M_1-k),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\alpha }_1+1\le k\le 2^{m}T-{\alpha }_2-1\\ \phi (2^{m}x+M_1-k)+{\displaystyle \sum _{j=2^{m}T+1}^{2^{m}T+{\alpha }_1}{T}_{R,2^{m}T-k}(\frac{j}{2^m})\phi (2^{m}x+M_1-j)}, 2^{m}T-{\alpha }_2\le k\le 2^{m}T\end{array}.$$

(4)

In definition (4) of the modified wavelet basis, $\phi (x)$ is the generalized Coiflet-type orthogonal scaling function, $M_1={\displaystyle \int x\phi (x)dx}$ is first-order moment of the scaling function, N is the number of vanishing moments of the corresponding wavelet function, and parameters ${\alpha }_1=M_1-1$ and ${\alpha }_2=3N-2-M_1$. We note that the generalized Coiflet-type wavelet with N = 6 and M₁ = 7 is employed in all the following numerical examples. In addition, the functions

$$T_{L,j}(x)={\displaystyle \sum _{i=0}^{N-1}{2}^{im}\frac{{\zeta }_{0,i,j}}{i!}{\left(x\right)}^i} \mathrm{and} T_{R,j}(x)={\displaystyle \sum _{i=0}^{N-1}{2}^{im}\frac{{\zeta }_{1,i,j}}{i!}{(x-T)}^i},$$

(5)

where the coefficients ${\zeta }_{0,i,k}$ and ${\zeta }_{1,i,k}$ are calculated through the relations $P_0=\{{\zeta }_{0,i,k}\}={(I-B_0)}^{-1}{A}_0$, and $P_1=\{{\zeta }_{1,i,k}\}={(I-B_1)}^{-1}{A}_1$ along with the matrices

$$\begin{array}{l}{A}_0=\{{\phi }^{(i)}(M_1-k)\}, A_1=\{{\phi }^{(i)}(k+M_1)\},\\ B_0=\{{\displaystyle \sum _{l=-{\alpha }_1}^{-1}\frac{1}{i!}{l}^i{\phi }^{(j)}(-l+M_1)}\}, \mathrm{and} B_1=\{{\displaystyle \sum _{l=1}^{M_1-1}\frac{1}{j!}{l}^j{\phi }^{(i)}(-l+M_1)}\}.\end{array}$$

(6)

The scaling function $\phi (x)$ and its derivatives ${\phi }^{(i)}(x)$ used in Equation (6) can be obtained by using the algorithm given in our previous works [32,34].

Making n-tuple integral of Equation (3) yields

$$f^{\left(-n\right)}\left(x\right)={\displaystyle {\int }_0^x{f}^{\left(-n+1\right)}(x^{\prime }})d{x}^{\prime }\approx {\displaystyle \sum _{k=0}^{2^{m}T}f\left(k/2^m\right)}{\Phi }_{m,k}^{\left(-n\right)}\left(x\right), x\in [0, T],$$

(7)

with

$${\Phi }_{m,k}^{\left(-n\right)}(x)=\{\begin{array}{l}{\phi }_{m,k-M_1}^{\left(-n\right)}(x)+{\displaystyle \sum _{j=-{\alpha }_2}^{-1}{T}_{L,k}(\frac{j}{2^m}){\phi }_{m,k-M_1}^{\left(-n\right)}(x)},\hspace{1em}\hspace{1em}\hspace{1em}0\le k\le {\alpha }_1\\ {\phi }_{m,k-M_1}^{\left(-n\right)}(x),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\alpha }_1+1\le k\le 2^{m}T-{\alpha }_2-1\\ {\phi }_{m,k-M_1}^{\left(-n\right)}(x)+{\displaystyle \sum _{j=2^{m}T+1}^{2^{m}T+{\alpha }_1}{T}_{R,2^{m}T-k}(\frac{j}{2^m}){\phi }_{m,k-M_1}^{\left(-n\right)}(x)},\hspace{1em}\hspace{1em}\hspace{1em}{2}^{m}T-{\alpha }_2\le k\le 2^{m}T\end{array},$$

(8)

in which the n-tuple integral of the modified wavelet basis can be calculated by the relation

$${\phi }_{m,k}^{\left(-n\right)}\left(x\right)={\displaystyle {\int }_0^x{\displaystyle {\int }_0^{{\xi }_n}\dots }}{\displaystyle {\int }_0^{{\xi }_2}\phi \left(2^{m}x-k\right)d{\xi }_1}d{\xi }_2\dots d{\xi }_n=\frac{1}{2^{mn}}\left({\phi }^{\left(-n\right)}\left(2^{m}x-k\right)-{\displaystyle \sum _{l=1}^n\frac{\left(2^{m}x\right)}{\left(n-l\right)!}{\phi }^{\left(-l\right)}\left(-k\right)}\right).$$

(9)

The special algorithm for exactly calculating the n-tuple integrals ${\phi }^{\left(-n\right)}\left(x\right)$ of the scaling function can be found in our previous work [31].

The accuracy of the wavelet approximation (7) on multiple integrals of the continuous function f(x) can be estimated as [31]

$${\Vert f^{\left(-n\right)}-f_{P_j}^{\left(-n\right)}(x)\Vert }_{\infty }\le P_{0,\infty }{2}^{-mN}, \mathrm{and} {\Vert f^{\left(-n\right)}-f_{P_j}^{\left(-n\right)}(x)\Vert }_{{\mathbf{L}}^2\left[0,T\right]}\le P_{0,L}{2}^{-mN},$$

(10)

in which the constants $P_{0,L}$ and $P_{0,\infty }$ are dependent on $d^{N}f(x)/d{x}^N$ and the number of integration n but independent of the resolution level m.

The set of scaling bases for 2-D space can be directly obtained by the tensor product of one-dimensional (1-D) wavelet bases. Thus, the n-tuple integral of the 2-D continuous function $f(x,y)\in {[0,T]}^2$ can be approximated by

$$f^{\left(-n,-p\right)}(x,y)\approx f_{P_j}^{\left(-n,-p\right)}(x,y)={\displaystyle \sum _{k=0}^{2^{m}T}{\displaystyle \sum _{l=0}^{2^{m}T}f(\frac{k}{2^{m}T},\frac{l}{2^{m}T}){\Phi }_{m,k}^{\left(-n\right)}(x){\Phi }_{m,l}^{\left(-p\right)}(y)}}+O(2^{-mN}).$$

(11)

3. Solution Procedure

Based on the wavelet approximation (11) and with the application of the boundary conditions, the unknown function u(x, y) in Equation (1) and its first-order partial derivatives $u_1=\partial u/\partial x$ and $v_1=\partial u/\partial y$ can be approximately expressed by their second-order partial derivatives $u_2={\partial }^{2}u/{\partial }^{2}x$ and $v_2={\partial }^{2}u/{\partial }^{2}y$ at nodes through the relations

$$\{\begin{array}{l}{u}_1={\displaystyle \sum _{k=0}^{2^{m}T}{\displaystyle \sum _{l=0}^{2^{m}T}{u}_2(\frac{k}{2^m},\frac{l}{2^m},t){\Phi }_{m,k}^{\left(-1\right)}(x){\Phi }_{m,l}\left(y\right)+u_1\left(0,y,t\right)}}\\ v_1={\displaystyle \sum _{k=0}^{2^{m}T}{\displaystyle \sum _{l=0}^{2^{m}T}{v}_2(\frac{k}{2^m},\frac{l}{2^m},t){\Phi }_{m,k}\left(x\right){\Phi }_{m,l}^{\left(-1\right)}(y)+v_1\left(x,0,t\right)}}\\ u={\displaystyle \sum _{k=0}^{2^{m}T}{\displaystyle \sum _{l=0}^{2^{m}T}{u}_2(\frac{k}{2^m},\frac{l}{2^m},t){\Phi }_{m,k}^{\left(-2\right)}(x){\Phi }_{m,l}\left(y\right)+x{u}_1\left(0,y,t\right)+h\left(0,y,t\right)}}\\ u={\displaystyle \sum _{k=0}^{2^{m}T}{\displaystyle \sum _{l=0}^{2^{m}T}{v}_2(\frac{k}{2^m},\frac{l}{2^m},t){\Phi }_{m,k}\left(x\right){\Phi }_{m,l}^{\left(-2\right)}(y)+y{v}_1\left(x,0,t\right)+h\left(x,0,t\right)}}\end{array}$$

(12)

in which

$$\{\begin{array}{l}{u}_1\left(0,y,t\right)=h\left(T,y,t\right)-h\left(0,y,t\right)-{\displaystyle \sum _{k=0}^{2^{m}T}{\displaystyle \sum _{l=0}^{2^{m}T}{u}_2(\frac{k}{2^m},\frac{l}{2^m},t){\Phi }_{m,k}^{\left(-2\right)}(T){\Phi }_{m,l}\left(y\right)}}\\ v_1\left(x,0,t\right)=h\left(x,T,t\right)-h\left(x,0,t\right)-{\displaystyle \sum _{k=0}^{2^{m}T}{\displaystyle \sum _{l=0}^{2^{m}T}{v}_2(\frac{k}{2^m},\frac{l}{2^m},t){\Phi }_{m,l}\left(x\right){\Phi }_{m,l}^{\left(-2\right)}(T)}}\end{array}.$$

(13)

Substituting Equation (12) into Equation (1) and then taking $x=k^{\prime }/2^m$ and $y=l^{\prime }/2^m$ for $k^{\prime },l^{\prime }=0,1,\dots ,2^{m}T$, respectively, one can obtain

$$\begin{array}{c}\alpha \mathbf{E}\frac{d^2{\mathbf{u}}_2\left(t\right)}{d{t}^2}+\beta \mathbf{E}\frac{d{\mathbf{u}}_2\left(t\right)}{dt}-\gamma \left(\left(\mathbf{B}-\mathbf{A}\right)\frac{d{\mathbf{u}}_2\left(t\right)}{dt}+\frac{d{\mathbf{f}}_2\left(t\right)}{dt}\right)-\lambda \left(\left(\mathbf{B}-\mathbf{A}\right){\mathbf{u}}_2+{\mathbf{f}}_2\left(t\right)\right)+\\ \rho \left(\left(\mathbf{F}+\mathbf{GA}\right){\mathbf{u}}_2{-\mathbf{Gf}}_2\left(t\right)\right)\approx {\mathbf{Bf}}_1\left(t\right)+\mathbf{B}g\left({\mathbf{u}}_2\right)\end{array},$$

(14)

in which g(u₂) indicates that the nonlinear term acts on each coordinate of the vector u₂ and the matrices E = AB, F = BC, and G = BDB⁻¹ with

$$\{\begin{array}{l}\mathbf{A}=\left\{a_{op}={\Phi }_{m,k}^{\left(-2\right)}\left(\frac{k^{\prime }}{2^m}\right){\Phi }_{m,l}\left(\frac{l^{\prime }}{2^m}\right)-\frac{k^{\prime }}{2^m}{\Phi }_{m,k}^{\left(-2\right)}\left(T\right){\Phi }_{m,l}\left(\frac{l^{\prime }}{2^m}\right)\right\},\mathbf{B}=\left\{b_{op}={\Phi }_{m,k}\left(\frac{k^{\prime }}{2^m}\right){\Phi }_{m,l}^{\left(-2\right)}\left(\frac{l^{\prime }}{2^m}\right)-\frac{l^{\prime }}{2^m}{\Phi }_{m,k}\left(\frac{k^{\prime }}{2^m}\right){\Phi }_{m,l}^{\left(-2\right)}\left(T\right)\right\}\\ \mathbf{C}=\left\{a_{op}={\Phi }_{m,k}^{\left(-1\right)}\left(\frac{k^{\prime }}{2^m}\right){\Phi }_{m,l}\left(\frac{l^{\prime }}{2^m}\right)-{\Phi }_{m,k}^{\left(-2\right)}\left(T\right){\Phi }_{m,l}\left(\frac{l^{\prime }}{2^m}\right)\right\},\mathbf{D}=\left\{b_{op}={\Phi }_{m,k}\left(\frac{k^{\prime }}{2^m}\right){\Phi }_{m,l}^{\left(-1\right)}\left(\frac{l^{\prime }}{2^m}\right)-{\Phi }_{m,k}\left(\frac{k^{\prime }}{2^m}\right){\Phi }_{m,l}^{\left(-2\right)}\left(T\right)\right\}\end{array}$$

(15)

The vectors $u_2={\left\{u_{2,p}=u_2\left(k/2^m,l/2^m\right)\right\}}^{\mathrm{T}}$, $v_2={\left\{v_{2,p}=v_2\left(k/2^m,l/2^m\right)\right\}}^{\mathrm{T}}$, and

$$\begin{array}{cc}\hfill f_1\left(t\right)=\{f_{1,o}=& f\left({\mathrm{Au}}_2(\frac{k}{2^m},\frac{l}{2^m},t)+k^{\prime }/2^m\left(h\left(T,\frac{l^{\prime }}{2^m},t\right)-h\left(0,\frac{l^{\prime }}{2^m},t\right)\right)+h\left(0,\frac{l^{\prime }}{2^m},t\right),\frac{l^{\prime }}{2^m},\frac{l^{\prime }}{2^m},t\right)\hfill \\ & -\rho \left(h\left(T,\frac{l^{\prime }}{2^m},t\right)-h\left(0,\frac{l^{\prime }}{2^m},t\right)\right)-\alpha d^2\left(\frac{k^{\prime }}{2^m}\left(h\left(T,\frac{l^{\prime }}{2^m},t\right)-h\left(0,\frac{l^{\prime }}{2^m},t\right)\right)+h\left(0,\frac{l^{\prime }}{2^m},t\right)\right)/d{t}^2\hfill \\ & -\beta d\left(\frac{k^{\prime }}{2^m}\left(h\left(T,\frac{l^{\prime }}{2^m},t\right)-h\left(0,\frac{l^{\prime }}{2^m},t\right)\right)+h\left(0,\frac{l^{\prime }}{2^m},t\right)\right)/dt{\}}^{\mathrm{T}}\hfill \end{array},$$

(16)

$$f_2\left(t\right)={\{f_{2,o}=\frac{l^{\prime }}{2^m}\left(h\left(\frac{k^{\prime }}{2^m},T,t\right)-h\left(\frac{k^{\prime }}{2^m},0,t\right)\right)+h\left(\frac{k^{\prime }}{2^m},0,t\right)-\frac{k^{\prime }}{2^m}\left(h\left(T,\frac{l^{\prime }}{2^m},t\right)-h\left(0,\frac{l^{\prime }}{2^m},t\right)\right)-h\left(0,\frac{l^{\prime }}{2^m},t\right)\}}^{\mathrm{T}},$$

(17)

and the subscripts $o=(2^m+1)k^{\prime }+l^{\prime }$, $p=(2^m+1)k+l$, and $k,l,k^{\prime },l^{\prime }=0,1,\dots ,2^{m}T$.

Notice that all the matrices A, B, C, D, E, F, and G in the semi-discretization system (14) of the nonlinear wave Equation (1) are constant matrices that are completely independent of the unknown vector U₂(t) and the time t. Therefore, no matrix generated in the spatial discretization has to be updated in the subsequent time integration to solve the nonlinear ODEs (14). Moreover, the proposed wavelet formulation holds a fully decoupling between spatial and temporal discretization.

Finally, the linear multi-step method constructed based on the Coiflet-type orthogonal wavelet with M₁ = 7 and N = 6 is employed to solve the nonlinear ODEs (14). First, the second-order ODEs are rewritten into the first-order equations as follows:

$$\{\begin{array}{l}\frac{d{\mathbf{u}}_2\left(t\right)}{dt}={\mathbf{w}}_2\\ \alpha \mathbf{E}\frac{d{\mathbf{w}}_2\left(t\right)}{dt}=\mathbf{H}\left({\mathbf{u}}_2,{\mathbf{w}}_2,t\right){=\mathbf{Qw}}_2\left(t\right)+{\mathbf{Ru}}_2+\left(\gamma +\lambda +\rho \mathbf{G}\right)d{\mathbf{f}}_2\left(t\right)/dt{+\mathbf{Bf}}_1\left(t\right)+\mathbf{B}g\left({\mathbf{u}}_2\right)\end{array},$$

(18)

in which $\mathbf{Q}=\left(\gamma \left(\mathbf{B}-\mathbf{A}\right)-\beta \mathbf{E}\right)$ and $\mathbf{R}=\lambda \left(\mathbf{B}-\mathbf{A}\right)-\rho \left(\mathbf{F}+\mathbf{GA}\right)$. Next, by applying the wavelet linear multi-step method [32], one can obtain

$$\{\begin{array}{l}{\mathbf{u}}_2\left(t_{i+1}\right)\approx {\mathbf{u}}_2\left(t_i\right)+\Delta t{\displaystyle \sum _{k=i-{\alpha }_2}^{i+1}{\omega }_k{\mathbf{w}}_2\left(t_k\right)}\\ \alpha {\mathbf{E}\mathbf{w}}_2\left(t_{i+1}\right)\approx \alpha {\mathbf{E}\mathbf{w}}_2\left(t_i\right)+\Delta t{\displaystyle \sum _{k=i-{\alpha }_2}^{i+1}{\omega }_k\mathbf{H}\left(t_k,u_2\left(t_k\right),{\mathbf{w}}_2\left(t_k\right)\right)}\end{array},$$

(19)

where $\Delta t$ is the time step. The coefficients ${\omega }_{-1},{\omega }_0,\dots ,{\omega }_9$ are listed in

Table 1. Coefficients for the wavelet linear multi-step method [32].

K	ωK	K	ωK	K	ωK
−1	0.328691834525214	3	−0.149711341365277	7	2.30481019094508 × 10−5
0	0.998606741614628	4	0.0345899116285495	8	2.08826733683054 × 10−10
1	−0.57539605262846	5	−0.00453827543632113	9	−9.43911615536308 × 10−13
2	0.367253112532719	6	0.000481043784287261

.

Applying Equation (21) recursively and using the initial condition directly in Equation (1), we can obtain the nodal values of the second-order partial derivatives of unknown functions. The unknown function can be further reconstructed through Equation (12).

4. Numerical Results

In this section, several widely considered test problems are studied numerically to demonstrate the validity of the proposed WICM. To quantificationally estimate the accuracy of numerical solutions, we introduce the error norms [3,27,28,29,30,35]

$$L_{\infty }\left(t\right)=\underset{1\le i\le N}{\max }\left|e_i\right|,\hspace{1em}{L}_2\left(t\right)=\sqrt{{\displaystyle \sum _{i=1}^N\left|{\left(e_i\right)}^2\right|}},\hspace{1em}\mathrm{and} \mathrm{RMS}\left(t\right)=L_2\left(t\right)/\sqrt{M},$$

(20)

in which $e_i=u_{\mathrm{exact}}(x_i)-u_{\mathrm{num}}(x_i)$ with the exact solution u_exact and the numerical solution u_num, and M is the total number of nodes in space. In addition, the rate of convergence in space is defined as [3,27,28,29,30,35]

$$C_{\mathrm{order}}=\log [L_2(N_2)/L_2(N_1)]/\log (N_2/N_1),$$

(21)

where L₂(N₂) and L₂(N₁) are the L₂ error norms of the numerical solution using N₂ and N₁ nodes, respectively.

Example 1.

We first consider the 1-D Klein–Gordon equation with cubic nonlinearity [28,35]

$$\{\begin{array}{l}\frac{{\partial }^{2}u}{\partial t^2}+\alpha \frac{{\partial }^{2}u}{\partial x^2}+\beta u+\gamma u^3=0, x\in \left[0,1\right]\\ u(x,0)=B\tan (Kx),\partial u/\partial t{|}_{t=0}=BcK{\sec }^2(Kx)\end{array},$$

(22)

with the Dirichlet boundary conditions extracted from the exact solution $u(x,t)=B\tan [K(x+ct)]$. Here, the parameters $\alpha =-2.5, \beta =1, \gamma =1.5$,$B=\sqrt{\beta /\gamma }$, and $K=\sqrt{-\beta /2(\alpha +c^2)}$.

Table 2. Error norms of numerical solutions at different times t for Example 1 with c = 0.5.

t	RBFCM (M = 50) [35]			LWSCM (M = 24) [28]			WICM (M = 15)
	L∞	L2	RMS	L∞	L2	RMS	L∞	L2	RMS
1	6.0 × 10−6	4.1 × 10−5	4.1 × 10−6	6.1 × 10−6	1.1 × 10−5	3.6 × 10−6	1.2 × 10−10	2.0 × 10−10	5.1 × 10−11
2	2.2 × 10−5	1.6 × 10−4	1.6 × 10−5	2.2 × 10−5	4.1 × 10−5	1.4 × 10−5	1.7 × 10−9	2.6 × 10−9	6.7 × 10−10
3	9.1 × 10−5	6.5 × 10−4	6.4 × 10−5	9.1 × 10−5	1.6 × 10−4	5.4 × 10−5	6.7 × 10−8	9.4 × 10−8	2.4 × 10−8

and

Table 3. Error norms of numerical solutions at different times t for Example 1 with c = 0.05.

t	RBFCM (M = 50) [35]			LWSCM (M = 24) [28]			WICM (M = 15)
	L∞	L2	RMS	L∞	L2	RMS	L∞	L2	RMS
1	3.6 × 10−7	1.8 × 10−6	1.8 × 10−7	1.1 × 10−7	1.7 × 10−7	5.6 × 10−8	1.0 × 10−11	2.0 × 10−11	5.3 × 10−12
2	3.9 × 10−7	1.5 × 10−6	1.5 × 10−7	2.0 × 10−7	3.6 × 10−7	1.2 × 10−7	1.2 × 10−11	2.2 × 10−11	5.8 × 10−12
3	4.2 × 10−7	1.7 × 10−6	1.7 × 10−7	2.4 × 10−7	4.2 × 10−7	1.4 × 10−7	1.6 × 10−11	2.9 × 10−11	7.5 × 10−12

display a comparison of numerical solutions obtained respectively by using the present WICM, radial basis function collocation method (RBFCM) [35], and the Legendre wavelets spectral collocation method (LWSCM) [28]. The relation between the error norms of the wavelet solutions at t = 4 and the spatial mesh size $\Delta x=2^{-m}$ is plotted in Figure 1, in which c = 0.5 and the time step $\Delta t=2^{-10}$ is used.

Example 2.

We consider the 1-D dissipative nonlinear wave equation as follows [27]

$$\{\begin{array}{l}\frac{{\partial }^{2}u}{\partial t^2}=\frac{{\partial }^{2}u}{\partial x^2}-2u\frac{\partial u}{\partial t}+\left({\pi }^2-1-2\sin \left(\pi x\right)\sin \left(t\right)\right)\times \sin \left(\pi x\right)\cos \left(t\right),\hspace{1em}x\in \left[0,1\right]\\ u\left(x,0\right)=\sin \left(\pi x\right),u_t\left(x,0\right)=0,u(0,t)=u(1,t)=0\end{array},$$

(23)

whose exact solution is $u\left(x,t\right)=\sin \left(\pi x\right)\cos \left(t\right)$. The error norms of the present wavelet solution (WICM) are shown in

Table 4. Error norms of numerical solutions for Example 2.

t	PDQM (M = 21, Δt = 0.001) [27]			MCB-DQM (M = 21, Δt = 0.001 [27]			WICM (M = 15, Δt = 1/210)
	L∞	L2	RMS	L∞	L2	RMS	L∞	L2	RMS
0.5	2.2 × 10−5	5.1 × 10−6	1.1 × 10−6	1.8 × 10−5	8.9 × 10−5	1.4 × 10−5	6.6 × 10−11	2.0 × 10−10	5.3 × 10−11
1	2.5 × 10−5	6.6 × 10−6	1.4 × 10−5	5.1 × 10−5	2.6 × 10−4	4.0 × 10−5	1.2 × 10−10	2.0 × 10−10	5.1 × 10−11
1.5	5.2 × 10−5	1.4 × 10−4	2.9 × 10−5	6.7 × 10−5	2.8 × 10−4	4.4 × 10−5	4.3 × 10−10	8.0 × 10−10	2.1 × 10−10
2	4.9 × 10−5	1.3 × 10−4	2.8 × 10−5	3.0 × 10−5	1.2 × 10−4	1.8 × 10−5	1.7 × 10−9	2.6 × 10−9	6.7 × 10−10
3	3.3 × 10−4	8.6 × 10−5	1.8 × 10−5	4.9 × 10−5	2.2 × 10−4	3.4 × 10−5	6.7 × 10−8	9.4 × 10−8	2.4 × 10−8

. For comparison, those for the polynomial differential quadrature method (PQDM) and modified cubic B-spline based differential quadrature method (MCB-DQM) are shown as well.

Example 3.

For a 2-D example, we consider the sine–Gordon equation as follows [3,36,37,38,39,40,41]

$$\frac{{\partial }^{2}u}{\partial t^2}=\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}-\sin (u),\hspace{1em}-7\le x,y\le 7,$$

(24)

subjected to the Dirichlet boundary conditions. Three cases of different initial conditions are studied here as follows:

$$\mathrm{Case} \mathrm{I}: u\left(x,y,0\right)=4{\tan }^{-1}\left(\exp \left(x+y\right)\right) \mathrm{and} \partial u\left(x,y,t\right)/\partial t{|}_{t=0}=-2\mathrm{sech}(x+y),$$

(25)

$$\mathrm{Case} \mathrm{II}: u\left(x,y,0\right)=4{\tan }^{-1}(\exp (3-\sqrt{x^2+y^2})) \mathrm{and} \partial u\left(x,y,t\right)/\partial t{|}_{t=0}=0,$$

(26)

$$\mathrm{Case} \mathrm{III}: u\left(x,y,0\right)=4{\tan }^{-1}(\exp (3-\sqrt{{\left(x-y\right)}^2/3+{\left(x+y\right)}^2/2})) \mathrm{and} \partial u\left(x,y,t\right)/\partial t{|}_{t=0}=0.$$

(27)

The exact solution of Equation (24) with initial conditions (25) can be expressed as $u\left(x,y,t\right)=4{\tan }^{-1}\left(\exp \left(x+y-t\right)\right)$. For this case, the present wavelet solution at t = 1, which is obtained by using spatial mesh 31 × 31, and $\Delta t=2^{-10}$, is given in Figure 2. One can see that the wavelet solution is in good agreement with the exact one.

Table 5. Error norms of numerical solutions at different time t for Case I of Example 3.

t	FDM [3] (M = 56 × 56, Δt = 0.001)		RBFCM [3] (M = 56 × 56, Δt = 0.001)		WICM (M = 31 × 31, Δt = 1/210 ≈ 0.001)
	L2	L∝	L2	L∝	L2	L∝
1	0.7221	0.0350	0.2860	0.0670	0.0162	0.0032
3	0.7877	0.0431	0.5872	0.0834	0.0216	0.0086
5	0.5167	0.0404	0.8288	0.1015	0.0279	0.0104

lists the error norms of numerical solutions obtained by using the FDM [3], RBFCM [3], and proposed WICM. Figure 3 shows the error norms of the present wavelet solution at t = 1 as a function of the spatial mesh size Δx.

The numerical results of Case II are shown in Figure 4 and Figure 5, where the spatial mesh N = 31 × 31 and time step Δt = 1/2¹⁰ are used. From Figure 4 and Figure 5, we can see that a circular ring soliton caused by the nonlinear term, sin(u/2), appears. Such a soliton shrinks at the initial stage, becomes unstable with oscillations and radiations, and is formed again at the end of t = 12.5. This evolution, observed in the present study, is in good agreement with the existing results in [36,37,38,39,40,41]. Figure 6 and Figure 7 show the numerical results for Case III, which are also obtained by using the spatial mesh N = 31 × 31 and time step Δt = 1/2¹⁰. One can see that an elliptical ring soliton shrinks first and then explodes. This observation agrees well with the existing results [36,38,39,40].

Example 4.

Consider the 2-D nonlinear generalized Benjamin–Bona–Mahony-Burgers equation [29,30]:

$$\frac{\partial u}{\partial t}-\frac{\partial }{\partial t}\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)-\frac{{\partial }^{2}u}{\partial x^2}-\frac{{\partial }^{2}u}{\partial y^2}+\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=\cos \left(u\right)\frac{\partial u}{\partial x}+\cos \left(u\right)\frac{\partial u}{\partial y}+f\left(x,y,t\right),-2<x,y<2,$$

(28)

in which the initial and Dirichlet boundary conditions as well as the source f(x, y, t) are taken, so that the exact solution is$u\left(x,y,t\right)=\exp (t)\left[{\mathrm{sech}}^2\left(x+y-r\right)+{\mathrm{sech}}^2\left(x+y+r\right)\right]$.

The solutions at t = 1 for different r obtained by the proposed wavelet method with Δt = 1/2⁸ and Δx = Δy = 1/8 are presented in Figure 8. A comparison of the accuracy and convergence rate in space for the present WICM, the RBFCM [29], and the interpolating element-free Galerkin (EFG) method [30] is shown in

Table 6. Error norms and convergence rate in space for various numerical methods for Example 4.

h	RBF [29]			EFG [30]			Present Method
	RMS	L∝	C	RMS	L∝	C	RMS	L∝	C
1/4	2.3 × 10−3	8.9 × 10−3	-	4.1 × 10−4	1.5 × 10−3	-	2.1 × 10−4	2.9 × 10−4	-
1/8	8.9 × 10−4	4.1 × 10−3	1.1	2.1 × 10−4	6.5 × 10−4	1.2	2.9 × 10−6	2.6 × 10−6	5.9
1/16	4.5 × 10−4	1.2 × 10−3	1.8	1.1 × 10−4	7.4 × 10−4	0.2	3.0 × 10−8	2.2 × 10−8	6.4

, in which t = 1 and r = 3 are used. The relation between the error norms of the present wavelet solution and the spatial mesh size Δx(=Δy) is shown in Figure 9.

Table 2, Table 3, Table 4, Table 5 and Table 6 show that the proposed wavelet method offers excellent numerical accuracy, which is much better than that of many existing methods, such as RBFCM [3,29,35], LWSCM [28], FDM [3], PDQM [27], MCB-DQM [27], and the interpolating EFG method [30]. Most interestingly, Figure 1, Figure 3 and Figure 9 clearly show that the convergence rate in space of the present WICM is always about order 6 for various nonlinearities. In addition, Figure 2, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 demonstrate that the proposed wavelet method offers a good capacity for tracing complicated nonlinear phenomena, such as the evolution of solitons.

5. Conclusions

In this study, WICM was employed to execute the spatial discretization of nonlinear wave equations. In the resulting semi-discrete system, all the matrices representing the spatial discretization were constant over time and, therefore, never needed to be recalculated in the subsequent time integration, implying that a full decoupling between spatial and temporal discretization was achieved in the proposed wavelet method. The numerical results demonstrated that the present WICM possesses much better accuracy than many existing methods, and showed a sixth order convergence in space for problems with various nonlinearities. The numerical tests also showed that the proposed wavelet method can effectively capture complicated nonlinear phenomena, such as the evolution of solitons.

However, the proposed method was only verified by nonlinear wave problems in regular spatial domains. For a future study, one may combine the algorithm for wavelet approximation of functions bounded in irregular domain, as we have developed previously [42,43], into the proposed method to solve real engineering problems with irregular shapes/domains.