A compact finite difference scheme with absorbing boundary condition for forced KdV equation

Abstract

Studying the long-time solution behavior of the Korteweg-de Vries (KdV) type equation with a periodic force acting at one end of the long channel is important for simulating the blood flow in artery driven by heart pulses. It is of great interest to develop an accurate numerical method for solving the forced KdV problem. In this article, we present the following methods to obtain an accurate approximation to the solution of KdV problem.

• •An accurate compact finite difference scheme is proposed for solving the above forced KdV problem with fourth-order accuracy.
• •An absorbing boundary condition at the right end of the interval is used to avoid the wave reflection.
• •The stability of scheme is proved by the von Neumann method and then tested by three examples. Results show that the method provides an accurate solution, and the wave propagates without reflection.

Graphical abstract

Specifications table

Subject area:	Mathematics and Statistics
More specific subject area:	Numerical Methods for PDEs
Name of your method:	Compact Finite Difference Scheme
Name and reference of original method:	S.K. Lele, Compact finite difference schemes with spectral-like resolution. Journal of Computational Physics, 103 (1992) 16–42.https://doi.org/10.1016/0021-9991(92)90,324-R
Resource availability:	N/A

Method details

Background

The Korteweg-de Vries (KdV) equation is often used in simulating the water wave propagation along a channel [1], particularly, in nonlinear studies including balance weak nonlinear and dispersive effects [2], [3], [4]. Such nonlinearity phenomena are interesting in applied mathematics, physics, and other disciplines [5], [6], [7], [8]. Recently, the KdV equation has been applied for modeling the unidirectional propagation of small-amplitude long waves in a nonlinear dispersive medium, such as heat transformation in crystal lattice, ion-acoustic wave in plasmas, blood flow in vessels, and cosmology [9], [10], [11], [12].

In [3], the authors proposed a wave-maker model using the initial boundary value problem with a KdV type equation to simulate the propagation of water wave under a periodic wave maker on the left boundary as follows:

$$\{\begin{array}{c}{u}_t+\alpha u_x+\beta u{u}_x+\gamma u_{xxx}=0,\hfill \\ u\left(x,0\right)=0,\hfill \\ u\left(0,t\right)=f\left(t\right),\hfill \end{array}$$ (1)

where $x=0$ corresponds the left boundary at which the water wave to be generated. Here, $u\left(x,t\right)$ is a wave height measured from an undisturbed water level, $\alpha$, $\beta$, $\gamma$ are given parameters, $f\left(t\right)$ is a periodic function at the left boundary. Experiments have shown that when nonlinear, dispersive waves are forced periodically at one end of the long channel, the signal eventually becomes temporally periodic at each spatial point [13]. For this initial boundary value problem, we often consider an infinite domain, i.e., $x\in \left[0,+\infty \right)$. In this case, boundary conditions are chosen to be $u\left(+\infty ,t\right)=u_x\left(+\infty ,t\right)=0$, this is similar to a situation that the water wave does not reach the end of channel. If there is a reflection when the wave arrives the right end, such KdV model is not appropriate.

It should be pointed out that without the force term $f\left(t\right)$, the KdV equation is completely integrable and has its own wave solution, which can be derived analytically [14], [15], [16], [17]. However, with the force term $f\left(t\right)$ on the boundary, the analytical solution for the forced KdV problem will be difficult to obtain, if not impossible. Therefore, developing an accurate numerical method for solving the above forced KdV problem is of great interest. Up to date, there are several numerical methods that have been developed, such as the local discontinuous Galerkin method [18], the Chebyshev spectral collocation method [19], the implicit-explicit BDF method [20], the lattice Boltzmann method (LBM) [21], and the meshless method with radial basis functions [22], [23], [24], [25].

Note that the interval in space is $x\in \left[0,+\infty \right)$ for this problem, the interval must be truncated into a finite computational domain when simulating numerically. Thus, a suitable boundary condition at the other end of the interval should be introduced in the computation. In previous works, several artificial boundary conditions were developed for nonlinear waves, such as the perfect matched layer (PML) [26], [27], [28], [29], the exact nonreflecting boundary conditions [30], [31], [32], the Haar wavelet method [33], and absorbing boundary conditions [34], [35], [36], [37], [38], [39].

Motivated by the above results, we consider a forced KdV type equation with initial value boundary conditions as follows:

$$\begin{array}{c}\hfill \{\begin{array}{c}{u}_t+\alpha u_x+\beta u{u}_x+\gamma u_{\mathit{xxx}}=0,x\in [0,+\infty ),t>0,\hfill \\ u(x,0)=\varphi \left(x\right),x\in [0,+\infty ),\hfill \\ u(0,t)=f\left(t\right),t>0,\hfill \end{array}\end{array}$$ (2)

where $f\left(t\right)$ is a periodic function with time $t$ and $\varphi \left(x\right)$ is a smooth function, $\alpha$, $\beta$, $\gamma$ are non-negative constants. We also impose a boundary condition at infinity to ensure the well-poseness of this system.

Compact finite difference scheme

In this section, we propose a fourth-order compact finite difference scheme for the IBVP (2). We consider $\left[0,L\right]$ to be our bounded computational domain, where the boundary condition $f\left(t\right)$ is given at $x=0$. At $x=L$, we design an appropriate absorbing boundary condition to avoid the wave reflection.

To derive our accurate compact finite difference scheme for solving the forced KdV Eq. (2), the following notations are used throughout this paper. For a positive integer $N$, let the time step be $\tau =\frac{T}{N}$ and $t_n=n\tau$ for $n=0,1,2,\dots ,N$. For a positive integer $J$, let the space step be $h=\frac{L}{J}$ and $x_j=jh$, $j=0,1,2,\dots ,J$. With this discretization, we denote $u_j^n=u(x_j,t_n)$ and $U_j^n$ to be the approximation of $u(x_j,t_n)$.

From [40], we have the following fourth-order finite difference schemes for $u_{xxx}$ and $u_x$:

$$\frac{1}{10}{\left(u_{xxx}\right)}_{j-1}^n+{\left(u_{xxx}\right)}_j^n+\frac{1}{10}{\left(u_{xxx}\right)}_{j+1}^n=\frac{6}{5h^2}\left[{\left(u_x\right)}_{j+1}^n-2{\left(u_x\right)}_{j-1}^n+{\left(u_x\right)}_{j-1}^n\right]+O\left(h^4\right),$$ (3)

and

$${\left(u_x\right)}_{j+1}^n+4{\left(u_x\right)}_j^n+{\left(u_x\right)}_{j-1}^n=\frac{3}{h}\left[u_{j+1}^n-u_{j-1}^n\right]+\mathrm{O}\left(h^4\right),$$ (4)

for $j=1,2,\dots ,J-1$. To develop a fourth-order finite difference scheme, we sum the first equation in (2) at grid points $x_{j-1},x_j,x_{j+1}$ with the same coefficients in Eq. (3). This gives:

$$\begin{array}{c}\frac{1}{10}\left[{\left(u_t\right)}_{j-1}^n+\alpha {\left(u_x\right)}_{j-1}^n+\beta u_{j-1}^n{\left(u_x\right)}_{j-1}^n+\gamma {\left(u_{xxx}\right)}_{j-1}^n\right]\hfill \\ +\left[{\left(u_t\right)}_j^n+\alpha {\left(u_x\right)}_j^n+\beta u_j^n{\left(u_x\right)}_j^n+\gamma {\left(u_{xxx}\right)}_j^n\right]\hfill \\ +\frac{1}{10}\left[{\left(u_t\right)}_{j+1}^n+\alpha {\left(u_x\right)}_{j+1}^n+\beta u_{j+1}^n{\left(u_x\right)}_{j+1}^n+\gamma {\left(u_{xxx}\right)}_{j+1}^n\right]=0.\hfill \end{array}$$ (5)

Substituting Eq. (3) into Eq. (5), and then using the Crank-Nicolson method, we obtain a discrete equation at $x_j^n$ as follows:

$$\begin{array}{c}\frac{1}{10\tau }\left(u_{j-1}^{n+1}-u_{j-1}^{n+1}\right)+\frac{1}{\tau }\left(u_j^{n+1}-u_j^n\right)+\frac{1}{10\tau }\left(u_{j+1}^{n+1}-u_{j+1}^n\right)\hfill \\ +\frac{\alpha }{20}\left[{\left(u_x\right)}_{j-1}^{n+1}+{\left(u_x\right)}_{j-1}^n\right]+\frac{\alpha }{2}\left[{\left(u_x\right)}_j^{n+1}+{\left(u_x\right)}_j^n\right]+\frac{\alpha }{20}\left[{\left(u_x\right)}_{j+1}^{n+1}+{\left(u_x\right)}_{j+1}^n\right]\hfill \\ +\frac{\beta }{20}\left[u_{j-1}^{n+1}{\left(u_x\right)}_{j-1}^{n+1}+u_{j-1}^n{\left(u_x\right)}_{j-1}^n\right]+\frac{\beta }{2}\left[u_j^{n+1}{\left(u_x\right)}_j^{n+1}+u_j^n{\left(u_x\right)}_j^n\right]+\frac{\beta }{20}\left[u_{j+1}^{n+1}{\left(u_x\right)}_{j+1}^{n+1}+u_{j+1}^n{\left(u_x\right)}_{j+1}^n\right]\hfill \\ +\frac{3\gamma }{5h^2}\left[{\left(u_x\right)}_{j+1}^{n+1}-2{\left(u_x\right)}_j^{n+1}+{\left(u_x\right)}_{j-1}^{n+1}\right]+\frac{3\gamma }{5h^2}\left[{\left(u_x\right)}_{j+1}^n-2{\left(u_x\right)}_j^n+{\left(u_x\right)}_{j-1}^n\right]=O\left({\tau }^2+h^4\right).\hfill \end{array}$$ (6)

Since ${\left(u_x\right)}_j^{n+1}$ and ${\left(u_x\right)}_j^n$, $j=0,\cdots ,J$, are unknown, we further employ a fourth-order compact scheme Eq. (4) to resolve it. Note that there are only $J-1$ equations with $J+1$ unknowns, we need to either eliminate or find other schemes for ${\left(u_x\right)}_0^n$ and ${\left(u_x\right)}_J^n$.

Absorbing boundary condition

In previous subsection, we design an accurate compact finite difference scheme for the interior grid points. In this subsection, we will derive two different schemes for left and right boundaries, respectively. In the IBVP (2), there is a wave-maker $f\left(t\right)$ at the left boundary $x=x_0$. This gives:

$$f^{\text{'}}\left(t_n\right)+\alpha {\left(u_x\right)}_0^n+\beta u_0^n{\left(u_x\right)}_0^n+\gamma {\left(u_{\mathit{xxx}}\right)}_0^n=0.$$ (7)

At $x=x_0$, we design a third-order finite difference scheme for $u_{xxx}$,

$${\left(u_{xxx}\right)}_0^n=\frac{a}{h^3}\left[u_1^n-u_0^n\right]+\frac{b}{h^3}\left[u_2^n-u_0^n\right]+\frac{c}{h^2}{\left(u_x\right)}_0^n+\frac{d}{h^2}{\left(u_x\right)}_1^n+\frac{e}{h^2}{\left(u_x\right)}_2^n+\mathrm{O}\left(h^3\right),$$ (8)

for some coefficients $a,b,c,d$ and $e$. Using the Taylor series expansion at $x=x_0$, we obtain the following equations for each term in Eq. (8),

$$u_1-u_0={\left[h{u}_x+\frac{h^2}{2}{u}_{x^2}+\frac{h^3}{6}{u}_{x^3}+\frac{h^4}{24}{u}_{x^4}+\frac{h^5}{120}{u}_{x^5}\right]}_{\left(x_0,t\right)}+O\left(h^6\right),$$ (9)

$$u_2-u_0={\left[2h{u}_x+\frac{2^2{h}^2}{2}{u}_{x^2}+\frac{2^3{h}^3}{6}{u}_{x^3}+\frac{2^4{h}^4}{24}{u}_{x^4}+\frac{2^5{h}^5}{120}{u}_{x^5}\right]}_{\left(x_0,t\right)}+O\left(h^6\right),$$ (10)

$${\left(u_x\right)}_1={\left[u_x+h{u}_{x^2}+\frac{h^2}{2}{u}_{x^3}+\frac{h^3}{6}{u}_{x^4}+\frac{h^4}{24}{u}_{x^5}\right]}_{\left(x_0,t\right)}+O\left(h^5\right),$$ (11)

$${\left(u_x\right)}_2={\left[u_x+2h{u}_{x^2}+\frac{2^2{h}^2}{2}{u}_{x^3}+\frac{2^3{h}^3}{6}{u}_{x^4}+\frac{2^4{h}^4}{24}{u}_{x^5}\right]}_{\left(x_0,t\right)}+O\left(h^5\right).$$ (12)

Substituting Eqs. (9)–(12) into Eq. (8), and then matching the coefficients, we obtain a system of equations for solving all coefficients as

$$\left[\begin{array}{ccccc}1& 2& 1& 1& 1\\ 1& 4& 0& 2& 4\\ 1& 8& 0& 3& 12\\ 1& 16& 0& 4& 32\\ 1& 32& 0& 5& 80\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\\ d\\ e\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 6\\ 0\\ 0\end{array}\right]$$ (13)

which gives us a desirable result:

$${\left(u_{xxx}\right)}_0^n=-\frac{24}{h^3}\left[u_1^n-u_0^n\right]-\frac{51}{2h^3}\left[u_2^n-u_0^n\right]+\frac{39}{2h^2}{\left(u_x\right)}_0^n+\frac{48}{h^2}{\left(u_x\right)}_1^n+\frac{15}{2h^2}{\left(u_x\right)}_2^n+O\left(h^3\right).$$ (14)

From Eq. (4) when $x=x_0$, we have:

$${\left(u_x\right)}_0^n=\frac{3}{h}\left[u_2^n-u_0^n\right]-4{\left(u_x\right)}_1^n-{\left(u_x\right)}_2^n+O\left(h^4\right).$$ (15)

Substituting Eqs. (14) and (15) into Eq. (7), we obtain a finite difference scheme on the left boundary point as

$$\begin{array}{c}\hfill \begin{array}{c}(30\gamma +4{\mathrm{h}}^2[\alpha +\beta \mathrm{f}\left({\mathrm{t}}_{\mathrm{n}}\right)]){\left({\mathrm{u}}_{\mathrm{x}}\right)}_1^{\mathrm{n}}+(12\gamma +{\mathrm{h}}^2[\alpha +\beta \mathrm{f}\left({\mathrm{t}}_{\mathrm{n}}\right)]){\left({\mathrm{u}}_{\mathrm{x}}\right)}_2^{\mathrm{n}}\hfill \\ ={\mathrm{h}}^2{\mathrm{f}}^{\text{'}}\left({\mathrm{t}}_{\mathrm{n}}\right)+(11\gamma +{\mathrm{h}}^2[\alpha +\beta \mathrm{f}\left({\mathrm{t}}_{\mathrm{n}}\right)])\frac{3}{\mathrm{h}}[{\mathrm{u}}_2^{\mathrm{n}}-\mathrm{f}\left({\mathrm{t}}_{\mathrm{n}}\right)]-\frac{24\gamma }{\mathrm{h}}[{\mathrm{u}}_1^{\mathrm{n}}-\mathrm{f}\left({\mathrm{t}}_{\mathrm{n}}\right)]+\mathrm{O}\left({\mathrm{h}}^3\right).\hfill \end{array}\end{array}$$ (16)

It should be pointed out that the values of $u$ and $u_x$ on the right boundary are still unknown. Since we truncate the domain to be a closed interval, the wave might rebound once it arrives the right end-point. We now develop an absorbing boundary condition to avoid this situation.

Following the idea in [41], we first interpolate a quadratic function $g\left(x\right)=a{x}^2+bx+c$ by using function values at $x_{J-2},x_{J-1},x_J$, where the coefficients can be derived by:

$$\{\begin{array}{c}a=\frac{1}{2h^2}\left[u_J^n-2u_{J-1}^n+u_{J-2}^n\right],\hfill \\ b=\frac{1}{h}\left[u_{J-1}^n-u_{J-2}^n-ah\left(x_{J-1}+x_{J-2}\right)\right],\hfill \\ c=u_{J-2}^n-b{x}_{J-2}-a{x}_{J-2}^2.\hfill \end{array}$$ (17)

We simply let $g\left(x\right)=u_{J-1}^{n-1}$ and solve it for $x$ using the bisection method to obtain the root $x=p\in \left[x_{J-1},x_J\right]$. The purpose of this step is to determine how far the solution $u_{J-1}^{n-1}$ propagates after one time-step $\tau$. Once the distance is determined, we may assume that the solution at $t^n$ will move the same distance after a time step. Thus, we obtain an estimated distance $q=x_J+x_{J-1}-p$ and the values of $u_J^{n+1}$ and ${\left(u_x\right)}_J^{n+1}$ can be approximated by

$$u_J^{n+1}=g\left(q\right),{\left(u_x\right)}_J^{n+1}=2aq+b.$$ (18)

Computational algorithm

We now summarize our finite difference scheme. Let $U_j^n$ be the approximation of $u_j^n$. We drop all the truncation errors in Eqs. (6) and (16), and obtain:

$$\begin{array}{c}\left[\frac{1}{10}+\frac{\beta \tau }{20}{\left(U_x\right)}_{j-1}^{n+1}\right]U_{j-1}^{n+1}+\left[1+\frac{\beta \tau }{2}{\left(U_x\right)}_j^{n+1}\right]U_j^{n+1}+\left[\frac{1}{10}+\frac{\beta \tau }{20}{\left(U_x\right)}_{j+1}^{n+1}\right]U_{j+1}^{n+1}\hfill \\ =\frac{1}{10}\left(U_{j-1}^n+10U_j^n+U_{j+1}^n\right)-\frac{\beta \tau }{20}\left[U_{j-1}^n{\left(U_x\right)}_{j-1}^n+10U_j^n{\left(U_x\right)}_j^n+U_{j+1}^n{\left(U_x\right)}_{j+1}^n\right]\hfill \\ -\frac{\alpha \tau }{20}\left\{\left[{\left(U_x\right)}_{j-1}^{n+1}+{\left(U_x\right)}_{j-1}^n\right]+10\left[{\left(U_x\right)}_j^{n+1}+{\left(U_x\right)}_j^n\right]+\left[{\left(U_x\right)}_{j+1}^{n+1}+{\left(U_x\right)}_{j+1}^n\right]\right\}\hfill \\ -\frac{3\gamma \tau }{5h^2}\left[{\left(U_x\right)}_{j+1}^{n+1}-2{\left(U_x\right)}_j^{n+1}+{\left(U_x\right)}_{j-1}^{n+1}+{\left(U_x\right)}_{j+1}^n-2{\left(U_x\right)}_j^n+{\left(U_x\right)}_{j-1}^n\right]\hfill \end{array},$$ (19)

and

$$\begin{array}{c}\left(30\gamma +4h^2\left[\alpha +\beta f\left(t_{n+1}\right)\right]\right){\left(U_x\right)}_1^{n+1}+\left(12\gamma +h^2\left[\alpha +\beta f\left(t_{n+1}\right)\right]\right){\left(U_x\right)}_2^{n+1}\hfill \\ =h^2{f}^{\prime }\left(t_{n+1}\right)+\left(11\gamma +h^2\left[\alpha +\beta f\left(t_{n+1}\right)\right]\right)\frac{3}{h}\left[U_1^{n+1}-f\left(t_{n+1}\right)\right]-\frac{24\gamma }{h}\left[U_2^{n+1}-f\left(t_{n+1}\right)\right].\hfill \end{array}$$ (20)

Note that the first scheme is used for interior grid points and the second one is designed for the left boundary point. We also replace $u$ with $U$ in Eq. (4),

$${\left(U_x\right)}_{j+1}^{n+1}+4{\left(U_x\right)}_j^{n+1}+{\left(U_x\right)}_{j-1}^{n+1}=\frac{3}{h}\left[U_{j+1}^{n+1}-U_{j-1}^{n+1}\right]$$ (21)

for $j=2,3,\dots ,J-1$.

The computational procedure from time level n to time level n + 1 for solving the forced KdV problem can be described as follows:

• Step 1. Initialize the values $\left\{U_j^{n+1}\right\}$ and solve the diagonal-dominated tridiagonal linear system for $\left\{{\left(U_x\right)}_j^{n+1}\right\}$ based on Eqs. (20)–(21) and Eq. (18).

• Step 2. Once $\left\{{\left(U_x\right)}_j^{n+1}\right\}$ are obtained, $\left\{U_j^{n+1}\right\}$can be obtained based on Eq. (19).

Repeat the above two steps until the convergent $\left\{U_j^{n+1}\right\}$ are obtained.

Stability analysis

We now analyze the stability of the present scheme. For wave propagation cases, the solutions are often bounded, i.e., $\left|u\left(x,t\right)\right|\le C$. For simplicity, we substitute $|U_j^n|\le C$ into the nonlinear term in Eq. (19) and consider the scheme only at interior points. Thus, we can simply use the von Neumann analysis for Eqs. (19) and (21).

To analyze the stability, we let

$$U_j^n={\lambda }^n{e}^{\mathit{Ij}\theta h},{\left(U_x\right)}_j^n=G^n{e}^{\mathit{Ij}\theta h},$$ (22)

where $\lambda$ and $G$ are the amplification factors for $U_j^n$, ${\left(U_x\right)}_j^n$, respectively, and $I=\sqrt{-1}$. Substituting them into Eq. (21), we obtain

$$G^n{e}^{I\left(j-1\right)\theta h}+4G^n{e}^{Ij\theta h}+G^n{e}^{I\left(j+1\right)\theta h}=\frac{3}{h}\left[{\lambda }^n{e}^{I\left(j+1\right)\theta h}-{\lambda }^n{e}^{I\left(j-1\right)\theta h}\right],$$ (23)

which gives

$$G^n=\frac{3\sin \left(\theta h\right)}{h\left[\cos \left(\theta h\right)+2\right]}{\lambda }^{n}I.$$ (24)

Substituting Eqs. (22) and (24) into Eq. (19) gives

$$\lambda -1+\frac{3\sin \left(\theta h\right)\Delta t\left(\alpha +\beta C\right)}{2h\left[\cos \left(\theta h\right)+2\right]}\left(\lambda +1\right)I-\frac{9\sin \left(\theta h\right)\gamma \left[\cos \left(\theta h\right)-1\right]}{5h^3\left[\cos \left(\theta h\right)+2\right]\left[\cos \left(\theta h\right)+5\right]}\left(\lambda +1\right)I=0$$ (25)

implying that

$$\left|\lambda \right|=\left|\frac{1-I\frac{3\sin \left(\theta \mathrm{h}\right)\Delta t}{h\left[\cos \left(\theta \mathrm{h}\right)+2\right]}\left\{\frac{\left(\alpha +\beta \mathrm{C}\right)}{2}-\frac{3\left[\cos \left(\theta \mathrm{h}\right)-1\right]}{5h^2\left[\cos \left(\theta \mathrm{h}\right)+5\right]}\right\}}{1+I\frac{3\sin \left(\theta \mathrm{h}\right)\Delta t}{h\left[\cos \left(\theta \mathrm{h}\right)+2\right]}\left\{\frac{\left(\alpha +\beta \mathrm{C}\right)}{2}-\frac{3\left[\cos \left(\theta \mathrm{h}\right)-1\right]}{5h^2\left[\cos \left(\theta \mathrm{h}\right)+5\right]}\right\}}\right|.$$ (26)

This indicates that the scheme in Eqs. (19) and (21) is unconditionally stable.

Numerical examples

We give three numerical examples to test the accuracy and stability of our compact scheme.

Example 1. Consider a simple KdV problem:

$$\begin{array}{c}\hfill \{\begin{array}{c}{u}_t+u{u}_x+u_{\mathit{xxx}}=0,x\in (-\infty ,+\infty ),t\in [0,+\infty ),\hfill \\ u(x,0)=\mathit{Asec}{h}^2(\mathit{kx}-x_0),x\in (-\infty ,+\infty ),\hfill \\ u(+\infty ,t)=u_x(+\infty ,t)=0,t\in [0,+\infty ),\hfill \\ u(-\infty ,t)=u_x(-\infty ,t)=0,t\in [0,+\infty ),\hfill \end{array}\end{array}$$ (27)

where $A=12k^2$.

The exact solution is $u\left(x,t\right)=A\text{sec}{\mathrm{h}}^2\left(kx-\omega t-x_0\right)$, where $\omega =4k^3$. In our computation, we considered the computational domain for space to be $\left[-30,30\right]$ and $0\le t\le 2$. The maximum error was calculated based on $E_{max}^n=max\left|U_j^n-u_j^n\right|.$ Assume that $E_{\text{max}}^n\left(\tau ,h\right)=\mathrm{O}\left({\tau }^p+h^q\right)$. When $\tau$ is very small, $E_{max}^n\left(\tau ,h\right)=O\left({\tau }^p+h^q\right)\approx O\left(h^q\right)$. Thus, by choosing different grid sizes $h_1$ and $h_2$, we have

$$\frac{E_{\text{max}}^n\left(\tau ,h_1\right)}{E_{\text{max}}^n\left(\tau ,h_2\right)}\approx {\left(\frac{h_1}{h_2}\right)}^q,$$ (28)

and hence obtain the convergence rate,

$$\mathrm{q}\approx \ln \left(\frac{E_{\text{max}}^n\left(\tau ,h_1\right)}{E_{\text{max}}^n\left(\tau ,h_2\right)}\right)/\text{ln}\left(\frac{h_1}{h_2}\right).$$ (29)

In this example, we chose $k=0.3$, $x_0=0$, $\tau ={10}^{-7}$, and the grid size $h=0.2,0.3,0.4,0.6,0.8,$ respectively, to obtain the convergence rate with respect to the spatial variable. Maximum errors and corresponding convergence rates were computed, as listed in Table 1. We can see from Table 1 that the convergence rate with respect to $x$ is approximately 3.9. Fig. 1 shows (a) the profiles of both exact and numerical solutions at $t=1$ and (b) the error between these two solutions, implying that there is no significant difference between the exact solution and the numerical solution.

Table 1.

Maximum error and convergence rate with $\tau ={10}^{-7}$.

h	$E_{max}^n\left(\tau ,h\right)$	Convergence Rate
0.8	$4.04938591\times {10}^{-4}$	N/A
0.6	$1.21542586\times {10}^{-4}$	4.18
0.4	$2.41861418\times {10}^{-5}$	3.98
0.3	$7.66367875\times {10}^{-6}$	3.99
0.2	$1.59507088\times {10}^{-6}$	3.87

Fig. 1.

(a) the profiles of both exact and numerical solutions at $t=1$ and (b) the error between two solutions.

Example 2. Consider a forced KdV problem:

$$\begin{array}{c}\hfill \{\begin{array}{c}{u}_t+0.9u_x+0.001u{u}_x+0.00001u_{\mathit{xxx}}=0,x\in [0,1],t\in [0,2],\hfill \\ u(x,0)=0,x\in [0,1],\hfill \\ u(0,t)=\sin \left(20\pi t\right),t\in [0,2].\hfill \end{array}\end{array}$$ (30)

In our computation, we chose the computational interval for $x$ to be $\left[0,1\right],$ $h=0.01$, $\tau =0.00001$, and $t$ within $0\le t\le 2$. Here, a small $\gamma ={10}^{-5}$ in Eq. (30) was chosen to ensure that the wave does not diminish before reaching the right end of the interval. For the same reason, the small interval $\left[0,1\right]$ was chosen so that the wave propagates to the right end of the interval quickly. We used the present scheme to solve this problem. For the right-end boundary condition, we used the absorbing boundary condition, and compared the numerical solution with that obtained based on the boundary condition $u_1^n=0$ and ${\left(u_x\right)}_1^n=0$, i.e., without the absorbing boundary condition.

Figs. 2 and 3 show the wave propagations simulated by using the present scheme coupled with and without the absorbing boundary condition, respectively. As we see from Fig. 3, without using the absorbing boundary condition, the wave shows a reflection and diverges eventually. On the other hand, the numerical solution with the absorbing boundary condition shows the wave propagating out the computational domain smoothly, as shown in Fig. 2.

Fig. 2.

Forced KdV equation with absorbing boundary condition at $t=0.5,1.0,1.5,2.0.$.

Fig. 3.

Forced KdV equation without absorbing boundary condition at $t=0.5,1.0,1.5,2.0$.

Example 3. Consider a forced KdV problem:

$$\begin{array}{c}\hfill \{\begin{array}{c}{u}_t+u_x+u{u}_x+u_{\mathit{xxx}}=0,x\in [0,2],t\in [0,2],\hfill \\ u(x,0)=0,x\in [0,2],\hfill \\ u(0,t)=\sin \left(2\pi t\right),t\in [0,2].\hfill \end{array}\end{array}$$ (31)

In our computation, we chose the computational interval for $x$ to be $\left[0,2\right]$, $h=0.04$, $\tau ={10}^{-5}$, and $0\le t\le 2$. For this case, the wave decays during the propagation. Fig. 4 shows the solution profiles vs. time $t$ at various points in the interval, which were obtained based on the present scheme coupled with (a) the absorbing boundary condition, and (b) the boundary condition $u_2^n=0$ and ${\left(u_x\right)}_2^n=0,$ respectively. From Fig. 4, one may see that using the absorbing boundary condition, the solution shows the wave propagates and decays gradually throughout the interval. On the other hand, using the boundary condition $u_2^n=0$ and ${\left(u_x\right)}_2^n=0$ the wave diminishes quickly before reaching the boundary. Fig. 5 shows that the 3D profile of the numerical solution with respect to time $t$ and space $x$, which was obtained based on the present scheme coupled with the absorbing boundary condition.

Fig. 4.

Solution profiles changing with time t at various points in the domain obtained using the present scheme coupled (a) with the absorbing boundary condition and (b) without the absorbing boundary condition.

Fig. 5.

3D profile of the numerical solution.

Conclusion

In this paper, we have developed an accurate finite difference scheme for solving a forced KdV problem with an absorbing boundary condition at the right end of a given interval to avoid wave reflection. The scheme is obtained based on the high-order compact finite difference method for solving the third-order derivative $u_{xxx}$ and the first-order derivative $u_x$ in the KdV type equation, which is coupled with the Crank-Nicolson method. In particular, the third-order derivative $u_{xxx}$ at the left end of the interval is approximated using a third-order compact finite difference scheme. On the other hand, $u$ and $u_x$ at the right end of the interval are obtained based on an absorbing boundary condition. Three examples have tested the scheme. Results show that the method provides an accurate solution, and the wave propagates without reflection.

Ethics statements

N/A

CRediT author statement

Jiaqi Chen contributed theoretical analysis, computation, preparing and editing for the manuscript, and Weizhong Dai contributed the concept, numerical method, reviewing and editing the manuscript.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data Availability

• Data will be made available on request.