Finite-Difference Hermite WENO Scheme for Degasperis-Procesi Equation

Abstract

We present a fifth-order finite-difference Hermite weighted essentially non-oscillatory (HWENO) method for solving the Degasperis–Procesi (DP) equation in this paper. First, the DP equation can be rewritten as a system of equations consisting of hyperbolic equations and elliptic equations by introducing an auxiliary variable, since the equations contain nonlinear higher order derivative terms. Then, the auxiliary variable equations are solved using the Hermite interpolation, while the HWENO scheme is performed for the hyperbolic equations. Compared with the popular WENO-type scheme, the most important feature of the HWENO scheme mentioned in this paper is the compactness of its spatial reconstruction stencil, which can achieve the fifth-order accuracy of the expected design with only three points, while the WENO method requires five points. Finally, we demonstrate the effectiveness of the HWENO method in various aspects by conducting some benchmark numerical tests.

1. Introduction

In this paper, we focus on the Degasperis–Procesi (DP) equation of the form:

$$u_t-u_{txx}+4f{\left(u\right)}_x=f{\left(u\right)}_{xxx},$$

(1)

where $f\left(u\right)=u^2/2$. We develop a new finite-difference fifth-order Hermite weighted essentially non-oscillatory (HWENO) schemes for solving (1).

The nonlinear DP equation can effectively simulate the propagation of dispersive waves and was first proposed by Degasperis and Procesi [1] in 1999. They studied the following equations:

$$u_t+a_1{u}_x+a_2{u}_{xxx}-a_3^2{u}_{txx}={(b_1{u}^2+b_2{u}_x^2+b_{3}u{u}_{xx})}_x,$$

(2)

where $a_1, a_2, a_3, b_1, b_2$ and $b_3$ are real constants. The equation satisfies the asymptotic integrability condition only when these coefficients satisfy some specific requirements. For example, when $a_3=b_2=b_3=0$, the equation is called the Korteweg–de Vries (KDV) equation; when $b_1=-\frac{3b_3}{2a_3^2},b_2=\frac{b_3}{2}$, the equation is called the Cahn–Hilliard (CH) equation; and when $b_1=-\frac{3b_3}{a_3^2},b_2=b_3$, the equation is called the Degasperis–Procesi (DP) equation.

Degasperis and Procesi found the DP equation while looking for integrable equations that resemble the Camassa–Holm equation in form; however, it was later confirmed that it has many important applications in physics. The DP equation is a common model for the evolution of shallow-water waves with small amplitude and long wavelength [2,3,4,5]. It can be solved using methods from soliton theory, which is a branch of mathematics that studies special solutions of nonlinear equations that behave like particles. The Degasperis–Procesi equation belongs to a family of third-order nonlinear dispersive partial differential equations that have a certain form involving derivatives of different orders and products of the unknown function and its derivatives. It can be seen as a model for shallow-water dynamics, where the unknown function represents the elevation of the water surface above a flat bottom. The equation captures some important features of water waves, such as nonlinearity, dispersion and breaking. It also has complete integrability, which means that it can be solved exactly using inverse scattering techniques or other methods. Not only peaked solutions [6], for example, $u(x,t)=c{e}^{-\mid x-ct\mid }$, but also shock waves solutions [7,8], for example, $u(x,t)=-\frac{1}{t+c}sign\left(x\right)e^{-\mid x\mid }$, are among the important features of the DP equation. It also satisfies those conservation laws that cannot be bound by the $H^1-$ norm of the solution, which is another feature of DP-type equations. While the explicit solution of this equation is currently constructed by very few and the construction of the global smooth solution is even more difficult, many are needed in practical applications; this has led people to start shifting their focus to the numerical solution and the construction of numerical schemes. After years of development, various numerical schemes have been designed one after another, and the following is a brief introduction to some developments of numerical schemes.

As strong discontinuities will appear in the solution of the equation, the designed high-order numerical scheme should have shock-capturing ability to avoid numerical oscillations. There are many numerical studies on the DP equation, such as the particle method [9] and operator splitting finite-difference methods [10,11]. Xu and Shu [12] used the LDG methods to solve this equation and proved the $L^2$ stability. Liu et al. [13] applied the direct DG methods to solve the DP equation. After years of research on the DP equation, many explicit solutions have been constructed, including multi-peakon solutions and multi-shock solutions [14,15]. Numerical schemes are often evaluated using these solutions. The WENO scheme, as a very popular high-precision numerical scheme of recent years, is also applied to the solution of DP equation. Here is a brief introduction to the WENO scheme.

WENO schemes (weighted essentially non-oscillatory schemes) are a kind of high-order high-resolution generalized Godunov scheme for solving hyperbolic conservation laws, which is suitable for solving problems with strong discontinuities and complex flow structures such as high-density ratio fluid-interface instability. The basic idea of WENO schemes is to change the method of using only the smoothest interpolation region to provide the approximate value of the numerical flux at the grid section in ENO schemes to the weighted average of the numerical flux at the grid interface provided by each possible interpolation region. Specifically, each possible interpolation region is assigned a weight, which determines the contribution of the interpolation region to the final numerical flux at the grid interface. They use some optimal weights to achieve higher order accuracy in smooth regions, and they give almost zero weight to those interpolation regions that have discontinuities. The framework of WENO schemes was first proposed by Jiang and Shu [16] in 1996. Due to its simple structure and high efficiency, this scheme has attracted the attention of many researchers. The design idea of the WENO scheme is to combine multiple low-order approximations through nonlinear weights, which can achieve consistent high-order accuracy in smooth regions and reduce the order in non-smooth regions to avoid numerical oscillations. Subsequently, more effective WENO schemes have been proposed [17,18,19]. The WENO scheme was first proposed to solve hyperbolic conservation-law problems, but it is very effective for all kinds of problems with discontinuous solutions, so it was soon extended to a variety of equations with discontinuous solutions, such as the Navier–Stokes equation, Hamilton–Jacobi equation and so on. As the WENO scheme shows good properties in all kinds of problems, its development speed was amazing; there are more than a hundred of its improved forms. Not only in academia, but also in industry, the high-order WENO scheme has been widely applied, such as when calculating the internal flow of aeroengines. In addition, the WENO scheme has also been integrated into industrial software in recent years and has received good evaluation and feedback.

In 2017, Xia and Xu [20] used the finite-difference and finite-volume WENO scheme to solve the DP equation for the first time. They introduced two auxiliary variables, m and p, and the DP equation was rewritten as a hyperbolic-elliptic system. Recently, Lin [21] developed a finite-difference unequal-sized WENO scheme and the multi-resolution WENO scheme for solving the DP equation. They achieve fifth-order and seventh-order accuracy and simplify the process of solving DP equations. Many other valid numerical schemes [22,23,24,25] have been constructed one after another. Since this paper designs an HWENO scheme to solve the DP equation, a brief introduction to the development of HWENO is given.

To enhance the compactness of the WENO scheme, Qiu and Shu [26,27] constructed the finite volume HWENO scheme by adding auxiliary equations, which greatly increased the compactness of the scheme. The HWENO scheme originates from essentially non-oscillatory and weighted essentially non-oscillatory schemes, which have been popular methods for solving nonlinear hyperbolic conservation laws in recent decades. The original HWENO schemes employ different polynomials for spatial discretizations of the original partial differential equation and its derivative equations. The schemes are compact, but they face some problems when simulating some benchmark problems such as the double Mach and the forward-step problems. Zhu and Qiu [28] tackled this issue by designing a new procedure to reconstruct the derivative terms. Liu and Qiu [29,30] introduced a new finite-difference HWENO framework in 2015, but needed an additional positivity-preserving flux-limiter methodology to ensure its robustness. Zahran and Abdalla applied a similar method to improve the accuracy of the finite-difference HWENO scheme from fifth order to higher order. Ma and Wu [31] devised a compact HWENO scheme, which uses the compact difference method to solve the first derivative values of the solution, while Cai et al. [32] used the strong stability-preserving (SSP) multi-step temporal discretization procedure for the HWENO schemes. The solution of the PDE and its derivatives are the unknown variables solved by these finite-volume and finite-difference HWENO schemes. Soon after, Li [33] borrowed the ideas from the MR-WENO scheme and constructed the finite-difference multi-resolution HWENO scheme. However, these finite-difference HWENO schemes will lose precision in two-dimensional problems, so Zhao [34] constructed the modified HWENO scheme by modifying the derivative value on each stencil. Zhu [35] designed a new method that is different from the traditional dimension-by-dimension discrete, and solved this problem. The HWENO scheme uses a smaller stencil than the WENO scheme, but it requires more CPU time to run. At present, there are many methods to speed up the calculation speed of HWENO scheme, such as using a hybrid HWENO scheme to avoid the complicated HWNEO reconstruction process in smooth parts, or using a fast-sweeping HWENO method to speed up the convergence of the HWENO scheme. We will not go into details here.

In this paper, following the idea of the HWENO scheme proposed by Liu and Qiu in [29], we construct an HWENO scheme for the DP equation. We adopt Lin’s method [21] to rewrite the DP equation as a hyperbolic-ellipse system. The first equation can be solved using the traditional HWENO schemes; the key is how to solve the second elliptic equation. For the second derivative of the equation, the compactness of the stencil will be destroyed using linear polynomial reconstruction, so we design a new linear Hermite interpolation to deal with this item. The first derivative of the first hyperbolic equation is discretized by the fifth-order HWENO method, and the time term is discretized by the Runge–Kutta method. It is the first time that the HWENO scheme has been applied to solve nonlinear high-order derivatives (>2) and the HWENO method we propose performs well for DP equations.

This paper is structured as follows: Section 2 presents the reconstruction procedure of the finite-difference HWENO scheme for solving the DP equation in detail. Section 3 shows extensive numerical tests to demonstrate the accuracy and the performance of the HWENO scheme for the DP equation. Section 4 provides concluding remarks.

2. HWENO Scheme for the DP Equation

In this section, we develop the fifth-order finite-difference HWENO scheme for the below DP equation:

$$\left\{\begin{array}{cc}{u}_t-u_{txx}+4f{\left(u\right)}_x=f{\left(u\right)}_{xxx},\hfill & \\ u(x,0)=u_0\left(x\right).\hfill & \end{array}\right.$$

(3)

To simplify, we use a uniform cell mesh $x_i, i=1,\cdots ,N$, with uniform mesh size $\Delta x=x_{i+1}-x_i$. We denote the cell by $I_i=[x_{i-1/2},x_{i+1/2}]$ and its center by $x_i=\frac{1}{2}(x_{i-1/2}+x_{i+1/2})$. By introducing the auxiliary variables r, the DP equation can be rewritten as a hyperbolic equation and an elliptic equation

$$\left\{\begin{array}{c}{u}_t+f{\left(u\right)}_x+r=0,\hfill \\ r-r_{xx}=3f{\left(u\right)}_x,\hfill \\ u(x,0)=u_0\left(x\right).\hfill \end{array}\right.$$

(4)

This can be easily proved by substituting $r=-u_t-f{\left(u\right)}_x$ into $r-r_{xx}=3f{\left(u\right)}_x$, and after simplification, the original equation can be obtained. Although an additional variable r is added and an additional auxiliary equation is increased, this can bring many benefits. First, it reduces the high-order derivatives to low-order derivatives. Second, it eliminates the mixed derivatives of time and space, which facilitates the construction of numerical schemes. The two equations, after processing, can be designed with independent numerical formats according to their respective properties. We set $R=r_x$ and $v=u_x$, Equation (4) can be rewritten as

$$\left\{\begin{array}{cc}{u}_t+f{\left(u\right)}_x+r=0,\hfill & \\ v_t+g{(u,v)}_x+R=0,\hfill & \\ u(x,0)=u_0\left(x\right),v(x,0)=u_0^{{}^{\prime }}\left(x\right),\hfill \end{array}\right.$$

(5)

and

$$\left\{\begin{array}{cc}r-r_{xx}=3f{\left(u\right)}_x,\hfill & \\ R-R_{xx}=3g{(u,v)}_x,\hfill & \\ u(x,0)=u_0\left(x\right),v(x,0)=u_0^{{}^{\prime }}\left(x\right),\hfill \end{array}\right.$$

(6)

where $g(u,v)=f^{\prime }\left(u\right)v$. The purpose of introducing the auxiliary equation is to use the point value of the function and the value of the first derivative when we reconstruct the numerical fulx. The auxiliary equation is constructed by simply taking the derivative of the original equation. To solve the system of Equation (5), we need to first solve the point values of $r_i$ and $R_i$ through Equation (6), and then solve the values of $u_i$ and $v_i$ through time discretization. Then we discretize Equation (6), and then obtain the approximations of $r\left(x_i\right)$ and $R\left(x_i\right)$, respectively. Equation (6) can be written in the following conservation form

$$\left\{\begin{array}{cc}{r}_i-\frac{{\widehat{r}}_{i+1/2}-{\widehat{r}}_{i-1/2}}{{\Delta x}^2}=3\frac{{\widehat{f}}_{i+1/2}-{\widehat{f}}_{i-1/2}}{\Delta x},\hfill & \\ R_i-\frac{{\widehat{R}}_{i+1/2}-{\widehat{R}}_{i-1/2}}{{\Delta x}^2}=3\frac{{\widehat{g}}_{i+1/2}-{\widehat{g}}_{i-1/2}}{\Delta x},\hfill & \end{array}\right.$$

(7)

The numerical approximations $r_i$ and $R_i$ correspond to the point values $r\left(x_i\right)$ and $R\left(x_i\right)$, respectively. To solve this system, an approximation of the flux needs to be obtained, and then the resulting linear system solved. Now, we will describe in detail the specific steps to solve ${\widehat{r}}_{i+1/2}$, ${\widehat{R}}_{i+1/2}$, ${\widehat{f}}_{i+1/2}$ and ${\widehat{g}}_{i+1/2}$, respectively. The rest of the fluxes can be obtained in the same way and are, therefore, omitted here.

Step 1. A function $s\left(x\right)$ is introduced for reconstruction, which satisfies

$$\frac{1}{\Delta x^2}{\int }_{x-\frac{\Delta x}{2}}^{x+\frac{\Delta x}{2}}{\int }_{\chi -\frac{\Delta x}{2}}^{\chi +\frac{\Delta x}{2}}s\left(\theta \right)d\theta d\chi =r\left(x\right).$$

(8)

The derivation of this formula is described in detail in [36]. This is the function that the WENO scheme used to approximate the second derivative, and we also used it to carry out the Hermite interpolation. By differentiating both sides of the above equation, the following equation can be easily obtained:

$$r{\left(x\right)}_{xx}=\frac{s(x+\Delta x)-2s\left(x\right)+s(x-\Delta x)}{\Delta x^2}.$$

(9)

We can construct a quintic polynomial $\phi \left(x\right)$ that satisfies

$$\frac{1}{\Delta x^2}{\int }_{x_j-\frac{\Delta x}{2}}^{x_j+\frac{\Delta x}{2}}\left({\int }_{\chi -\frac{\Delta x}{2}}^{\chi +\frac{\Delta x}{2}}\phi \left(\theta \right)d\theta \right)d\chi =r\left(x_j\right),x_j\in \{x_{i-1},x_i,x_{i+1},x_{i+2}\},$$

(10)

$$\frac{1}{\Delta x^2}{\int }_{x_j-\frac{\Delta x}{2}}^{x_j+\frac{\Delta x}{2}}\left({\int }_{\chi -\frac{\Delta x}{2}}^{\chi +\frac{\Delta x}{2}}{\phi }^{\prime }\left(\theta \right)d\theta \right)d\chi =R\left(x_j\right),x_j\in \{x_{i-1},x_{i+2}\}.$$

(11)

The stencil used in the second equation is the two points $i-1$ and $i+2$, instead of choosing i and $i+1$. This is because we found that the numerical simulation effect of this stencil selection would be better after numerical simulation of all the cases, especially for problems with strong discontinuities. This selection will enhance the stability of the HWENO scheme. Of course, if the stencil is chosen as i and i + 1, the numerical scheme is still valid. It is just that for problems with strong discontinuities, some weak numerical oscillations will appear in the simulation at the discontinuity. Thus, the numerical flux ${\widehat{r}}_{i+1/2}$ can be obtained by:

$$\begin{array}{cc}\hfill {\widehat{r}}_{i+1/2}=& \phi (x_i+\Delta x)-\phi \left(x_i\right)\hfill \\ \hfill =& \frac{1}{324}\left(24\Delta x(R_{i-1}+R_{i+2})+79r_{i-1}-513r_i+513r_{i+1}-79r_{i+2}\right).\hfill \end{array}$$

(12)

The flux ${\widehat{r}}_{i+1/2}$ is computed by a high-order linear reconstruction directly, without adding any limiter. Due to the elliptic nature of the equation, it will not cause numerical oscillation. For $R_{xx}$, even though it is the third derivative of r, because of the nature of the equation, the central difference or higher order linear reconstruction does not cause oscillation either. Thus, we can perform the same thing as ${\widehat{r}}_{i+1/2}$ which is an important reason why the HWENO format can be used for the DP equation. If WENO reconstruction or HWENO reconstruction is still needed for the second derivative like the nonlinear degenerate parabolic equation, this elliptic equation cannot be transformed into a linear system of equations. This will cause great difficulty in solving the equation.

Step 2. In the same wave, a seventh-degree polynomial $\psi \left(x\right)$ can be obtained from the following equations

$$\frac{1}{\Delta x^2}{\int }_{x_j-\frac{\Delta x}{2}}^{x_j+\frac{\Delta x}{2}}\left({\int }_{\chi -\frac{\Delta x}{2}}^{\chi +\frac{\Delta x}{2}}\psi \left(\theta \right)d\theta \right)d\chi =r\left(x\right),x_j\in \{x_{i-1},x_i,x_{i+1},x_{i+2}\},$$

(13)

$$\frac{1}{\Delta x^2}{\int }_{x_j-\frac{\Delta x}{2}}^{x_j+\frac{\Delta x}{2}}\left({\int }_{\chi -\frac{\Delta x}{2}}^{\chi +\frac{\Delta x}{2}}{\psi }^{\prime }\left(\theta \right)d\theta \right)d\chi =R\left(x\right),x_j\in \{x_{i-1},x_i,x_{i+1},x_{i+2}\}.$$

(14)

The numerical flux ${\widehat{R}}_{i+1/2}$ has the form

$$\begin{array}{cc}\hfill {\widehat{R}}_{i+1/2}& ={\psi }^{\prime }(x_i+\Delta x)-{\psi }^{\prime }\left(x_i\right)\hfill \\ \hfill & =\frac{1}{4}\left(\Delta x(-R_{i-1}-9R_i+9R_{i+1}+R_{i+2})-4r_{i-1}+4r_i+4r_{i+1}-4r_{i+2}\right).\hfill \end{array}$$

(15)

The reconstruction of the flux ${\widehat{r}}_{i+1/2}$ and ${\widehat{R}}_{i+1/2}$ is obtained. The sixth-order approximation of ${\widehat{r}}_{i+1/2}$ is different from the traditional linear reconstruction which uses six points; only four points are used here. Since no flux splitting is required here, the stencil remains compact and does not increase in width. The stencil used to solve the eighth-order approximation of ${\widehat{R}}_{i+1/2}$ is $S=\{x_{i-1},x_i,x_{i+1},x_{i+2}\}$, which is still a compact stencil with only four points.

Step 3. To maintain stability, the upwind quality of the schemes is required. First, both the flux f(u) and the flux g(u, v) can be split into two parts, respectively: $f\left(u\right)=f{\left(u\right)}^{+}+f{\left(u\right)}^{-}$, $g(u,v)=g{(u,v)}^{+}+g{(u,v)}^{-}$ with $\frac{d{f}^{+}\left(u\right)}{du}\ge 0$, $\frac{d{f}^{-}\left(u\right)}{du}\le 0$ and $\frac{d{g}^{+}(u,v)}{dv}\ge 0$, $\frac{d{g}^{-}(u,v)}{dv}\le 0$. Here, we use a simple Lax–Friedrichs splitting:

$$\begin{array}{cc}\hfill & f^{+}\left(u\right)=\frac{1}{2}(f\left(u\right)+\alpha u),f^{-}\left(u\right)=\frac{1}{2}(f\left(u\right)-\alpha u);\hfill \\ \hfill & g^{+}(u,v)=\frac{1}{2}(g(u,v)+\alpha v),g^{-}(u,v)=\frac{1}{2}(g(u,v)-\alpha v),\hfill \end{array}$$

(16)

where $\alpha ={max}_u\mid f^{\prime }\left(u\right)\mid$. At the same time, ${\widehat{f}}_{i+1/2}$ and ${\widehat{g}}_{i+1/2}$ also split into two parts: ${\widehat{f}}_{i+1/2}={\widehat{f}}_{i+1/2}^{+}+{\widehat{f}}_{i+1/2}^{-}$ and ${\widehat{g}}_{i+1/2}={\widehat{g}}_{i+1/2}^{+}+{\widehat{g}}_{i+1/2}^{-}$.

Step 3.1. We construct three quadratic polynomials $p_1\left(x\right)$, $p_2\left(x\right)$, $p_3\left(x\right)$ which satisfy

$$\begin{array}{cc}\hfill \frac{1}{\Delta x}{\int }_{I_m}{p}_1\left(x\right)dx=f^{+}\left(u_m\right),& m=i-1,i,\frac{1}{\Delta x}{\int }_{I_{i-1}}{p}_1^{\prime }\left(x\right)dx=g^{+}(u_{i-1},v_{i-1}),\hfill \\ \hfill \frac{1}{\Delta x}{\int }_{I_m}{p}_2\left(x\right)dx=f^{+}\left(u_m\right),& m=i,i+1,\frac{1}{\Delta x}{\int }_{I_{i+1}}{p}_2^{\prime }\left(x\right)dx=g^{+}(u_{i+1},v_{i+1}),\hfill \\ \hfill \frac{1}{\Delta x}{\int }_{I_m}{p}_3\left(x\right)dx=f^{+}\left(u_m\right),& m=i-1,i,i+1.\hfill \end{array}$$

(17)

We want to approximate $p_j\left(x\right)$ at the half nodes $x_{i+\frac{1}{2}}$, so we have

$$\begin{array}{cc}\hfill p_1\left(x_{i+\frac{1}{2}}\right)=& \frac{13}{6}{f}^{+}\left(u_i\right)-\frac{7}{6}{f}^{+}\left(u_{i-1}\right)-\frac{2}{3}\Delta x{g}^{+}(u_{i-1},v_{i-1}),\hfill \\ \hfill p_2\left(x_{i+\frac{1}{2}}\right)=& \frac{1}{6}{f}^{+}\left(u_i\right)+\frac{5}{6}{f}^{+}\left(u_{i+1}\right)-\frac{1}{3}\Delta x{g}^{+}(u_{i+1},v_{i+1}),\hfill \\ \hfill p_3\left(x_{i+\frac{1}{2}}\right)=& \frac{5}{6}{f}^{+}\left(u_i\right)-\frac{1}{6}{f}^{+}\left(u_{i-1}\right)+\frac{1}{3}{f}^{+}\left(u_{i+1}\right),\hfill \end{array}$$

(18)

Step 3.2. Then, we use the traditional smoothness indicator [29]

$${\beta }_n=\sum _{l=1}^3\Delta x^{2l-1}{\int }_{x_i}^{x_{i+1}}{\left(\frac{d^l{p}_n\left(x\right)}{d{x}^l}\right)}^{2}dx,n=1,2,3,$$

(19)

to evaluate the smoothness of the stencil and construct the nonlinear weights ${\omega }_n$

$${\omega }_n=\frac{{\overline{\omega }}_n}{{\sum }_{l=1}^3{\overline{\omega }}_l},{\overline{\omega }}_n=\frac{d_n}{{({\beta }_n+\epsilon )}^2},n=1,2,3,$$

(20)

where $d_1=\frac{9}{80},d_2=\frac{42}{80}$ and $d_3=\frac{29}{80}$ are linear weights and $\epsilon ={10}^{-6}$, which is introduced to enhance the stability of the scheme. The method of solving linear weights here is the same as the traditional WENO or HWENO format. Through linear weights, the third-order approximation reconstructed on the three stencils can be combined into a fifth-order approximation. In smooth regions, the value of nonlinear weights will approach the value of linear weights, and in non-smooth regions, nonlinear weights can automatically give more weight to smoother stencils, thus achieving the effect of essential non-oscillation.

Step 3.3. The final reconstruction is

$$\begin{array}{c}\hfill {\widehat{f}}_{i+1/2}^{+}=\sum _{n=1}^3{\omega }_n{p}_n\left(x_{i+1/2}\right).\end{array}$$

(21)

Step 4. Next, the construction process of the flux ${\widehat{g}}_{i+1/2}^{+}$ is introduced in detail.

Step 4.1. We first construct three cubic polynomials $p_1\left(x\right)$, $p_2\left(x\right)$ and $p_3\left(x\right)$ that satisfy

$$\begin{array}{cc}\hfill \frac{1}{\Delta x}{\int }_{I_m}{p}_1\left(x\right)dx=f^{+}\left(u_m\right),& \frac{1}{\Delta x}{\int }_{I_m}{p}_1^{\prime }\left(x\right)dx=g^{+}(u_m,v_m),m=i-1,i,\hfill \\ \hfill \frac{1}{\Delta x}{\int }_{I_m}{p}_2\left(x\right)dx=f^{+}\left(u_m\right),& \frac{1}{\Delta x}{\int }_{I_m}{p}_2^{\prime }\left(x\right)dx=g^{+}(u_m,v_m),m=i,i+1,\hfill \\ \hfill \frac{1}{\Delta x}{\int }_{I_m}{p}_3\left(x\right)dx=f^{+}\left(u_m\right),& m=i-1,i,i+1,\frac{1}{\Delta x}{\int }_{I_i}{p}_3^{\prime }\left(x\right)dx=g^{+}(u_i,v_i).\hfill \\ \hfill \end{array}$$

(22)

Step 4.2. Then, we take the same form smoothness indicator as in [16,37]

$${\beta }_n=\sum _{l=1}^3\Delta x^{2l-1}{\int }_{x_i}^{x_{i+1}}{\left(\frac{d^l{p}_n\left(x\right)}{d{x}^l}\right)}^{2}dx,n=1,2,3,$$

(23)

and construct the nonlinear weight ${\omega }_n$

$${\omega }_n=\frac{{\overline{\omega }}_n}{{\sum }_{l=1}^3{\overline{\omega }}_l},{\overline{\omega }}_n=\frac{d_n}{{({\beta }_n+\epsilon )}^2},n=1,2,3,$$

(24)

where $d_1=\frac{1}{18}$, $d_2=\frac{15}{18}$ and $d_3=\frac{2}{18}$ are linear weights, and $\epsilon ={10}^{-6}$ which is introduced to to enhance the stability of the scheme.

Step 4.3. The final reconstruction is

$$\begin{array}{c}\hfill {\widehat{g}}_{i+1/2}^{+}=\sum _{n=1}^3{\omega }_n{p}_n^{\prime }\left(x_{i+1/2}\right).\end{array}$$

(25)

The HWENO reconstruction proposed in Step 3 and Step 4 is the same as the fifth-order finite-difference HWENO scheme proposed by Liu [29], and we have not made any changes. Other HWENO schemes are also available here.

Step 5. To write Equation (6) in vector form, let us introduce the following vectors. We set $\overrightarrow{r}={(r_1,\cdots ,r_N)}^T$, $\overrightarrow{R}={(R_1,\cdots ,R_N)}^T$, $\overrightarrow{f}={(3f{\left(u\right)}_x{\mid }_{x=x_1},\cdots ,3f{\left(u\right)}_x{\mid }_{x=x_N})}^T$ and $\overrightarrow{h}={(3g{(u,v)}_x{\mid }_{x=x_1},\cdots ,3g{(u,v)}_x{\mid }_{x=x_N})}^T$. Then, Equation (6) can be written as the following matrix form

$$\left\{\begin{array}{cc}A\overrightarrow{r}+B\overrightarrow{R}=\overrightarrow{f},\hfill & \\ C\overrightarrow{r}+D\overrightarrow{R}=\overrightarrow{h},\hfill & \end{array}\right.$$

(26)

or

$$\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)·\left(\begin{array}{c}\overrightarrow{r}\\ \overrightarrow{R}\end{array}\right)=\left(\begin{array}{c}\overrightarrow{f}\\ \overrightarrow{h}\end{array}\right)$$

Here, the matrices A, B, C and D have the following forms:

$$\left[\begin{array}{cccccccccccccc}{a}_3& a_4& a_5& \cdots & \cdots & a_1& a_2& b_3& b_4& b_5& \cdots & \cdots & b_1& b_2\\ .& .& .& .& .& .& .& .& .& .& .& .& .& .\\ a_4& a_5& \cdots & \cdots & a_1& a_2& a_3& b_4& b_5& \cdots & \cdots & b_1& b_2& b_3\\ c_3& c_4& c_5& \cdots & \cdots & c_1& c_2& d_3& d_4& d_5& \cdots & \cdots & d_1& d_2\\ .& .& .& .& .& .& .& .& .& .& .& .& .& .\\ c_4& c_5& \cdots & \cdots & c_1& c_2& c_3& d_4& d_5& \cdots & \cdots & d_1& d_2& d_3\end{array}\right]\left(\begin{array}{c}{r}_1\\ .\\ r_N\\ R_1\\ .\\ R_N\end{array}\right)=\left(\begin{array}{c}3f_x\left(x_1\right)\\ .\\ 3f_x\left(x_N\right)\\ 3h_x\left(x_1\right)\\ .\\ 3h_x\left(x_N\right)\end{array}\right),$$

(27)

where

$$\begin{array}{cccccccccc}\hfill a_1& =\frac{79}{324\Delta x^2}\hfill & \hfill a_2& =-\frac{592}{324\Delta x^2}\hfill & \hfill a_3& =\frac{1026}{324\Delta x^2}+1\hfill & \hfill a_4& =-\frac{592}{324\Delta x^2}\hfill & \hfill a_5& =\frac{79}{324\Delta x^2}\hfill \\ \hfill b_1& =\frac{24}{324\Delta x}\hfill & \hfill b_2& =-\frac{24}{324\Delta x}\hfill & \hfill b_3& =0\hfill & \hfill b_4& =\frac{24}{324\Delta x}\hfill & \hfill b_5& =-\frac{24}{324\Delta x}\hfill \\ \hfill c_1& =-\frac{1}{\Delta x^3}\hfill & \hfill c_2& =\frac{2}{\Delta x^3}\hfill & \hfill c_3& =0\hfill & \hfill c_4& =-\frac{2}{\Delta x^3}\hfill & \hfill c_5& =\frac{1}{\Delta x^3}\hfill \\ \hfill d_1& =-\frac{1}{4\Delta x^2}\hfill & \hfill d_2& =-\frac{8}{4\Delta x^2}\hfill & \hfill d_3& =\frac{18}{4\Delta x^2}+1\hfill & \hfill d_4& =-\frac{8}{4\Delta x^2}\hfill & \hfill d_5& =-\frac{1}{4\Delta x^2}.\hfill \end{array}$$

This linear equation can be easily solved using the LU decomposition method to obtain $\overrightarrow{r}$ and $\overrightarrow{R}$. As it is not a simple tridiagonal matrix, it is still difficult to solve its inverse directly. Using the LU decomposition method to pretreat the matrix, the time-consuming LU decomposition can be avoided in each step.

Next, we solve Equation (5) and discretize to its conservation form:

$$\left\{\begin{array}{cc}\frac{d{u}_i\left(t\right)}{dt}+\frac{{\widehat{f}}_{i+1/2}-{\widehat{f}}_{i-1/2}}{\Delta x}+r_i=0,\hfill & \\ \frac{d{v}_i\left(t\right)}{dt}+\frac{{\widehat{g}}_{i+1/2}-{\widehat{g}}_{i-1/2}}{\Delta x}+R_i=0,\hfill & \end{array}\right.$$

(28)

where $u_i\left(t\right)$ and $v_i\left(t\right)$ denote the numerical approximations of $u(x_i,t)$ and $v(x_i,t)$ at the grid points, respectively. Each term in the equation is known and we can rewrite it as the below form:

$$\frac{d}{dt}\left(\begin{array}{c}{u}_i\left(t\right)\\ v_i\left(t\right)\end{array}\right)=\left(\begin{array}{c}\frac{{\widehat{f}}_{i-1/2}-{\widehat{f}}_{i+1/2}}{\Delta x}-r_i\\ \frac{{\widehat{g}}_{i-1/2}-{\widehat{g}}_{i+1/2}}{\Delta x}-R_i\end{array}\right)$$

(29)

Let us set $U={(u,v)}^T$, and the above equation can be written as follows: $\frac{dU}{dt}=L\left(U\right)$. Where the operator L represents the right end of Equation (29). Using the TVD Runge–Kutta method [38], these equations can be solved.

3. Numerical Experiments

To test the performance of the fifth-order Hermite WENO scheme described in the previous section, some benchmark numerical examples are presented in this section. For all the examples in this paper, the time step is set as $\Delta t=0.6\Delta x$, unless otherwise specified. We only tested this scheme and did not compare it with other numerical schemes such as the WENO, US-WENO, MR-WENO, DG, etc., schemes. For the DP equation, the numerical results obtained using different numerical schemes are very similar; it is difficult to see the obvious difference. Therefore, other studies on numerical solutions of the DP equation have not been compared with other numerical schemes, but only with exact solutions to verify the effectiveness of numerical schemes. Hence, we only compare the numerical results with the exact solutions in the following.

Example 1. 

Accuracy test for the single-soliton solution.

Since there is no smooth real exact solution to this equation, many researchers use its virtual exact solution for precision analysis. We use the smooth part of its solitary wave solution to test the accuracy. Now we consider the following peak traveling wave solution:

$$\begin{array}{c}\hfill u(x,t)=c{e}^{-\mid x-ct\mid },\end{array}$$

(30)

where c is the wave speed. We also adopt the initial condition in [20] to test the accuracy of the scheme. The computation domain is $[-25,25]$. $c=0.25$ is chosen to test the accuracy in smooth parts, which is 1/10 of the computational domain away from the peak. The final time is $t=1$. The time step is small enough to make the spatial discretization error dominant in the simulation. In addition,

Table 1. Example 1. $L_1$ and $L_{\infty }$ errors at $t=1$.

	HWENO
$\mathbf{N}$	${\mathit{L}}_{\mathit{1}}$Error	Order	${\mathit{L}}_{\mathit{\infty }}$Error	Order
40	2.15 × 10−6		l2.60 × 10−5
80	7.00 × 10−7	1.62	1.16 × 10−5	1.17
160	2.03 × 10−8	5.11	3.91 × 10−7	4.88
320	5.92 × 10−10	5.10	1.23 × 10−8	5.00
640	1.72 × 10−11	5.11	3.69 × 10−10	5.05

shows the numerical errors and orders of the HWENO scheme presented. The expected design’s fifth-order accuracy is achieved by this method.

Example 2. 

Single peakon and anti-peakon solutions.

The wave propagation of peakon $u(x,t)=e^{-|x-t|},$ and anti-peakon solutions $u(x,t)=-e^{-|x+t|}$ is tested in this example. The traveling speed is 1 and the computational domain is [−40, 40]. We show the numerical solution with $N=640$ grid points at $t=4,8,12,16$ in Figure 1 and Figure 2. The peakon and anti-peakon profiles match well with the exact solution. There is no obvious near-discontinuous oscillation in the numerical results. We find that there is a slight numerical oscillation far away from the isolated wave, which is found to appear in the numerical simulation of the WENO scheme and DG scheme. The reason for the numerical oscillation is not clear at present. There are no articles studying the numerical oscillation, nor are there any articles that give a theoretical explanation. It is hoped that this can be improved in the future. This weak numerical oscillation also occurs in Example 3.

Example 3. 

Two-peakon interaction.

The simulation of the interaction of two peakons for the DP equation with the initial condition $u=c{e}^{-\mid x-a\mid }+d{e}^{-\mid x-b\mid }$ is performed for this example with the parameters $a=-13.792$, $b=-4$, $c=2$ and $d=1$. The computational domain is $[-40,40]$, which is divided into $N=640$ grid points. The numerical solution computed by the proposed HWENO scheme is shown in Figure 3. The property of a soliton is that it retains its original shape and speed until and after it encounters other solitons. It is shown by the numerical results that this characteristic is fully met. Although this example has no exact solution, its numerical results are similar to those obtained by other numerical schemes.

Example 4. 

Shock peak solution.

In addition to the soliton solution, there is also a shock solution $u(x,t)=-\frac{x{e}^{-\mid x\mid }}{(t+1)\mid x\mid }$ for the Degasperis–Procesi equation. The solution has a discontinuity at $x=0$ which poses a great challenge to the shock-capture capability of the numerical scheme. For such strong discontinuity problems, the HWENO scheme is more difficult to deal with because it not only uses point values, but also the values of first-order derivatives. The computational domain is $[-25,25]$ which is divided into $N=640$ grid points. Due to the shock discontinuity in the solution, the numerical oscillations could appear for the linear schemes. We show the shock peakon profile of numerical results and exact solution at $t=3$ and 6 in Figure 4. The figure shows that the numerical results match the exact solution well, and no significant numerical oscillation is observed. The proposed HWENO scheme can handle the problem with discontinuities effectively, as shown by the example.

Example 5. 

Peak and anti-peakon interaction.

This example demonstrates the interaction between a peakon and anti-peakon. We use the following initial condition for the DP equation: $u(x,0)=e^{-\mid x+5\mid }-e^{-\mid x-5\mid }.$ We divide the computation domain $[-20,20]$ into $N=640$ grid points. Although the two peaks do not affect each other after collision in Example 3, a shock peak will be generated when peak and anti-peak collide at $t\approx 5$. Figure 5 shows the numerical solution at $T=0,4,5$ and 7. The shock is completely resolved in the numerical solution.

Example 6. 

Triple interaction.

This example involves a more complex interaction of the DP equation, with peakon, anti-peakon, and stationary-shock peakon. We use the following initial condition: $u(x,0)=sign\left(x\right)e^{-\mid x\mid }-e^{-\mid x-5\mid }+e^{-\mid x+5\mid }.$ We divide the computational domain $[-20,20]$ into $N=320$ grid points. The exact solution has a triple collision at $t\approx 5.32$. Figure 6 shows the numerical solutions at $t=0,2,5.32$ and 7. None of the solutions have numerical oscillations.

Example 7. 

Wave-breaking phenomena.

This example demonstrates a special phenomenon of the Degasperis–Procesi equation, namely, the wave-breaking phenomena. We assume the initial condition $u(x,0)\in H^s\left(R\right)$ and $s>\frac{1}{2}$, and that there is a point $x_0\in R$ where the momentum density $m_0\left(x\right)=u(x,0)-u_{xx}(x,0)$ changes its sign from positive to negative. Based on the theoretical analysis in [39], if the initial value satisfies these conditions, the solution will break down in finite time and shock waves will typically emerge later. We use two initial conditions, given by

$$u(x,0)=e^{0.5x^2}sin\left(\pi x\right),$$

(31)

and

$$u(x,0)=sec{h}^2\left(\frac{x+50}{10}\right).$$

(32)

to check this theory. For initial condition (31), the computation domain is $[-2,2]$. We divided it into $N=640$ grid points. Figure 7 shows the numerical solutions at $t=0$, $0.18$, $0.5$ and $1.1$; we can see the shock appear at $t>0.18$. For the other initial condition, we plot the numerical solutions at $t=0$, 15, 30 and 60 in the interval $-100$ to 100 which is divided into $N=1280$ grid points. We can see from Figure 8 the shock is fully resolved in the numerical solution.

4. Concluding Remarks

We propose a fifth-order finite-difference Hermite weighted essentially non-oscillatory scheme for the DP equation in this paper. The main benefit of this scheme is that it has a compact reconstruction process and requires only three points to attain the fifth-order accuracy, compared with the WENO method in [21]. A drawback of the finite-difference HWENO scheme is that it will lose its designed accuracy order when dealing with two-dimensional problems. Since the DP equations are all one-dimensional, we do not need to think about this. Numerical experiments show that the HWENO scheme can achieve the expected design accuracy for smooth regions and can achieve the characteristic of being essentially non-oscillatory in the vicinity of discontinuous. The presented method is not only applicable to the DP equation involved in this paper, but also to the CH equation or KDV equation mentioned above; thus, this will be the follow-up work of this paper. The numerical scheme constructed in this paper uses the fifth-order finite-difference HWENO scheme proposed by Liu [29] and is also valid for using other types of HWENO schemes, such as the HWENO scheme proposed by Li [33] and Zhao [34]. We construct the fifth-order finite-difference HWENO scheme in this paper. The higher order HWENO scheme and the finite-volume HWENO scheme can be designed and constructed in a similar way. In addition, since the second derivative is discretized in the scheme, it has good reference significance for solving equations containing the second derivative, such as nonlinear degenerate parabolic equations. Moreover, the $\mu DP$ equation is an important class of DP-type equations, whose properties are more complex and also one of the topics of our future research.