A New Fifth-Order Finite Difference Compact Reconstruction Unequal-Sized WENO Scheme for Fractional Differential Equations

Abstract

This paper designs a new finite difference compact reconstruction unequal-sized weighted essentially nonoscillatory scheme (CRUS-WENO) for solving fractional differential equations containing the fractional Laplacian operator. This new CRUS-WENO scheme uses stencils of different sizes to achieve fifth-order accuracy in smooth regions and maintain nonoscillatory properties near discontinuities. The fractional Laplacian operator of order $\beta$$(0<\beta <1)$ is split into the integral part and the first derivative term. Using the Gauss–Jacobi quadrature method to solve the integral part of the fractional Laplacian operators, a new finite difference CRUS-WENO scheme is presented to discretize the first derivative term of the fractional equation. This new CRUS-WENO scheme has the advantages of a narrower large stencil and high spectral resolution. In addition, the linear weights of the new CRUS-WENO scheme can be any positive numbers whose sum is one, which greatly reduces the calculation cost. Some numerical examples are given to show the effectiveness and feasibility of this new CRUS-WENO scheme in solving fractional equations containing the fractional Laplacian operator.

1. Introduction

In this paper, we design a new finite difference compact reconstruction unequal-sized weighted essentially nonoscillatory (CRUS-WENO) scheme. It is the first time that we can use the new CRUS-WENO scheme to solve fractional differential equations containing the fractional Laplacian operator. We first mention a few features and advantages of this CRUS-WENO scheme. This new scheme has the advantages of a small number of stencils and high spectral resolution. Different from the classic WENO scheme [1,2], the linear weights of the new CRUS-WENO scheme can be any positive numbers whose sum is one, which greatly reduces the calculation cost. Furthermore, the CRUS-WENO scheme only uses a three-point stencil and two two-point stencils to approximate the classical first derivative. Finally, this new CRUS-WENO scheme can achieve fifth-order accuracy on the narrower stencils and suppress oscillations near discontinuities.

In recent decades, the WENO schemes have been extensively developed to solve hyperbolic conservation law equations. Since Liu et al. [3] first constructed the WENO scheme in 1994, various WENO schemes have been continuously proposed and applied to different fields [4,5,6,7]. In 1996, Jiang et al. [2] constructed the finite difference WENO format for the first time and extended it to multidimensional situations. In 2000, Balsara and Shu [8] designed a high-order WENO scheme, which can achieve any high-order accuracy. In the procedure of constructing the traditional WENO scheme, the linear weight may have negative values. Therefore, special processing is required for linear weights, which increases the computational complexity. In 2016, Zhu et al. [9] overcame this shortcoming and proposed a finite difference US-WENO scheme for the first time. Although these WENO schemes can achieve high-order accuracy, their stencils become wider and wider as the accuracy increases. As early as 1992, Lele [10] proposed a compact scheme that achieved high-order accuracy on smaller stencils. In 2001, Jiang et al. [11] innovatively proposed a nonlinear weighted compact scheme for shock capturing. Based on [11], Ghosh et al. [12] proposed the finite difference compact reconstruction WENO scheme in 2012. In 2019, Shi and Guo [13] designed a finite difference alternative Compact-WENO scheme. The use of the compact reconstruction scheme reduced the number of stencils and improved the spectral resolution. However, due to the limitations of the classic WENO scheme [1,2], additional calculation costs were added during the execution of the compact reconstruction scheme. Therefore, we design a new finite difference CRUS-WENO scheme in this article. This scheme has the advantages of a small number of stencils and high spectral resolution. In addition, the linear weight of the new CRUS-WENO scheme can be any positive number whose sum is one, which greatly reduces the calculation cost.

Introducing the concept of fractional calculus into the existing integer-order model can solve the shortcomings of the integer-order itself and better adapt to the changing real world. In recent years, fractional differential equations have appeared in many application fields, such as gas explosion [14], complex network [15,16], geography [17,18], physical systems [19], and signal processing [20]. The definition of the fractional differential operator is not unique. In 2018, Kaur et al. [21] analyzed a new form of the (3 + 1)-dimensional generalized Kadomtsev-Petviashvili(KP)-Boussinesq equation for exploring lump solutions by making use of its Hirota bilinear form. In 2019, Kaur et al. [22] explored bright–dark lump wave solutions for a new form of the (3 + 1)-dimensional BKP-Boussinesq equation. In the same year, Wazwaz et al. [23] examined a family of Boussinesq equations of distinct structures and dimensions. In 2021, Hosseini et al. [24] explored a (2 + 1)-dimensional nonlinear model with the beta time derivative describing the wave propagation in the Heisenberg ferromagnetic spin chain. This article mainly studies the fractional Laplacian operator [25,26]. In the financial mathematical model, there have been some results in the research on this operator [27,28]. However, there has not been much research on higher-order numerical schemes of conservation laws with fractional Laplacian operators. In 2010, Droniou [26] first proposed the finite difference method to solve the fractal conservation laws equations. In 2011, Cifani et al. [29] used the discontinuous Galerkin method to solve the fractal conservation laws equations and gave numerical results. However, Xu et al. [30] pointed out in 2014 that their numerical test [29] failed to achieve the expected results and then proposed a local discontinuous Galerkin method to solve the fractional convection–diffusion equations. Furthermore, they mentioned that for this fractional convection–diffusion equation, when $\beta <1$, the solution was usually not smooth and shocks may appear even for the smooth initial datum. Based on [30], Safdari et al. [31] proposed the local discontinuous Galerkin method of spline interpolation in 2021 to solve the nonlinear fractional convection–diffusion equation. In 2013, Deng et al. [32] innovatively proposed using the high-order finite difference WENO scheme to discretize the Caputo fractional differential equation whose initial values may be discontinuous. In 2021, Zhang et al. [33] designed a new sixth-order US-WENO scheme to solve the Caputo fractional differential equation. Based on previous research foundations, this paper proposes a new high-order CRUS-WENO method to solve fractional differential equations containing the fractional Laplacian operator.

In this article, we propose a new fifth-order finite difference compact reconstruction unequal-sized weighted essentially nonoscillatory scheme (CRUS-WENO) for solving fractional differential equations containing the fractional Laplacian operator. This scheme mainly uses the information defined on a three-point stencil and two two-point stencils to approximate the classical first derivative in the fractional differential equation. However, for the integral parts in fractional differential equations, we use the Gauss–Jacobi quadratures method to solve them. When there are discontinuities in the large stencil, the scheme will automatically replace the information of the large stencil with the information of either of the two small stencils. We can construct a tridiagonal linear equation system based on the final form of the new CRUS-WENO scheme. Furthermore, the value of the numerical flux can be obtained by solving this linear equation directly by using the LU decomposition method. Finally, we use the third-order TVD Runge–Kutta method [34] to carry out the time discretization.

The rest of this article is organized as follows. In Section 2, we design a new finite difference CRUS-WENO scheme to solve the fractional Laplacian equation. In Section 3, several numerical examples are given to illustrate the feasibility and effectiveness of this new CRUS-WENO scheme. We give a simple conclusion in Section 4.

2. New CRUS-WENO Scheme

The fractional differential equation is

$$\left\{\begin{array}{c}\frac{\partial u(x,t)}{\partial t}+\frac{\partial f\left(u\right)}{\partial x}=\nu (-{(-\Delta )}^{\frac{\beta }{2}})u(x,t),(x,t)\in \mathbb{R}\times (0,T],\beta \in (0,1),\hfill \\ u(x,0)=u_0\left(x\right),x\in \mathbb{R},\hfill \end{array}\right.$$

(1)

in which f is a Lipschitz continuous function, and $(-{(-\Delta )}^{\frac{\beta }{2}})$ is a fractional Laplacian operator that can be defined by the Fourier transform [25,35,36,37,38]

$$\widehat{-{(-\Delta )}^{\frac{\beta }{2}}u\left(\zeta \right)}={\left(2\pi \right)}^{\beta }{\left|\zeta \right|}^{\beta }\widehat{u}\left(\zeta \right).$$

(2)

The fractional Laplacian operator can also be defined by the concept of the fractional calculus [30,31,39]

$$-{(-\Delta )}^{\frac{\beta }{2}}u\left(x\right)=\frac{{\partial }^{\beta }u\left(x\right)}{{\partial \left|x\right|}^{\beta }}=-\frac{{}_{-\infty }\mathrm{D}_x^{\beta }u\left(x\right)+{}_x\mathrm{D}_{\infty }^{\beta }u\left(x\right)}{2cos\left(\frac{\beta \pi }{2}\right)},$$

(3)

where ${}_{-\infty }\mathrm{D}_x^{\beta }$ and ${}_x\mathrm{D}_{\infty }^{\beta }$ are the left and right Riemann–Liouville fractional derivatives of order $\beta$, respectively, and are defined as [40] in the following

$${}_{-\infty }\mathrm{D}_x^{\beta }u\left(x\right)=\frac{1}{\mathrm{\Gamma }(1-\beta )}\frac{d}{dx}{\int }_{-\infty }^x\frac{u\left(\xi \right)}{{(x-\xi )}^{\beta }}d\xi ,\beta \in (0,1),$$

(4)

and

$${}_x\mathrm{D}_{\infty }^{\beta }u\left(x\right)=-\frac{1}{\mathrm{\Gamma }(1-\beta )}\frac{d}{dx}{\int }_x^{\infty }\frac{u\left(\xi \right)}{{(\xi -x)}^{\beta }}d\xi ,\beta \in (0,1).$$

(5)

The high-order CRUS-WENO scheme specified in this paper is considered on a finite domain $\Omega =[a,b]\subset \mathbb{R}$. Therefore,

$$-{(-\Delta )}^{\frac{\beta }{2}}u\left(x\right)=-\frac{{}_a\mathrm{D}_x^{\beta }u\left(x\right)+{}_x\mathrm{D}_b^{\beta }u\left(x\right)}{2cos\left(\frac{\beta \pi }{2}\right)}.$$

(6)

Combining the ideas of (4) and (5), (1) can be rewritten as

$$\frac{\partial u(x,t)}{\partial t}+\frac{\partial f\left(u\right)}{\partial x}=-\nu \frac{\frac{1}{\mathrm{\Gamma }(1-\beta )}\frac{d}{dx}{\int }_a^x\frac{u(\xi ,t)}{{(x-\xi )}^{\beta }}d\xi -\frac{1}{\mathrm{\Gamma }(1-\beta )}\frac{d}{dx}{\int }_x^b\frac{u(\xi ,t)}{{(\xi -x)}^{\beta }}d\xi }{2cos\left(\frac{\beta \pi }{2}\right)}.$$

(7)

By applying a simple linear transformation, (7) can be rewritten as

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial u(x,t)}{\partial t}+\frac{\partial f\left(u\right)}{\partial x}=}& -\nu \frac{\frac{d}{dx}\left[{\left(\frac{x-a}{2}\right)}^{1-\beta }{\int }_{-1}^1\frac{u(\frac{a+x}{2}+\frac{x-a}{2}\zeta ,t)}{{(1-\zeta )}^{\beta }}d\zeta \right]}{2cos\left(\frac{\beta \pi }{2}\right)·\mathrm{\Gamma }(1-\beta )}\hfill \\ \hfill {\displaystyle }& +\nu \frac{\frac{d}{dx}\left[{\left(\frac{b-x}{2}\right)}^{1-\beta }{\int }_{-1}^1\frac{u(\frac{b+x}{2}+\frac{b-x}{2}\zeta ,t)}{{(1+\zeta )}^{\beta }}d\zeta \right]}{2cos\left(\frac{\beta \pi }{2}\right)·\mathrm{\Gamma }(1-\beta )},\hfill \end{array}$$

(8)

in which the two integral parts on the right hand side of (8) are solved by the Gauss–Jacobi quadratures.

Next, this article focuses on the construction procedure of the new fifth-order CRUS-WENO scheme for solving the first spatial derivative term on the left hand side of (8). For the convenience of the following description, the classic first derivative problem is attributed as

$$u_t=g{\left(u\right)}_x.$$

(9)

We define a space uniform mesh $\Delta x=\frac{b-a}{M}$ with cell $I_r=\{x_{r-\frac{1}{2}},x_{r+\frac{1}{2}}\},x_{r+\frac{1}{2}}=x_r+\frac{\Delta x}{2}$. $u_r$ is the numerical approximation of $u(x,t)$ at point $x_r$. The semi-discrete finite difference equation of (9) is

$$\frac{d{u}_r}{dt}=\frac{{\widehat{g}}_{r+\frac{1}{2}}-{\widehat{g}}_{r-\frac{1}{2}}}{\Delta x},$$

(10)

where ${\widehat{g}}_{r+\frac{1}{2}}$ is the numerical flux, and it satisfies

$$\frac{{\widehat{g}}_{r+\frac{1}{2}}-{\widehat{g}}_{r-\frac{1}{2}}}{\Delta x}=g{\left(u\right)}_x{|}_{x_r}+\mathcal{O}(\Delta x^k).$$

(11)

The flux $g\left(u\right)$ can be split into $g\left(u\right)=g^{+}\left(u\right)+g^{-}\left(u\right)$ with $\frac{d{g}^{+}\left(u\right)}{du}\ge 0$ and $\frac{d{g}^{-}\left(u\right)}{du}\le 0$. The Lax-Friedrichs splitting [41] is used in this paper. For simplicity, the superscript is omitted when constructing the new fifth-order CRUS-WENO scheme in the following.

Similar to [9], one three-cell stencil $S_1=\{I_{r-1},I_r,I_{r+1}\}$ and two two-cell stencils $S_2=\{I_{r-1},I_r\},S_3=\{I_r,I_{r+1}\}$ are selected. Then one quartic polynomial and two quadric polynomials are defined on these stencils, respectively. Then, one fifth-order and two second-order compact approximations are constructed by

$$\frac{3}{10}{\widehat{g}}_{r-\frac{1}{2}}+\frac{6}{10}{\widehat{g}}_{r+\frac{1}{2}}+\frac{1}{10}{\widehat{g}}_{r+\frac{3}{2}}=\frac{1}{30}{g}_{r-1}+\frac{19}{30}{g}_r+\frac{10}{30}{g}_{r+1},$$

(12)

$$\frac{2}{3}{\widehat{g}}_{r-\frac{1}{2}}+\frac{1}{3}{\widehat{g}}_{r+\frac{1}{2}}=\frac{1}{6}{g}_{r-1}+\frac{5}{6}{g}_r,$$

(13)

$$\frac{2}{3}{\widehat{g}}_{r+\frac{1}{2}}+\frac{1}{3}{\widehat{g}}_{r+\frac{3}{2}}=\frac{1}{6}{g}_r+\frac{5}{6}{g}_{r+1}.$$

(14)

Different from the classic compact reconstruction WENO scheme [12], the linear weights $d_i,i=1,2,3$ of the new CRUS-WENO scheme can be set as any positive numbers with only one requirement that ${\sum }_{i=1}^3{d}_i=1$. This advantage greatly simplifies the difficulty in the spatial reconstruction procedure and effectively saves calculation cost. In the next section, for example, three different types of linear weights are set as (1) $d_1=0.98$, $d_2=0.01$, and $d_3=0.01$; (2) $d_1=1/3$, $d_2=1/3$, and $d_3=1/3$; and (3) $d_1=0.01$, $d_2=0.495$, and $d_3=0.495$. The new smooth indicators ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ [2] are given as

$$\begin{array}{c}{\beta }_1=\frac{1}{14400}{(-80g_{r-1}+10g_{r-2}+80g_{r+1}-10g_{r+2})}^2+\frac{781}{2880}(2g_{r-1}-g_{r-2}\hfill \\ -2g_{r+1}+g_{r+2}{)}^2+\frac{1}{15600}{(174g_{r-1}-11g_{r-2}-326g_r+174g_{r+1}-11g_{r+2})}^2\hfill \\ +\frac{1421461}{1310400}{(-4g_{r-1}+g_{r-2}+6g_r-4g_{r+1}+g_{r+2})}^2,\hfill \end{array}$$

(15)

$${\beta }_2={(g_{r-1}-g_r)}^2,$$

(16)

and

$${\beta }_3={(g_r-g_{r+1})}^2.$$

(17)

Then, the nonlinear weights [4,42] are given as

$${\omega }_j=\frac{{\alpha }_j}{{\sum }_{i=1}^3{\alpha }_i},{\alpha }_j=d_j(1+\frac{\tau }{\epsilon +{\beta }_j}),j=1,2,3,$$

(18)

where $\tau ={\left(\frac{|{\beta }_1-{\beta }_2|+|{\beta }_1-{\beta }_3|}{2}\right)}^2$, and $\epsilon ={10}^{-6}$ in all numerical tests. To sustain the nonoscillatory performance of the new CRUS-WENO scheme, we use the nonlinear weights in (12)–(14)

$$\begin{array}{c}{\omega }_1\left(\frac{1}{d_1}(\frac{3}{10}{\widehat{g}}_{r-\frac{1}{2}}+\frac{6}{10}{\widehat{g}}_{r+\frac{1}{2}}+\frac{1}{10}{\widehat{g}}_{r+\frac{3}{2}})-\frac{d_2}{d_1}(\frac{2}{3}{\widehat{g}}_{r-\frac{1}{2}}+\frac{1}{3}{\widehat{g}}_{r+\frac{1}{2}})-\frac{d_3}{d_1}(\frac{2}{3}{\widehat{g}}_{r+\frac{1}{2}}+\frac{1}{3}{\widehat{g}}_{r+\frac{3}{2}})\right)\\ +{\omega }_2(\frac{2}{3}{\widehat{g}}_{r-\frac{1}{2}}+\frac{1}{3}{\widehat{g}}_{r+\frac{1}{2}})+{\omega }_3(\frac{2}{3}{\widehat{g}}_{r+\frac{1}{2}}+\frac{1}{3}{\widehat{g}}_{r+\frac{3}{2}})\hfill \\ ={\omega }_1\left(\frac{1}{d_1}(\frac{1}{30}{g}_{r-1}+\frac{19}{30}{g}_r+\frac{10}{30}{g}_{r+1})-\frac{d_2}{d_1}(\frac{1}{6}{g}_{r-1}+\frac{5}{6}{g}_r)-\frac{d_3}{d_1}(\frac{1}{6}{g}_r+\frac{5}{6}{g}_{r+1})\right)\hfill \\ +{\omega }_2(\frac{1}{6}{g}_{r-1}+\frac{5}{6}{g}_r)+{\omega }_3(\frac{1}{6}{g}_r+\frac{5}{6}{g}_{r+1}).\hfill \end{array}$$

(19)

We then simplify (19) as

$$\begin{array}{cc}\hfill {\displaystyle }& \left({\omega }_1(\frac{3}{10d_1}-\frac{d_2}{3d_1})+\frac{2}{3}{\omega }_2\right){\widehat{g}}_{r-\frac{1}{2}}+\left({\omega }_1(\frac{6}{10d_1}-\frac{d_2}{3d_1}-\frac{2d_3}{3d_1})+\frac{1}{3}{\omega }_2+\frac{2}{3}{\omega }_3\right){\widehat{g}}_{r+\frac{1}{2}}+\hfill \\ \hfill {\displaystyle }& \left({\omega }_1(\frac{1}{10d_1}-\frac{d_3}{3d_1})+\frac{1}{3{\omega }_3}\right){\widehat{g}}_{r+\frac{3}{2}}=\left({\omega }_1(\frac{1}{30d_1}-\frac{d_2}{6d_1})+\frac{1}{6}{\omega }_2\right)g_{r-1}+\hfill \\ \hfill {\displaystyle }& \left({\omega }_1(\frac{19}{30d_1}-\frac{5d_2}{6d_1}-\frac{d_3}{6d_1})+\frac{5}{6}{\omega }_2+\frac{1}{6}{\omega }_3\right)g_r+\left({\omega }_1(\frac{10}{30d_1}-\frac{5d_3}{6d_1})+\frac{5}{6}{\omega }_3\right)g_{r+1}.\hfill \end{array}$$

(20)

(20) is the final high-order spatial reconstruction procedure. We can construct a tridiagonal system according to (20) and solve it directly by using the LU decomposition method. The construction procedures of the numerical fluxes ${\widehat{g}}_{r-\frac{1}{2}}$ and ${\widehat{g}}_{r+\frac{1}{2}}$ are similar, we will omit them here for simplicity. After performing the high-order spatial discretization of (1), we use the third-order TVD Runge–Kutta time discretization method [34] to solve the ODE. Finally, the fully discrete scheme both in space and time is designed on structured meshes.

3. Numerical Tests

In this section, the numerical results of some benchmark examples of the new CRUS-WENO scheme specified in the previous section, in comparison with the classical finite difference WENO scheme [2,6] called the WENO-JS scheme, are presented. The CFL number is 0.6 for all numerical simulations.

Example 1.

We first consider the following fractional differential equation

$$\left\{\begin{array}{c}\frac{\partial u(x,t)}{\partial t}=-{(-\Delta )}^{\frac{\beta }{2}}u(x,t)+\phi (x,t),x\in [0,1],\hfill \\ u(x,0)=u_0\left(x\right),\hfill \end{array}\right.$$

with the source term $\phi (x,t)=e^{-t}(-u_0\left(x\right)+{(-\Delta )}^{\frac{\beta }{2}}{u}_0\left(x\right))$. Furthermore, the initial condition is $u(x,0)=x^6{(1-x)}^6$. Its exact solution is $u(x,t)=e^{-t}{x}^6{(1-x)}^6$. The errors and orders of the accuracy computed by the new CRUS-WENO scheme and the WENO-JS scheme are shown in

Table 1. The errors and orders of the accuracy of Example 1 at $t=0.1$.

$\mathit{\beta }$ = 0.2
	CRUS-WENO (1)				WENO-JS
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
10	5.25 × 10${}^{-13}$		1.21 × 10${}^{-12}$		6.55 × 10${}^{-11}$		2.23 × 10${}^{-10}$
20	8.39 × 10${}^{-15}$	5.97	2.79 × 10${}^{-14}$	5.44	1.34 × 10${}^{-13}$	8.93	6.36 × 10${}^{-13}$	8.46
30	9.23 × 10${}^{-16}$	5.44	3.82 × 10${}^{-15}$	4.90	7.87 × 10${}^{-15}$	6.99	3.69 × 10${}^{-14}$	7.02
40	2.15 × 10${}^{-16}$	5.06	6.98 × 10${}^{-16}$	5.91	1.20 × 10${}^{-15}$	6.53	6.30 × 10${}^{-15}$	6.14
	CRUS-WENO (2)				CRUS-WENO (3)
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
10	5.25 × 10${}^{-13}$		1.21 × 10${}^{-12}$		5.25 × 10${}^{-13}$		1.21 × 10${}^{-12}$
20	8.39 × 10${}^{-15}$	5.97	2.79 × 10${}^{-14}$	5.44	8.39 × 10${}^{-15}$	5.97	2.79 × 10${}^{-14}$	5.44
30	9.22 × 10${}^{-16}$	5.45	3.80 × 10${}^{-15}$	4.92	9.22 × 10${}^{-16}$	5.45	3.80 × 10${}^{-15}$	4.92
40	2.16 × 10${}^{-16}$	5.05	6.72 × 10${}^{-16}$	6.02	2.16 × 10${}^{-16}$	5.05	6.73 × 10${}^{-15}$	6.02
$\mathbf{\beta }$= 0.4
	CRUS-WENO (1)				WENO-JS
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
10	8.74 × 10${}^{-13}$		1.94 × 10${}^{-12}$		2.70 × 10${}^{-10}$		1.01 × 10${}^{-9}$
20	1.46 × 10${}^{-14}$	5.90	4.80 × 10${}^{-14}$	5.34	3.55 × 10${}^{-13}$	9.57	1.38 × 10${}^{-12}$	9.51
30	1.55 × 10${}^{-15}$	5.55	7.58 × 10${}^{-15}$	4.55	2.15 × 10${}^{-14}$	6.91	8.82 × 10${}^{-14}$	6.78
40	2.70 × 10${}^{-16}$	6.07	1.66 × 10${}^{-15}$	5.27	4.43 × 10${}^{-15}$	5.50	5.15 × 10${}^{-14}$	1.87
	CRUS-WENO (2)				CRUS-WENO (3)
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
10	8.74 × 10${}^{-13}$		1.94 × 10${}^{-12}$		8.74 × 10${}^{-13}$		1.94 × 10${}^{-12}$
20	1.46 × 10${}^{-14}$	5.90	4.80 × 10${}^{-14}$	5.34	1.46 × 10${}^{-14}$	5.90	4.80 × 10${}^{-14}$	5.34
30	1.55 × 10${}^{-15}$	5.55	7.57 × 10${}^{-15}$	4.55	1.55 × 10${}^{-15}$	5.55	7.57 × 10${}^{-15}$	4.55
40	2.69 × 10${}^{-16}$	6.07	1.66 × 10${}^{-15}$	5.28	2.69 × 10${}^{-16}$	6.07	1.66 × 10${}^{-15}$	5.28
$\mathbf{\beta }$= 0.8
	CRUS-WENO (1)				WENO-JS
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
10	4.91 × 10${}^{-12}$		1.16 × 10${}^{-11}$		1.53 × 10${}^{-8}$		9.55 × 10${}^{-8}$
20	8.76 × 10${}^{-14}$	5.81	2.20 × 10${}^{-13}$	5.72	1.93 × 10${}^{-11}$	9.63	1.21 × 10${}^{-10}$	9.63
30	7.82 × 10${}^{-15}$	5.96	3.17 × 10${}^{-14}$	4.78	5.95 × 10${}^{-13}$	8.58	2.21 × 10${}^{-12}$	9.87
40	1.42 × 10${}^{-15}$	5.94	7.75 × 10${}^{-15}$	4.90	8.01 × 10${}^{-14}$	6.97	2.96 × 10${}^{-13}$	6.98
	CRUS-WENO (2)				CRUS-WENO (3)
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
10	4.91 × 10${}^{-12}$		1.16 × 10${}^{-11}$		4.91 × 10${}^{-12}$		1.16 × 10${}^{-11}$
20	8.76 × 10${}^{-14}$	5.81	2.20 × 10${}^{-13}$	5.72	8.76 × 10${}^{-14}$	5.81	2.20 × 10${}^{-13}$	5.72
30	7.82 × 10${}^{-15}$	5.96	3.17 × 10${}^{-14}$	4.78	7.82 × 10${}^{-15}$	5.96	3.17 × 10${}^{-14}$	4.78
40	1.42 × 10${}^{-15}$	5.94	7.75 × 10${}^{-15}$	4.90	1.42 × 10${}^{-15}$	5.94	7.75 × 10${}^{-15}$	4.90

. For several different β, we give the numerical results of three different linear weights, respectively.

Example 2.

We consider the following fractional Burgers’ equation

$$\left\{\begin{array}{c}\frac{\partial u(x,t)}{\partial t}+\frac{\partial }{\partial x}\left(\frac{u^2(x,t)}{2}\right)=\sigma (-{(-\Delta )}^{\frac{\beta }{2}})u(x,t)+\phi (x,t),x\in [-2,2],\hfill \\ u(x,0)=u_0\left(x\right),\hfill \end{array}\right.$$

with $\sigma =0.1$, and the source term is $\phi (x,t)=e^{-t}(-u_0\left(x\right)+e^{-t}{u}_0\left(x\right)u_0^{{}^{\prime }}\left(x\right)+\sigma {(-\Delta )}^{\frac{\beta }{2}}{u}_0\left(x\right))$. The initial condition is

$$u(x,0)=\left\{\begin{array}{c}\frac{{(1-x^2)}^4}{10},-1\le x\le 1,\hfill \\ 0,otherwise.\hfill \end{array}\right.$$

Its exact solution is

$$u(x,t)=\left\{\begin{array}{c}\frac{e^{-t}{(1-x^2)}^4}{10},-1\le x\le 1,\hfill \\ 0,otherwise.\hfill \end{array}\right.$$

The errors and orders of the accuracy computed by the new CRUS-WENO scheme and the WENO-JS scheme are shown in

Table 2. The errors and orders of the accuracy of Example 2 at $t=0.1$.

$\mathit{\beta }$ = 0.2
	CRUS-WENO (1)				WENO-JS
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
10	8.10 × 10${}^{-5}$		3.33 × 10${}^{-4}$		8.43 × 10${}^{-5}$		3.41 × 10${}^{-4}$
20	1.62 × 10${}^{-6}$	5.64	4.29 × 10${}^{-6}$	6.28	1.60 × 10${}^{-6}$	5.72	4.39 × 10${}^{-6}$	6.28
30	2.26 × 10${}^{-7}$	4.86	1.29 × 10${}^{-6}$	2.96	4.33 × 10${}^{-7}$	3.21	1.94 × 10${}^{-6}$	2.01
40	5.49 × 10${}^{-8}$	4.92	2.47 × 10${}^{-7}$	5.75	9.19 × 10${}^{-8}$	5.39	3.66 × 10${}^{-7}$	5.80
	CRUS-WENO (2)				CRUS-WENO (3)
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
10	7.42 × 10${}^{-5}$		3.20 × 10${}^{-4}$		7.32 × 10${}^{-5}$		3.19 × 10${}^{-4}$
20	2.08 × 10${}^{-6}$	5.16	7.05 × 10${}^{-6}$	5.50	2.20 × 10${}^{-6}$	5.06	8.09 × 10${}^{-6}$	5.30
30	2.30 × 10${}^{-7}$	5.43	1.29 × 10${}^{-6}$	4.18	2.32 × 10${}^{-7}$	5.55	1.30 × 10${}^{-6}$	4.52
40	5.49 × 10${}^{-8}$	4.98	2.47 × 10${}^{-7}$	5.76	5.49 × 10${}^{-8}$	5.00	2.47 × 10${}^{-7}$	5.77
$\mathbf{\beta }$= 0.6
	CRUS-WENO (1)				WENO-JS
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
10	8.11 × 10${}^{-5}$		3.33 × 10${}^{-4}$		8.31 × 10${}^{-5}$		3.41 × 10${}^{-4}$
20	1.43 × 10${}^{-6}$	5.83	5.21 × 10${}^{-6}$	6.00	1.53 × 10${}^{-6}$	5.76	4.60 × 10${}^{-6}$	6.21
30	2.17 × 10${}^{-7}$	4.65	1.34 × 10${}^{-6}$	3.35	4.94 × 10${}^{-7}$	2.79	2.01 × 10${}^{-6}$	2.05
40	4.95 × 10${}^{-8}$	5.13	1.88 × 10${}^{-7}$	6.84	9.79 × 10${}^{-8}$	5.62	3.82 × 10${}^{-7}$	5.76
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
10	7.97 × 10${}^{-5}$		3.35 × 10${}^{-4}$		8.07 × 10${}^{-5}$		3.33 × 10${}^{-4}$
20	2.44 × 10${}^{-6}$	5.03	8.70 × 10${}^{-6}$	5.27	2.74 × 10${}^{-6}$	4.88	1.02 × 10${}^{-5}$	5.03
30	2.27 × 10${}^{-7}$	5.86	1.20 × 10${}^{-6}$	4.89	2.30 × 10${}^{-7}$	6.11	1.20 × 10${}^{-6}$	5.26
40	4.95 × 10${}^{-8}$	5.30	1.88 × 10${}^{-7}$	6.44	4.95 × 10${}^{-8}$	5.34	1.88 × 10${}^{-7}$	6.46
$\mathbf{\beta }$= 0.8
	CRUS-WENO (1)				WENO-JS
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
10	7.47 × 10${}^{-5}$		2.97 × 10${}^{-4}$		7.60 × 10${}^{-5}$		3.25 × 10${}^{-4}$
20	3.88 × 10${}^{-6}$	4.27	1.85 × 10${}^{-5}$	4.01	1.56 × 10${}^{-6}$	5.60	4.85 × 10${}^{-6}$	6.07
30	2.54 × 10${}^{-7}$	6.73	1.15 × 10${}^{-6}$	6.85	5.41 × 10${}^{-7}$	2.61	2.09 × 10${}^{-6}$	2.07
40	3.49 × 10${}^{-8}$	6.89	1.92 × 10${}^{-7}$	6.22	9.58 × 10${}^{-8}$	6.02	3.96 × 10${}^{-7}$	5.79
	CRUS-WENO (2)				CRUS-WENO (3)
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
10	1.39 × 10${}^{-4}$		3.86 × 10${}^{-4}$		1.65 × 10${}^{-4}$		4.60 × 10${}^{-4}$
20	4.53 × 10${}^{-6}$	4.94	1.45 × 10${}^{-5}$	4.73	5.11 × 10${}^{-6}$	5.01	1.64 × 10${}^{-5}$	4.81
30	3.13 × 10${}^{-7}$	6.59	1.72 × 10${}^{-6}$	5.26	2.66 × 10${}^{-7}$	7.29	1.11 × 10${}^{-6}$	6.65
40	3.50 × 10${}^{-8}$	7.62	1.92 × 10${}^{-7}$	7.61	3.50 × 10${}^{-8}$	7.05	1.93 × 10${}^{-7}$	6.08

. For several different β, we give the numerical results of three different linear weights, respectively.

Example 3.

We first consider the following two-dimensional fractional differential equation

$$\left\{\begin{array}{c}\frac{\partial u(x,y,t)}{\partial t}=-{(-\Delta )}^{\frac{\beta }{2}}u(x,y,t)+\phi (x,y,t),x,y\in [0,1]\times [0,1],\hfill \\ u(x,y,0)=u_0(x,y),\hfill \end{array}\right.$$

with the source term $\phi (x,y,t)=e^{-t}(-u_0(x,y)+{(-\Delta )}^{\frac{\beta }{2}}{u}_0(x,y))$. Furthermore, the initial condition is $u(x,y,0)=x^6{y}^6{(1-x)}^6$. Its exact solution is $u(x,y,t)=e^{-t}{x}^6{y}^6{(1-x)}^6$. The errors and orders of the accuracy computed by the new CRUS-WENO scheme and the WENO-JS scheme are shown in

Table 3. The errors and orders of the accuracy of Example 3 at $t=0.1$.

$\mathit{\beta }$ = 0.4
	CRUS-WENO (1)				WENO-JS
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
10 × 10	1.57 × 10${}^{-13}$		1.94 × 10${}^{-12}$		2.83 × 10${}^{-11}$		1.01 × 10${}^{-9}$
20 × 20	2.36 × 10${}^{-15}$	6.06	4.80 × 10${}^{-14}$	5.34	2.76 × 10${}^{-14}$	10.00	1.38 × 10${}^{-12}$	9.51
30 × 30	2.39 × 10${}^{-16}$	5.64	7.58 × 10${}^{-15}$	4.55	1.51 × 10${}^{-15}$	7.16	8.82 × 10${}^{-14}$	6.78
40 × 40	4.09 × 10${}^{-17}$	6.14	1.66 × 10${}^{-15}$	5.27	3.14 × 10${}^{-16}$	5.47	5.32 × 10${}^{-14}$	1.76
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
10 × 10	1.57 × 10${}^{-13}$		1.94 × 10${}^{-12}$		1.57 × 10${}^{-13}$		1.94 × 10${}^{-12}$
20 × 20	2.36 × 10${}^{-15}$	6.06	4.82 × 10${}^{-14}$	5.33	2.36 × 10${}^{-15}$	6.06	4.84 × 10${}^{-14}$	5.33
30 × 30	2.40 × 10${}^{-16}$	5.64	7.73 × 10${}^{-15}$	4.52	2.40 × 10${}^{-16}$	5.64	7.81 × 10${}^{-15}$	4.50
40 × 40	4.12 × 10${}^{-17}$	6.12	1.86 × 10${}^{-15}$	4.96	4.13 × 10${}^{-17}$	6.11	2.03 × 10${}^{-15}$	4.69
$\mathbf{\beta }$= 0.8
	CRUS-WENO (1)				WENO-JS
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
10 × 10	8.83 × 10${}^{-13}$		1.16 × 10${}^{-11}$		1.56 × 10${}^{-9}$		9.55 × 10${}^{-8}$
20 × 20	1.41 × 10${}^{-14}$	5.97	2.20 × 10${}^{-13}$	5.72	1.50 × 10${}^{-12}$	10.02	1.21 × 10${}^{-10}$	9.63
30 × 30	1.21 × 10${}^{-15}$	6.05	3.17 × 10${}^{-14}$	4.78	4.11 × 10${}^{-14}$	8.87	2.21 × 10${}^{-12}$	9.87
40 × 40	2.15 × 10${}^{-16}$	6.01	7.75 × 10${}^{-15}$	4.90	5.20 × 10${}^{-15}$	7.18	2.96 × 10${}^{-13}$	6.98
	CRUS-WENO (2)				CRUS-WENO (3)
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
10 × 10	8.83 × 10${}^{-13}$		1.16 × 10${}^{-11}$		8.83 × 10${}^{-13}$		1.16 × 10${}^{-11}$
20 × 20	1.41 × 10${}^{-14}$	5.97	2.20 × 10${}^{-13}$	5.72	1.41 × 10${}^{-14}$	5.97	2.20 × 10${}^{-13}$	5.72
30 × 30	1.21 × 10${}^{-15}$	6.05	3.17 × 10${}^{-14}$	4.78	1.21 × 10${}^{-15}$	6.05	3.17 × 10${}^{-14}$	4.78
40 × 40	2.15 × 10${}^{-16}$	6.01	7.75 × 10${}^{-15}$	4.90	2.15 × 10${}^{-16}$	6.01	7.75 × 10${}^{-15}$	4.90

. For several different β, we give the numerical results of three different linear weights, respectively.

Example 4.

We consider the following fractional Burgers’ equation with a discontinuous initial condition

$$\left\{\begin{array}{c}\frac{\partial u(x,t)}{\partial t}+\frac{\partial }{\partial x}\left(\frac{u^2(x,t)}{2}\right)=\sigma (-{(-\Delta )}^{\frac{\beta }{2}})u(x,t),x\in [-2,2],\hfill \\ u(x,0)=u_0\left(x\right),\hfill \end{array}\right.$$

and $\sigma =0.04$. The initial condition is

$$u(x,0)=\left\{\begin{array}{c}1,-1\le x\le 0,\hfill \\ 0,otherwise.\hfill \end{array}\right.$$

Figure 1 shows the numerical solution of the fractional Burgers’ equation for different fractional derivatives $\beta =0.3$, $0.5$, and $0.7$ at $t=0.5$, respectively. Figure 2 shows the numerical solution of the fractional Burgers’ equation for fractional derivatives $\beta =0.5$ at $t=0.01$, $0.1$, and 1, respectively. At the same time, the numerical solutions of the CRUS-WENO scheme with the three different linear weights and the WENO-JS scheme are plotted in Figure 1 and Figure 2.

It can be seen from Table 1, Table 2, Table 3, that both the CRUS-WENO and WENO-JS schemes can reach their designed order of accuracy. However, the new CRUS-WENO scheme with different types of linear weights has smaller truncation errors than the WENO-JS scheme. As shown in Figure 1 and Figure 2, we can observe that neither the new CRUS-WENO scheme nor the WENO-JS scheme generate oscillations near the strong discontinuities. However, compared with the WENO-JS scheme, the new CRUS-WENO scheme does not require additional processing of linear weights, which reduces the amount of computation. Furthermore, this new CRUS-WENO scheme also uses fewer stencils. At the same time, the following conclusions can be drawn from the numerical results. Under the constraints of the new CRUS-WENO scheme, the selection of the linear weights has little effect on the numerical results. This further reflects the superiority of this new CRUS-WENO scheme.

4. Conclusions

In this article, we first designed a new finite difference fifth-order CRUS-WENO scheme for solving the fractional differential equations with the fractional Laplacian operators. In this method, we divided the order of the fractional Laplacian operators into an integral part and a first derivative term. After using the Gauss–Jacobi quadrature method to solve the integral part of the fractional Laplacian operators, a new CRUS-WENO scheme was presented to discretize the first derivative term of the fractional equation. Different from the classic WENO scheme [1,2], this new finite difference CRUS-WENO scheme saves the calculation procedure in the space reconstruction procedures. Furthermore, this new CRUS-WENO scheme can achieve fifth-order accuracy on smooth regions and suppress oscillations near discontinuities. This new scheme also has the advantages of a narrower large spatial stencil and high spectral resolution. It can be seen from the numerical results of some benchmark examples that this new CRUS-WENO scheme was successfully applied in solving the fractional differential equations with the fractional Laplacian operators.