**3. Numerical Examples**

In this section, we illustrate the effectiveness of the presented scheme which is abbreviated by ES4 by means of three typical examples. Numerical results include the convergence order and the capacity of dealing with discontinuous problems.

**Example 1.** *Consider the linear advection equation ut* + *ux* = 0 *on the domain* [−1, 1] *with two initial data u*(*x*, 0) = *sin*(*πx*) *and u*(*x*, 0) = *sin*4(*πx*)*. For a comparison, a third order entropy stable scheme (ES3) [12] is also implemented for this example. The numerical errors ( L*<sup>1</sup> *error and L*<sup>∞</sup> *error) and convergence orders are displayed in Tables 1 and 2. We can observe that the fourth order convergence of the proposed scheme is confirmed and ES4 performs better than ES3.*


**Table 1.** The numerical errors and convergence orders for *ut* + *ux* = 0 with *u*(*x*, 0) = *sin*(*πx*) at *t* = 8.

**Table 2.** The numerical errors and convergence orders for *ut* + *ux* = 0 with *u*(*x*, 0) = *sin*4(*πx*) at *t* = 1.


**Example 2.** *Consider the Burgers equation*

$$
u\_t + (\mu^2/2)\_x = 0\tag{13}$$

*subjected to the initial data*

$$\mu(\mathbf{x},0) = \begin{cases} 1, & \text{for } |\mathbf{x}| \le 1/3, \\\ -1, & \text{for } 1/3 < |\mathbf{x}| \le 1. \end{cases}$$

*For this problem, we can deduce the analytical solution that evolves a rarefaction fan and a stationary shock on the left-hand and right-hand side, respectively. Figure 1 presents the numerical result at time t* = 0.3 *on a mesh of* 100 *grids. Our scheme resolves the shock wave and the rarefaction wave very well.*

**Figure 1.** The numerical result for Burgers equation.

*Entropy* **2019**, *21*, 508

**Example 3.** *Consider the Euler equations from aerodynamics*

$$
\frac{
\partial
}{
\partial t
}
\begin{pmatrix}
\rho \\
\rho\mu \\
E
\end{pmatrix} + \frac{
\partial
}{
\partial x
}
\begin{pmatrix}
\rho\mu \\
\rho\mu^2 + p \\
\mu(E+p)
\end{pmatrix} = 0
\tag{14}
$$

*with ρ*, *μ*, *p and E being the density, velocity, pressure and total energy, respectively. For an idea gas, the total energy E is given by the relation*

$$E = \frac{p}{\gamma - 1} + \frac{1}{2}\rho\mu^2\tag{15}$$

*with the specific heats ratio γ* = 1.4*. Three Riemann problems are tested by the presented scheme.*

Case 1: Sod's shock tube problem. The initial data is given as

$$(\rho\_\prime \mu\_\prime p) = \begin{cases} (1,0,1), & \text{for } x < 0, \\\ (0.125,0,0.1), & \text{for } x > 0. \end{cases}$$

The numerical simulation is carried out on a mesh of 200 grids on [−0.5, 0.5] up to time *t* = 0.16. The computed density is plotted in Figure 2. We can see that the ES4 scheme performs well by capturing the shock, the contact discontinuity and the rarefaction wave accurately.

**Figure 2.** The density for Sod's shock tube problem.

Case 2: Toro's 123 problem. The initial data is given as

$$(\rho\_\prime \mu\_\prime p) = \begin{cases} \ (1, -2, 0.4)\_\prime & \text{for } x < 0, \\\ (1, 2, 0.4)\_\prime & \text{for } x > 0. \end{cases}$$

The difficulty for simulating this problem lies in the fact that the pressure between the evolved rarefactions is very small (near vacuum) and may bring about the blow-ups of the code if the numerical method is not robust. The numerical simulation is carried out on a mesh of 200 grids on [−0.5, 0.5] up to time *t* = 0.1. Figure 3 displays the computed density. It can be observed that the computed result by ES4 compares well with the reference solution.

**Figure 3.** The density for Toro's 123 problem.

Case 3: Shu–Osher problem. The initial data is given as

$$(\rho, \mu, p) = \begin{cases} (3.857, 2.629, 10.333), & \text{for } x < -4, \\ (1 + 0.2\sin(5x), 0, 1), & \text{for } x > -4. \end{cases}$$

This problem, also called the shock density-wave interaction problem, describes a moving Mach 3 shock interacting with sine waves in density. The numerical simulation is carried out on a mesh of 500 grids on [−5, 5] up to time *t* = 1.8. We present the results of density in Figure 4. It can be seen clearly that the ES4 scheme produces accurate results and captures the sine wave well.

**Figure 4.** The density for the Shu–Osher problem.

#### **4. Conclusions**

This paper presents a fourth order entropy stable scheme for solving one-dimensional hyperbolic conservation laws. Along the lines of [10,12], our scheme is also obtained by utilizing entropy conservative flux in conjunction with suitable numerical diffusion. We first select the existing fourth order entropy conservative scheme based on the combination of the two-point entropy conservative flux. The main novelty lies in the construction of numerical diffusion by presenting a fourth order non-oscillatory reconstruction possessing the sign property. Compared to other high order schemes, the main advantage of our scheme is the entropy stability. Some numerical results are displayed to show the accuracy and shock capturing capacity of our scheme. Ongoing work involves generalizing the idea of this paper to multidimensional cases and other hyperbolic systems such as shallow water equations and magnetohydrodynamic equations.

**Funding:** This research is supported by the National Natural Science Foundation of China (Grant No. 11601037) and the Natural Science Foundation of Shaanxi Province (Grant No. 2018JQ1027).

**Conflicts of Interest:** The author declares no conflict of interest.
