The linear advection equation is of the form

$ \begin{equation} u_t+a u_x=0, \end{equation} $

(1.1)

where $a$ is a constant. For any function $U(u)$ differentiable with respect to $u$, it is not difficult to obtain

$ \begin{equation} U(u)_t+a U(u)_x=0, \end{equation} $

(1.2)

which show that (1.1) possesses infinitely many conservation laws, i.e., any smooth function of $u$, including $u$ itself, is conservative.

In recent years, Mao and his co-workers developed a class of finite-volume schemes for linear advection equation, see [1-8]. As a scheme of the Godunov type, the schemes proceeded in the following three steps: reconstruction, evolution, cell-averaging. The entropy scheme was presented by Li and Mao in [3]. In [6], Cui and Mao revealed that the entropy scheme had an error self-canceling mechanism and showed that the scheme was super-convergent. However, when computing discontinuous solutions, the scheme produced spurious oscillations near discontinuities. To fix this problem, the authors of [3] used the TVD limiter of the Ultra-bee scheme to control the reconstruction step in each cell. See [9, 10] or [1] for the Ultra-bee scheme. The resulted scheme was thus a combination of the entropy scheme and the Ultra-bee scheme. In [3], Li and Mao chose $U(u)=u^2$ as entropy function. As is known to all, physics Entropy is $s=\log\frac{p}{\rho^\gamma}$. As a preparation for the extension of the scheme to Euler system, it needs to choose a function related to the function log.

In this paper, we follow the idea of [3] and develop a Entropy-Ultra-bee scheme with a entropy function $U(u)=u \log u$ for linear advection equation. We presents a number of numerical examples of linear advection equation, which show that the scheme has a very good quality in long-time numerical computation and has better resolution of continuities. The organization of the paper is as follows: Section 1 is the introduction. In Section 2, we describe the scheme. In Section 3, we present three numerical examples. Section 4 is the conclusion.

We will describe Entropy-Ultra-bee scheme with the new entropy function following [1]. We denote the cell size by $h$ and use a Cartesian grid with grid cells $(x_{j-1/2}, x_{j+1/2})$'s centered at $\{x_j\}$'s, where $x_{j \pm 1/2}=(j\pm 1/2)h$ and $x_j=j h$, $j=0, \pm 1, \pm 2, \cdots$. We use $\tau$ to denote the time increment. The Entropy-Ultra-bee scheme is a Godunov type scheme. However, different from the other Godunov type schemes, it involves two numerical entities, the numerical solution $u^n$, which is a cell-average approximation to the true solution,

$ \begin{equation} u^n_j \simeq {1 \over h} \int^{x_{j+1/2}}_{x_{j-1/2}}u(x, t_n)dx, \end{equation} $

(2.1)

and the numerical entropy $U^n$, which is a cell-average approximation to an entropy of the true solution,

$ \begin{equation} U^n_j \simeq {1 \over h} \int^{x_{j+1/2}}_{x_{j-1/2}}U(u(x, t_n))dx, \end{equation} $

(2.2)

where $U(u)$ is an entropy function of the solution.

For simplicity, we describe the scheme for (1.1) and (1.2) with $a=1$. As a scheme of the Godunov type, Entropy-Ultra-bee scheme with the new entropy function proceeds in the following three steps.

ⅰ) Step-Reconstruction  The solution is reconstructed in each cell as a step function involving two constant states,

$ \begin{equation} R(x;u^n, U^n)=u_j^n+\left\{ \begin{array}{ll} -d_j^n, ~& ~ {x_{j-1/2}< x\leq x_j}, \\ +d_j^n, ~& ~ {x_j< x\leq x_{j+1/2}}, \end{array} \right. \end{equation} $

(2.3)

where $d_j^n$ is called the half step (HS) of the reconstruction. It is obvious that

$ \begin{equation} \frac{1}{h}\int_{x_{j-1/2}}^{x_{j+1/2}}R(x;u^n, U^n)dx=u_j^n. \end{equation} $

(2.4)

To compute the HS $d^n_j$, we need first to compute an entropy HS $d^{n, e}_j$ by requiring

$ \begin{equation} \frac{1}{h}\int_{x_{j-1/2}}^{x_{j+1/2}}U(R(x;u^n, U^n))dx=U_j^n, \end{equation} $

(2.5)

i.e., the entropy cell-average of the reconstructed solution is equal to the numerical entropy in the cell. Equation (2.5) turns out to be an equation for $d^{n, e}_j$,

$ \begin{equation} \frac{1}{2}(u^n_j-d^{n, e}_j)\log(u^n_j-d^{n, e}_j)+\frac{1}{2}(u^n_j+d^{n, e}_j)\log(u^n_j+d^{n, e}_j)=U^n_j. \end{equation} $

(2.6)

The solvability of (2.6) was discussed in ${\rm{\S}}$2.2 in [1], which states that the equation will have two opposite roots with the same absolute value.

The Ultra-Bee HS is computed as

$ \begin{equation} d_j^{n, ub}=\left\{ \begin{array}{ll} 0, ~& ~ (u_j^n-u_{j-1}^n)(u_{j+1}^n-u_j^n)<0, \\ u_{j+1}^n-u_j^n, ~& ~ |u_j^n-u_{j-1}^n|\geq\frac{\nu}{1-\nu}|u_{j+1}^n-u_j^n|, \\ \frac{1-\nu}{\nu}(u_j^n-u_{j-1}^n), ~&~|u_j^n-u_{j-1}^n|<\frac{\nu}{1-\nu}|u_{j+1}^n-u_j^n|. \end{array} \right. \end{equation} $

(2.7)

The HS is then taken as the one smaller in absolute value of $d_j^{n, e}$ and $d_j^{n, ub}$ with the sign of $(u_{j+1}^n-u_{j-1}^n)$,

$ \begin{equation} d_j^n=\text{sgn}(u_{j+1}^n-u_{j-1}^n)\text{min} \{|d_j^{n, e}|, |d_j^{n, ub}|\}. \end{equation} $

(2.8)

ⅱ) Evolution  Solve the following initial value problem (IVP)

$ \begin{equation} \left\{ \begin{array}{ll} v_t+v_x=0, ~& ~ {t_n< t\leq t_{n+1}}, \\ v(x, t_n)=R(x;u^n, U^n). \end{array} \right. \end{equation} $

(2.9)

The problem can be exactly solved as

$ \begin{equation} v(x, t)=R(x-(t-t_n);u^n, U^n). \end{equation} $

(2.10)

ⅲ) Cell-averaging  Compute the cell-averages of the numerical solution at $t=t_{n+1}$ as

$ \begin{equation} u_j^{n+1}=\frac{1}{h}\int_{x_{j-1/2}}^{x_{j+1/2}}v(x, t_{n+1})dx \end{equation} $

(2.11)

and

$ \begin{equation} U_j^{n+1}=\frac{1}{h}\int_{x_{j-1/2}}^{x_{j+1/2}}U(v(x, t_{n+1}))dx. \end{equation} $

(2.12)

We can compute $u_j^{n+1}$ in the following flux form,

$ \begin{equation} u^{n+1}_j=u^n_j-\lambda ({\hat f}^n_{j+1/2}-{\hat f}^n_{j-1/2}), \end{equation} $

(2.13)

where

$ \begin{equation} {\hat f}^n_{j+1/2}=f(u^*(u_j^n+d_j^n, u_{j+1}^n-d_{j+1}^n)). \end{equation} $

(2.14)

We can also compute $U^{n+1}_j$ in the following flux form,

$ \begin{equation} U_{j}^{n+1}= U_{j}^n-\lambda(\hat{F}_{j+1/2}^{n}-\hat{F}_{j-1/2}^{n}), \end{equation} $

(2.15)

where the numerical entropy flux ${\hat F}^n_{j+1/2}$ is computed as

$ \begin{equation} {\hat F}^n_{j+1/2}=U(u^*(u_j^n+d_j^n, u_{j+1}^n-d_{j+1}^n)). \end{equation} $

(2.16)

Thus we complete the description of the scheme.

In this section, we will present three numerical examples to show the effectiveness of Entropy-Ultra-bee scheme with an entropy function $U(u)=u \log u$ for linear advection equation (1.1) taking $a=1$. The computational domain is [0, 1]. To show the accuracy of the scheme, numerical solutions are compared against the true solutions. The "new" stands for the numerical solution computed by the Entropy-Ultra-bee scheme with the entropy function $U(u)=u\log u$. We take $\lambda=0.45$.

Example 3.1  The initial value is

$ \begin{equation} u(x, 0)=\left\{{\begin{array}{ll} 1.0+\displaystyle\mbox{exp}\{-\frac{1}{1-16(x-\frac{1}{2})^{2}}\}, & \mbox{if}~~ x\in(\frac{1}{4}, \frac{3}{4}), \\ 1.0,&\mbox{else} \end{array}} \right. \end{equation} $

(3.1)

with periodic boundary conditions at the two ends. This example is come from [11]. We conduct the simulation on a grid of 200 cells, up to $t=1$, $t=10$ and the numerical results are displayed in Fig. 3.1. It is clearly see that the results at $t=1$ are in good agreement with the analytical solution. However, we notice a little defect near the corner at $t=10$.

Example 3.2  The initial conditions are

$ \begin{equation} u_0(x)=2.0+\mbox{exp}\{-100\times (x-\frac{1}{2})^2\}\mbox{sin}80x, ~~~~~~~~~~~~\mbox{if}~~0\leq x< 1 \end{equation} $

(3.2)

with periodic boundary conditions at the two ends. This example is the Wave-packet problem, see [11]. This is a very challenge problem. We conduct the simulation on a grid of 200 cells, up to $t=1$, $t=10$ and the numerical results are displayed in Fig. 3.2. We can see that the scheme give good results for this problem. As to our knowledge, no scheme up to date has ever got qualified numerical result for this example with the same grid beyond the time $t= 10$.

Example 3.3  The initial value is

$ \begin{equation} u(x, 0)=1.0+\mbox{exp}\{-200(x-0.3)^2\}+\left \{ \begin{array}{ll} 1,&\mbox{if}~~0.6<x<0.8, \\ 0,&\mbox{else}. \end{array} \right. \end{equation} $

(3.3)

This example is come from [11]. We conduct the simulation on a grid of 200 cells, up to $t=1$, $t=10$ and the numerical results are displayed in Fig. 3.3. We see that the numerical solution agrees quite well with the exact one in the smooth region, all the spurious oscillations near discontinuities are eliminated and jumps are kept quite sharp with at most one transition point for each.

In this paper, we present Entropy-Ultra-bee scheme with an entropy function $U(u) =u log u$ for linear advection equation. Numerical experiments show that the scheme has a very good quality in long-time numerical computation and has better resolution of continuities.