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$$u_t+\left(\frac{u^2}{2}\right)_x=0 \hspace*{1em}(0<x<1, \hspace*{1em}t>0),$$ (6)

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$$\left\{\begin{array}{ll} u(t,0)=u(t,1) & (t>0),\\ u(0,x)=u_0(x) & (0<x<1) \end{array}\right.$$ (7)

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$$\begin{eqnarray*} u^{n+1/2}_{j} & = & \frac{u^n_{j+1/2}+u^n_{j-1/2}}{2} - \fr... ...1,2,\ldots, j=0,1,2,\ldots, L-1, u^n_j=u(n\Delta t,j\Delta x)) \end{eqnarray*}$$

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$$\frac{\Delta x^2}{2\Delta t}u_{xx}$$

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$$\{\vert f'(u)\vert+T\vert g\vert\}\frac{\Delta t}{\Delta x}\leq 1$$

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