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$$Q = \{(t,x);\ t_1\leq t\leq t_2, \ a\leq x \leq b\}$$

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$$g(u)_x = \frac{\partial g(u)}{\partial x} = \frac{d g(u)}{d u}\frac{\partial u}{\partial x} = g'(u)u_x$$

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$$uu_x = \left(\frac{u^2}{2}\right)_x$$

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$$\int\hspace{-6pt}\int _Q g(t,x)\ dtdx = \int_{t_1}^{t_2}\le... ...\} dt = \int_a^b \left\{\int_{t_1}^{t_2} g(t,x)\ dt\right\} dx$$

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$$\begin{eqnarray*} \lefteqn{\int\hspace{-6pt}\int _Q u_t(t,x) dtdx} \\ & = & \... ...frac{u(t,b)^2}{2} dx - \int_{t_1}^{t_2} \frac{u(t,a)^2}{2} dx \end{eqnarray*}$$

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$$\int_a^b u(t_2,x) dx - \int_a^b u(t_1,x) dx + \int_{t_1}^{... ...c{u(t,b)^2}{2} dx - \int_{t_1}^{t_2} \frac{u(t,a)^2}{2} dx =0$$ (22)

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$$Q=\{(t,x);\ t_0\leq t\leq t_0+\Delta t, \ x_0-\Delta x\leq x \leq x_0+\Delta x\}$$

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$$u_{\ell} = u(t_0,x_0-0) = \lim_{x\rightarrow x_0-0} u(t_0,x)$$

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$$u_r = u(t_0,x_0+0) = \lim_{x\rightarrow x_0+0} u(t_0,x)$$

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$$\begin{eqnarray*} \lefteqn{\int_a^b u(t_2,x) dx - \int_a^b u(t_1,x) dx}\\ & =... ...\ell}^2}{2} dt \\ & = & \frac{1}{2}(u_r^2-u_{\ell}^2)\Delta t \end{eqnarray*}$$

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$$-u_r\Delta \phi + u_{\ell}\Delta \phi +\frac{1}{2}(u_r^2-u_{\ell}^2)\Delta t \approx 0$$

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$$(u_r- u_{\ell})\frac{\Delta \phi}{\Delta t} \approx\frac{1}{2}(u_r^2-u_{\ell}^2) = \frac{1}{2}(u_r-u_{\ell})(u_r+u_{\ell})$$

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$$\frac{\Delta \phi}{\Delta t} \approx\frac{u_r+u_{\ell}}{2}$$

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$$\phi'(t_0)=\frac{u_r+u_{\ell}}{2}$$ (23)

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$$u_{\ell} < \phi'(t_0) < u_r$$ (24)

$$u_{\ell} > \phi'(t_0) > u_r$$ (25)

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