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Consider a conservative scheme $$ u_{j}^{n+1}=u_{j}^{n}-\frac{\Delta t}{\Delta x}\left[g\left(u_{j+\frac{1}{2}}^{L}, u_{j+\frac{1}{2}}^{R}\right)-g\left(u_{j-\frac{1}{2}}^{L}, u_{j-\frac{1}{2}}^{R}\right)\right] $$ for solving the initial value problem $$ u_{t}+f(u)_{x}=0 $$ where $g(,)$ is a numerical flux function. How can we show that the total $u(x, t)$ of the numerical solution is conserved over time.

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    $\begingroup$ You want to show that $\sum_j u_j^{n+1} = \sum_j u_j^{n}$ so sum the equation over $j=1,2,\ldots,n$ and see what you get. Most terms will cancel (telescoping). For the remaining terms: boundary conditions. $\endgroup$ – Winther May 6 at 16:54

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