[[Differential form of the fluid equations]] # 1: 1D Linear Convection $ \frac{\partial u}{\partial t}+c\frac{\partial u}{\partial x}=0 $ use the following:f - space-time discretization - $i$: index of grid in $x$ - $n$: index of grid in $t$ - numerical scheme - forward difference in $t$ - backward difference in $x$ Write the partial derivatives out using discretized form: $ \begin{align} \frac{\partial u}{\partial t}&=\frac{u^{n+1}_i-u^n_i}{\Delta t} \\ \frac{\partial u}{\partial x}&=\frac{u^{n}_i-u^n_{i-1}}{\Delta x} \end{align} $ Plug into the original equation: $ \frac{u^{n+1}_i-u^n_i}{\Delta t}+c\frac{u^{n}_i-u^n_{i-1}}{\Delta x}=0 $ Rearrange for $u^{n+1}_i$ (velocity at the current index in $x$, at the next time step): $ u^{n+1}_i=u^n_i-c\frac{\Delta t}{\Delta x}u^n_i-u^n_{i-1} $ Add initial conditions: $ u=\begin{cases} 2&0.5\le x \le 1 \\ 1&\text{everywhere else} \end{cases} $ Boundary conditions: $ u=1 \quad @\quad x=0,2 $