2.2.8 Linear advection equation: discretization

The prototype of a hyperbolic PDE, the linear advection equation (63), can be discretized for discrete time-steps on a spatial grid

$$t^n = \Delta t n + t_0$$ (88)

$$x_i = \Delta x i + x_0$$ (89)

by replacing

$$\frac{\partial \rho}{\partial t} \rightarrow \frac{\rho^{n+1}_i-\rho^n_i}{\Delta t}$$ (90)

$$\frac{\partial \rho}{\partial x} \rightarrow \frac{\rho^n_{i+1}-\rho^n_{i-1}}{2 \Delta x} \enspace .$$ (91)

The result is the explicit Euler scheme (``naive'' scheme, FTCS scheme)

$$\rho^{n+1}_i = \rho^n_i - \frac{\Delta t}{2\Delta x} v_{} \left( \rho^n_{i+1}-\rho^n_{i-1} \right) \enspace .$$ (92)

This looks quite similar to the discretization of the heat equation (85).