# Advection

The archetypical linear model of advection in one dimension is the PDE

$$
\partial_t u + c \partial_x u = 0,
$$

where the constant $c$ is the velocity of travel for any initial condition. This is a **hyperbolic PDE**. It requires one initial condition and one boundary condition; the BC must represent an inflow condition, as we will see.

Another important linear hyperbolic PDE is the **wave equation**, which in 1D is 

$$
\partial_{tt} u = c^2 \partial_{xx}. 
$$

The wave equation allows solutions of velocities $c$ and $-c$ simultaneously. It requires two boundary conditions and initial conditions on both $u$ and $\partial_t u$, unless it is reformulated as **Maxwell's equations**, which in 1D are 

$$
\partial_t u &= c \partial_x w, \\ 
\partial_t w &= c \partial_x u. 
$$

Here $u$ and $w$ are essentially the electric and magnetic fields, and boundary conditions are often imposed on just one of them.