5. 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

\[\begin{split} \partial_t u &= c \partial_x w, \\ \partial_t w &= c \partial_x u. \end{split}\]

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