The linear 1-d advection equation

The linear 1-d advection equation is a simple model for the advection process. It models the propagation of a function \(u\) in one dimensionsal space \(x\) with speed \(c\).

()\[ \frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0 \]

Function \(u\) is a generic function that may be viewed as a geophysical quantity of interest such as temperature. The propagation speed \(c\) can be viewed as the speed with which a particular feature of the quantity of intereste moves across \(x\), such as the speed at which a temperature front moves in space or, if temperature is represented by a wavelike function, \(c\) may be regarded as the phase speed of the wavelike function.

The general solution of () is the initial condition \(u(x,t=0)=u0\) translating in space: \(u(x,t)=u0(x-ct)\).