1D Linear Advection

Time =
$$c$$ =


This tutorial aims to give you a general idea of strategies to solve the advection or convection equation. In 1D this is:

\frac{\partial \phi}{\partial t} + c \frac{\partial \phi}{\partial x} = 0

where \phi = \phi(x,t):[a,b]\times[0,T]\mapsto\mathbb{R} might be the concentration of a chemical or temperature being advected or convected by fluid with constant speed c. This is an example of a 1st order hyperbolic equation.

The initial condition $$\phi(x,0) = f(x)$$ leads to the solution $$\phi(x,t) = f(x - ct)$$ which is a wave whose initial shape remains unchanged as it moves in the $$x > 0$$ direction with speed $$c$$.

This example shows a wave given by:

\phi(x,t) = 5\sin(x-ct)

for different values of the wave speed $$c$$.