Advection Test Problem

Consider the advection equation:

\[a_t + u a_x = 0\]

This requires initial conditions, \(a(x, t=0)\), and a boundary condition. The solution to this is easy to write down:

\[a(x, t) = a(x - ut)\]

any initial profile \(a(\xi)\) is simply advected to the right (for \(u > 0\)) at a velocity \(u\).

Tip

To test our solver, we would like a problem with a known solution at any future point in time. For advection, this is easy, since the advection equation preserves any initial function and just moves it to the right (for \(u > 0\)) at a velocity \(u\).

Therefore, we can use periodic boundary conditions on a domain \([0, 1]\) and advect for a time \(1/u\), one period, and we should get back exactly what we started with.

Tophat Initial Conditions

Our first set of initial conditions is a tophat:

\[\begin{split} a(x, t=0) = \left \{ \begin{array}{ll} 0 & \mbox{if } x < \frac{1}{3}\\ 1 & \mbox{if } \frac{1}{3} \le x < \frac{2}{3} \\ 0 & \mbox{if } x \ge \frac{2}{3} \end{array} \right . \end{split}\]

This is discontinuous, so the derivative \(\partial a/\partial x\) is not defined a some regions.

Smooth Initial Conditions

Our second set of initial conditions is a Gaussian

\[a(x, t=0) = e^{-40 (x - 1/2)^2}\]