Exercises

1. Conservative vs. non-conservative form:

Let’s consider a finite difference approximation to Burgers’ equation. For simplicity, let’s assume that \(u > 0\) always. We can discretize it to first-order accuracy using upwinding in conservation form:

\[\frac{u_i^{n+1} - u_i^n}{\Delta t} = \frac{1}{\Delta x} \left (\frac{1}{2} (u_i^n )^2 - \frac{1}{2} (u_{i-1}^n)^2 \right )\]

or in the original differential form as:

\[\frac{u_i^{n+1} - u_i^n}{\Delta t} = u_i^n \frac{u_i^n - u_{i-1}^n}{\Delta x}\]

Now consider the follow set of initial conditions:

\[\begin{split}u(x, t=0) = \left \{ \begin{array}{c} 2 & \mbox{if}~ x < 1/2 \\ 1 & \mbox{if}~ x \ge 1/2 \end{array} \right .\end{split}\]

This will drive a rightward moving shock with a speed \(S = 3/2\).

Code up both of the above discretizations and measure the speed of the shock—which method gets it correct?