Homework 7

Note

You are free to discuss these questions with your classmates and on our class slack, but you must write your own solutions, including your own source code.

All code should be uploaded to Brightspace along with any analytic derivations, notes, etc.

1. Let’s consider two different first-order finite-difference methods for Burgers’ equation:

   \[u_t + u u_x = 0\]

   As we saw in class, this is a nonlinear equation which can be written in conservative form as:

   \[u_t + \left [ \frac{1}{2} u^2 \right ]_x = 0\]

   Although these forms are analytically equivalent, the numerical solutions will differ.

   We will look at two different first-order difference methods for solving this to understand why solving the equation in conservative form is important. As in class, we will assume outflow boundary conditions.

   To do this, we’ll consider initial conditions for a shock:

   \[\begin{split}u(x,t=0) = \left \{ \begin{array}{cc} 2 & \mathrm{if~} x < 0.5 \\ 1 & \mathrm{if~} x \ge 0.5 \\ \end{array} \right .\end{split}\]

   We will run for a maximum time of \(t = 0.2\).

   Your task

   • Using first-order upwinding, we can difference the first form of Burgers’ equation as:

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

     This is valid as long as \(u_i^n > 0\). Write a program to solve this equation with the above shock initial conditions:

     These are the initial conditions for a shock that we explored in class.

     Important

     While you can base your solution on the first-order advection example from class, you need to be sure to compute the timestep inside of the evolution, for each step, since \(u\) changes as we evolve.

   • Now difference the conservative form as:

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

     Write a program to solve this, and run it on the same initial data as in before.

     Note that both of these discretizations assume that \(u > 0\).

   • Measure the speed of the shock for each method. To do this simply difference the shock position at the final time and the initial time and estimate the shock speed as \(S = \Delta x / \Delta t\).

     Which method gets the shock speed correct? Why?

     Present your results at two different grid resolutions.