MA8103 Non-Linear Hyperbolic Conservation Laws

Spring term 2020

Exercises

The exercises (few and far between) are listed on the main page. This is just an overflow page, for information about the exercises that don't fit on the main page.

Exercise 3.10

The problem is partly wrong, but perhaps more misleading than wrong.

Part (a)

I can't explain what the problem is without giving you a hint, so here goes: The stated equality comes from expressing \(f(u^n_j)\) and \(f(u^n_{j-2})\) by a Taylor expansion around \(u^n_{j-1}\). The two function arguments \(\eta_{j-1/2}\) and \(\eta_{j-3/2}\) come from the remainder terms of these two Taylor expansions. But the notation indicate that “\(\eta_{j-1/2}\)” from one grid point equals “\(\eta_{j-3/2}\)” from its neigbour to the right, and that is not right, since they come from different Taylor expansions.

If you ignore this problem, or use less confusing notation, what you really want to prove is \[ v^{n+1}_j \le \bigl(1-\lambda f'(u^n_{j-1})\bigr)v^n_j+\lambda f'(u^n_{j-1})v^n_(j-1) -c\Delta t\bigl((v^n_j)^2+(v^n_{j-1})^2\bigr). \]

Part (b)

The “inductive assumption” is really not needed until you get to (c). You do, however, need the weaker inequality \[ v^n_j \le \frac1{2c\Delta t}. \] You are also going to need to assume that the CFL condition holds.

Part (c)

Hint: Find an inequality involving \(1/\hat v^{n+1}\) and \(1/\hat v^{n}\). This will make the induction step very easy.

Exercise 5.9

The p-system with \(p(v)=1/v\): \[ \begin{align} v_t-u_x&=0 \\ u_t+(1/v)_x&=0\end{align} \]

Parts (a) and (d)

The Riemann invariants are not unique! If \(w\) is a Riemann invariant, then so is \(g(w)\) where \(g\) is any function. (To be pedantic, we should be writing \(g \circ w\) instead for the function composition.)

Normally, this is not such a big deal, but part (d) will not work unless you pick the “right” Riemann invariants. There are only a couple of “natural” choices, and you can probably see which one works. If not, there is a hint in two transformations of the problem leaving the equations invariant:

  1. If you add a constant to \(u\), the equations are unchanged, and
  2. If you rescale \(v\) and \(t\) by the same multiplicative factor, the equations are also unchanged. Rescaling \(t\) does not change the wave curves, only the speeds of shocks and rarefactions. And rescaling \(v\) is the same as adding a constant to the logarithm of \(v\).
2020-04-21, Harald Hanche-Olsen