Numerical Methods for Hyperbolic Partial Differential Equations - 2019

Giulio 09.08.2019 11:00-11:30

Course taken as a Bachelor Student

He asked for preferred topic. I didn't have any so he started with asking multiple Riemann Problems which involved combinations of schocks and rarefactions. The fluxes were $$ f(u) = \frac{u^{2}}{2} $$ (Burger's equation) and $$ f(u) = u(1-u) $$ (Traffic flow flux). The aim was to solve them, not always completely and analitcally. He wants to know if the characteristics method are known. He asked also what the difference between conservative and non conservative form are. Wasn't sure, at the end I got that the conservative form enables discontinous solutions. Given a RP he then asked what Finite Volume Scheme we could use to solve it, I said Godunov and he asked why it works. Because it is a MCC scheme that converges toward an entropy solution. He then asked for another RP with defined initial conditions, the professor proposed the Numerical Flux $$ F(U_{j}^{n},U_{j+1}^{n})= F(a,b)= f(b)$$ (Left Flux) and he said if this scheme would work well. In general no, but given our initial condition the scheme becomes MCC for all times ( using that MCC Scheme implies Max/Min principle) hence converging to an entropy solution. The Professor then proceeds to approximate Burger's equation via a (new) scheme $$ \frac{U_{j}^{n+1}-U_{j}^{n}}{\Delta t} + U_{j}^{n} \frac{U_{j+1}^{n}-U_{j-1}^{n}}{2 \Delta x}. $$ He asked if it made sense as a discretization and yes we are just approximating derivatives via differences (but the points are to be understood in FVS frame, as cell averages). The professor asks why this scheme should work or not (without implementing it), I put it in update form without any advance. He gives me the Finite Volume form with numerical flux $$ F(U_{j}^{n},U_{j+1}^{n})= F(a,b)= \frac{ab}{2}$$ and we use a Lemma in the lecture notes to check for monotonicity of the scheme.

The professor welcomes you calling your name, making the exam less tense. The exam was totally relaxed for the whole time.