MATH76110/26/2018

1Conservation laws

Consider the conservation law

and the Riemann problem

for $t>0$,$-4\pi ⩽x⩽4\pi$.

The above problem is solved in Bearclaw for:

1. Advection equation $q_t+{(uq)}_x+{(vq)}_y=0$

2. Burgers equation $q_t+{\left(\frac{1}{2}{q}^2\right)}_x=0$

3. Wave equation ${\phi }_{tt}-c^2{\nabla }^2\phi =0$, transformed to system ${𝒒}_t+𝑨{𝒒}_x+𝑩{𝒒}_y=𝟎$, with notation $(u,r,s)=({\phi }_t,{\phi }_x,{\phi }_y)$

   ${\displaystyle 𝒒=\left(\begin{array}{l}u\\ r\\ s\end{array}\right),𝑨=-\left(\begin{array}{lll}0& c^2& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right),𝑩=-\left(\begin{array}{lll}0& 0& c^2\\ 0& 0& 0\\ 1& 0& 0\end{array}\right)}$

4. Euler equations of gas dynamics

   ${\displaystyle 𝒒=\left(\begin{array}{l}\rho \\ l\\ m\\ \epsilon \end{array}\right),𝒇\left(𝒒\right)=\left(\begin{array}{l}l\\ \frac{l^2}{\rho }+p\\ \frac{lm}{\rho }\\ lH\end{array}\right),𝒈\left(𝒒\right)=\left(\begin{array}{l}m\\ \frac{lm}{\rho }\\ \frac{m^2}{\rho }+p\\ mH\end{array}\right),}$

   with $l=\rho u$, $m=\rho v$, $\epsilon =\rho E$, $H=E+\frac{1}{2}(u^2+v^2)$, $p=\rho RT=(\gamma -1)(\epsilon -\frac{l^2+m^2}{\rho }).$

Each of the above problems will be solved in Bearclaw using literate programming techniques. See lab07 subdirectories for each problem.