MATH76110/26/2018

Problem. Consider the conservation law

For each of the cases considered below:

1. Find the analytical solution of the Riemann problem

   ${\displaystyle 𝒒(0,x,y)=\{\begin{array}{ll}{𝒒}_l& x<0\\ {𝒒}_r& x>0\end{array}.}$

   (1)

   for $t>0$,$-4\pi ⩽x⩽4\pi$. For each case:

   1. Find the eigenstructure of the linearized problem ${𝒒}_t+𝑨{𝒒}_x+𝑩{𝒒}_y=0$, $𝑨=\partial 𝒇/\partial 𝒒$, $𝑩=\partial 𝒈/\partial 𝒒$.

   2. Determine the magnitude of the jumps propagated by each eigenmode finding $\delta 𝒘$, $𝑨𝑿=𝑿{𝚲}^x$, $𝑩𝒀=𝒀{𝚲}^y$, $𝑿\delta 𝒘=\delta 𝒒$, $𝒀\delta 𝒘=\delta 𝒒$.

2. Use Bearclaw to solve the Riemann problem (1) using the following finite volume method variants:

   1. First-order Godunov method

   2. Second-order wave propagtion method without limiters

   3. Second-order wave propagation method with limiters

   4. Adaptive mesh refinement applied to first-order Godunov method

   5. Adaptive mesh refinement applied to wave propagation method with limiters

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