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{ \large \textbf{\textsc{Math 5588 -- Homework 10 (Due Thursday April 13)}}}
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\begin{enumerate}
\item Compute the six elemental stiffness values $k_{ij}^s=k_{ji}^s$ for an equilateral triangle.
\item Suppose your domain is triangulated by equilateral triangles. Calculate the nonzero values of $k_{ij}=\sum_{s=1}^M k_{ij}^s$ in the case that vertex $i$ is an interior vertex (that is, not on the boundary).
\item Solve Burger's equation $u_t + uu_x = 0$ with initial condition $u(x,0)=x$. 
\item Solve $u_t + u^2 u_x = 0$ with $u(x,0)=2+x$. Sketch the characteristics.
\item Solve Burger's equation $u_t + uu_x = 0$ with initial condition $u(x,0)=f(x)$ where
\[f(x) = \begin{cases}
2,&\text{if } x < -1\\
1,&\text{if } -1 < x < 0\\
0,&\text{if } x>0.
\end{cases}\]
[Hint: Proceed as we did in class by sketching the characteristics and then using the Rankine-Hugoniot Condition to determine the speed of the shock curves.]
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