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\textbf{\Large Maths 3018/6111 - Numerical Methods \\*[8mm]
Worksheet 7}\\*[.8cm]
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\section*{Theory}

\begin{enumerate}
\item Find the linear system that results from using central
  differencing to solve the elliptic equation
  \begin{equation*}
    u_{xx} + u_{yy} = \sin ( \pi y ) \left( 2 - 6 x - (\pi x)^2 (1 -
      x) \right).
  \end{equation*}
  Central differencing, trivial boundary conditions, and a $3 \times
  2$ grid should be used.
\item Explain the strategy for solving evolutionary PDEs using finite
  differencing methods.
\item Write out a FTCS method for the equations
  \begin{align*}
    \partial_t y - \partial_{xx} y & = \sin(x), & x & \in [0,1], &
    y(t, 0) & = 0 = y(t, 1), \\
    \partial_t y + \partial_{x} y & = e^{-(x-1/2)^2}, & x & \in [0,1], &
    y(t, 0) & = 0 = y(t, 1).
  \end{align*}
  In both cases the initial data should be taken to be $y(0, x) =
  f(x)$. Full details of the grid, the discrete initial and boundary
  conditions, and the discrete update algorithm should be given.
\item Using von Neumann analysis, find when the BTCS method is stable
  for the advection equation.
\end{enumerate}

\section*{Coding}

\begin{enumerate}
\item Implement the finite difference method above to solve the
  elliptic equation
  \begin{equation*}
    u_{xx} + u_{yy} = \sin ( \pi y ) \left( 2 - 6 x - (\pi x)^2 (1 -
      x) \right).
  \end{equation*}
  Check how it converges with resolution (the exact solution is $u =
  x^2 (1 - x) \sin(\pi y)$.
\item Implement a FTBS algorithm for the advection equation with
  periodic boundaries. Use a sine wave on $x \in [0,1]$ as initial
  data. Check how it converges with resolution.
\item{} [Additional] Apply the FTBS algorithm to the traffic flow equation
  \begin{equation*}
    \partial_t y + \partial_x \left( y ( 1 - y) \right) = 0.
  \end{equation*}
  Use initial data $y(0, x) = \exp(-x^2)$, $y(0, x) = 0.2 + 1.2 H(x)$
  and $y(0, x) = 1 - H(x)$ on $x \in [-10, 10]$ where $H(x)$ is the
  Heaviside step function. By comparing against the results for
  Burger's equation, explain your results. You may want to restrict to
  $t \leq 0.5$.
\end{enumerate}

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