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{\large\sc Delft University of Technology}\\
{\sc Faculty of Information Technology and Systems}\\%[2ex]
%{\bf Opgaven}\\
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{\bf Numerieke Wiskunde b.o. (wi4012)}\\
{\bf Numerieke Analyse CII (wi4014)}\\
{\bf Numerieke Methoden PDV (wi3001)}\\
{\bf 2000/2001 First Series.}
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\begin{enumerate}
\item Excercise 3.10 (page 48)
\item Instead of (3.17) you also could solve:
\[
- \frac{d}{dx} e^{-vx} \frac{dy}{dx} = 0\;, \;\;\; y(0) = 0\;, \;\;\; y(1) = 1
\] 
Using central differences (see excercise 3.3.a page 36) one obtains a positive
definite matrix $A$. 
\ben
\item Show that the solution of this problem does not exhibit
wiggles. Hint: supposing both $u_{i-1} > u_{i}$ and $u_{i+1} > u_{i}$ hold,
try to arrive at a contradiction.
\item Extend this method to non-constant velocities, that is:
\[
- y^{''} + v(x)y^{'} = f\;, \;\;\; y(0) = 0\;, \;\;\; y(1) = 1\;.
\]
Hint: consider  $\phi (x) = \int\limits_{0}^{x} v(t)\ dt.$
\een

Unfortunately this method does not work in two or three dimensions.

\item In the triangle (0, 0), (1, 0), (0, 1) (see figure) the PDE \\[2ex] 
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\[
   - \divg k \nabla \varphi + \bm{u} \cdot \nabla \varphi = f
\] 
holds, in which $\varphi, k, f$ and $\bm u$ are functions of $x$ and $y$. We
suppose, that the flow is incompressible:
\[
\divg \bm{u} = 0
\]
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\\[2ex]
At the boundaries we have the following boundary conditions:
\[
\begin{array}{llll}
x = 0\;: &\varphi = g_{1} &x - y = 1\;:& k\frac{\partial \varphi}{\partial
n} = 0\\
y = 0\;: &\sigma \varphi + k \frac{\partial \varphi}{\partial n} = g_{2} &
&
\end{array}
\] 
This problem is solved by a finite volume method with an equidistant rectangular
grid.
\ben
\item Derive an $O(h^2)$ discretization molecule for the inner region. Hint:
from the incompressibility condition, the equality
$$\bm u\cdot\nabla \varphi = \divg(\bm u\varphi)$$ 
follows.
\item Show how the boundary conditions at each of the three boundaries have to
be treated in order to maintain an accuracy of $O(h^2)$. Choose the grid in such
a way, that all boundaries contain nodes.
\een
\end{enumerate}
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{\bf wi4014 to be submitted before december 1, 2000}
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