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\lhead{Technische Universität Berlin\\Numerische Mathematik I - WS 2019/2020}
\rhead{Prof.~V.~Mehrmann\\M.~Froidevaux}
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\fancypagestyle{AbgabeDatum}{\fancyfoot[C]{Diese Aufgaben müßen spätestens am 24.10.19 in der Vorlesung abgegeben werden.}}
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\begin{document}
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%{\LARGE Numerische Mathematik}\\[1.5ex]
%%{\large \CurrentWeekNb. \"Ubungsblatt}\\ 
{\Large Exercise sheet for the \CurrentWeekNb. tutorial}\\ 
{Week 28.10.-1.11.19}\\
%\\{Diese Aufgaben müßen spätestens am 24.10.19 in der Vorlesung abgegeben werden.}
%{Abgabe:} Spätestens am 24.10.19 in der Vorlesung\\[0.5cm]
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%\hfill {Abgabe: Spätestens am 24.10.19 in der Vorlesung}\\[0.5cm]
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%%\def\exerciseList{001, 002, 003}
%\begin{enumerate}
%


\begin{enumerate}
%\item ...
 \item Let $h>0$ and $t_i = ih$, for $i=0, 1, 2, \ldots, N$. Heun's method applied to the problem
\begin{equation}
\label{eq:Anfangswertproblem}
\begin{split}
&\dot{y}(t) = f(t,y(t)),\ \text{ for } t>0 \\
&y(0)=y_0
\end{split}
\end{equation}
  yields the iteration
\begin{align*}
u_0 &= y_0\\
\tilde{u}_{i+1} &= u_i + h f(t_i, u_i)\\
u_{i+1} &= u_i + h\frac{f(t_i, u_i)+f(t_{i+1}, \tilde{u}_{i+1})}{2},
\end{align*}
where $u_0=y_0$.
\begin{enumerate}
\item Suppose that the function $f:\mathbb{R}\times\mathbb{R}\to \mathbb{R}$ is Lipschitz continuous in the second variable with Lipschitz constant $L$.
Show that the increment function\footnote{Defined through $u_{i+1}=u_i+h\Phi(t_i,u_i,h)$.} $\Phi(t,y,h)$ of Heun's method is also Lipschitz continuous in the second variable, and calculate the associated Lipschitz constant.
\item Show that Heun's iteration of the initial value problem \eqref{eq:Anfangswertproblem} with $f(t,y(t))=2y(t)$ and $y_0=2$ converges to the solution of the initial value problem.
\end{enumerate} 

\item Transform the following decimal numbers in their binary and hexadecimal representation (i.e.~$b=2,16$):
\begin{enumerate}
\item $x = 12.125$,
\item $x = 0.1$.
\end{enumerate}
\item Consider the so-called implicit Euler method
\[
u_{i+1}=u_i+hf(t_{i+1}, u_{i+1})
\]
for the solution of the initial value problem $\dot{y}=f(t,y)$, $y(t_0)=y_0$.
\begin{enumerate}
\item Which practical problem arises, that is not present in the case of explicit Euler and Heun's method?
\item Calculate the order of consistency\footnote{That is $p$, where $\tau_h=\mathcal O(h^p)$ is the local discretization error (defined in 3.c).}.
\item Suppose that $f$ is Lipschitz continuous in the second component, i.e.
\[
\abs*{f(t,y_1)-f(t,y_2)}\leq L\abs*{y_1-y_2}
\]
with $L$ constant for all $t$ and all $y_1,y_2$.
Show that the global error $e_i\coloneqq y(t_i)-u_i$ satisfies the inequality
%
\begin{equation}
\label{eq:globalenFehler}
\abs{e_{i+1}} \leq \frac{1}{1-Lh} (\abs{e_i}+h\abs{\tau_h}),
\end{equation}
%
for all $i\geq0$, where
\[
\tau_h\coloneqq \frac{y(t_{i+1})-y(t_i)}{h}-f(t_{i+1}, y(t_{i+1})).
\]
is the local discretization error.

\emph{Bonus:} From \eqref{eq:globalenFehler} it follows that the method also has convergence order 1. (You do not need to prove it here.)
\end{enumerate}
\end{enumerate}
\end{document}


