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\begin{document}
\title{Experiments with Schemes for Exponential Decay}
\author{Hans Petter Langtangen\footnote{
Center for Biomedical Computing, Simula Research Laboratory, and
Department of Informatics, University of Oslo.}}
\date{\today}

\begin{abstract}
This report investigates the accuracy of three finite difference
schemes for the ordinary differential equation $u'=-au$ with the
aid of numerical experiments. Numerical artifacts are in particular
demonstrated.
\end{abstract}

\tableofcontents

\section{Mathematical problem}
\label{math:problem}
\index{model problem}\index{exponential decay}

\begin{align}
u'(t) &= -au(t), \quad t \in (0,T], \label{ode}\\
u(0)  &= I,                         \label{initial:value}
\end{align}
where $a$, $I$, and $T$ are prescribed parameters, and $u(t)$ is
the unknown function to be estimated. This mathematical model
is relevant for physical phenomena featuring exponential decay
in time, e.g., vertical pressure variation in the atmosphere,
cooling of an object, and radioactiv decay.

\section{Numerical solution method}
\label{numerical:problem}
\index{mesh in time} \index{$\theta$-rule} \index{numerical scheme}
\index{finite difference scheme}

We introduce a mesh in time with points $0= t_0< t_1 \cdots < t_{N_t}=T$. For simplicity, we assume constant spacing $\Delta t$
between the mesh points: $\Delta t = t_{n}-t_{n-1}$,
$n=1,\ldots,N_t$. Let $u^n$ be the numerical approximation to the
exact solution at $t_n$.

The $\theta$-rule \cite{Iserles_2009}
is used to solve (\ref{ode}) numerically:

$u^{n+1} = \frac{1 - (1-\theta) a\Delta t}{1 + \theta a\Delta t}u^n,$
for $n=0,1,\ldots,N_t-1$. This scheme corresponds to

\begin{itemize}
\item The Forward Euler scheme when $\theta=0$
\item The Backward Euler scheme when $\theta=1$
\item The Crank-Nicolson scheme when $\theta=1/2$
\end{itemize}

\section{Implementation}

The numerical method is implemented in a Python function
\cite{Langtangen_2012}:

\begin{quote}
\begin{verbatim}
def solver(I, a, T, dt, theta):
"""Solve u'=-a*u, u(0)=I, for t in (0,T] with steps of dt."""
dt = float(dt)           # avoid integer division
Nt = int(round(T/dt))    # no of time intervals
T = Nt*dt                # adjust T to fit time step dt
u = zeros(Nt+1)          # array of u[n] values
t = linspace(0, T, Nt+1) # time mesh

for n in range(0, Nt):   # n=0,1,...,Nt-1
u[n+1] = (1 - (1-theta)*a*dt)/(1 + theta*dt*a)*u[n]
return u, t
\end{verbatim}
\end{quote}

\section{Numerical experiments}
\index{numerical experiments}

We define a set of numerical experiments where $I$, $a$, and $T$ are
fixed, while $\Delta t$ and $\theta$ are varied.
In particular, $I=1$, $a=2$, $\Delta t = 1.25, 0.75, 0.5, 0.1$.

\subsection{The Backward Euler method}
\index{BE}

\begin{center}  % inline figure
\centerline{\includegraphics[width=0.9\linewidth]{BE.png}}
\end{center}

\subsection{The Crank-Nicolson method}
\index{CN}

\begin{center}  % inline figure
\centerline{\includegraphics[width=0.9\linewidth]{CN.png}}
\end{center}

\subsection{The Forward Euler method}
\index{FE}

\begin{center}  % inline figure
\centerline{\includegraphics[width=0.9\linewidth]{FE.png}}
\end{center}

\subsection{Error vs $\Delta t$}
\index{error vs time step}

How $E$ varies with $\Delta t$ for $\theta=0,0.5,1$
is shown in Figure~\ref{fig:E}.

\paragraph{Observe:}
The data points for the three largest $\Delta t$ values in the
Forward Euler method are not relevant as the solution behaves
non-physically.

\begin{figure}[!ht]
\centerline{\includegraphics[width=0.9\linewidth]{error.png}}
\caption{
Error versus time step. \label{fig:E}
}
\end{figure}

\subsection{Summary}

\begin{enumerate}
\item $\theta =1$: $E\sim \Delta t$ (first-order convergence).
\item $\theta =0.5$: $E\sim \Delta t^2$ (second-order convergence).
\item $\theta =1$ is always stable and gives qualitatively corrects results.
\item $\theta =0.5$ never blows up, but may give oscillating solutions
if $\Delta t$ is not sufficiently small.
\item $\theta =0$ suffers from fast-growing solution if $\Delta t$ is
not small enough, but even below this limit one can have oscillating
solutions (unless $\Delta t$ is sufficiently small).
\end{enumerate}

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