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\begin{document}

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% ----------------- title -------------------------

\title{{\color{seccolor} Experiments with Schemes for Exponential Decay}}

% ----------------- author(s) -------------------------

\author{Hans Petter Langtangen\footnote{Email: \texttt{hpl@simula.no}. Center for Biomedical Computing, Simula Research Laboratory and Department of Informatics, University of Oslo.}}

% ----------------- end author(s) -------------------------

\date{Jul 2, 2016}
\maketitle

\begin{quote}\small
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{quote}

\tableofcontents

\vspace{1cm} % after toc

% !split
\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 radioactive 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 \href{{http://en.wikipedia.org/wiki/Forward_Euler_method}}{Forward Euler}
scheme when $\theta=0$

\item The \href{{http://en.wikipedia.org/wiki/Backward_Euler_method}}{Backward Euler}
scheme when $\theta=1$

\item The \href{{http://en.wikipedia.org/wiki/Crank-Nicolson}}{Crank-Nicolson}
scheme when $\theta=1/2$
\end{itemize}

\section{Implementation}

The numerical method is implemented in a Python function
\cite{Langtangen_2014} \texttt{solver} (found in the \href{{http://bit.ly/29ayDx3}}{\nolinkurl{model.py}} Python module file):

\begin{minted}[fontsize=\fontsize{9pt}{9pt},linenos=false,mathescape,baselinestretch=1.0,fontfamily=tt,xleftmargin=2mm]{python}
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{minted}

\section{Numerical experiments}

\index{numerical experiments}

A set of numerical experiments has been carried out,
where $I$, $a$, and $T$ are fixed, while $\Delta t$ and
$\theta$ are varied. In particular, $I=1$, $a=2$,
$\Delta t = 1.25, 0.75, 0.5, 0.1$.
Figure~\ref{fig:BE} contains four plots, corresponding to
four decreasing $\Delta t$ values. The red dashed line
represent the numerical solution computed by the Backward
Euler scheme, while the blue line is the exact solution.
The corresponding results for the Crank-Nicolson and
Forward Euler methods appear in Figures~\ref{fig:CN}
and~\ref{fig:FE}.

\index{Backward Euler method}

\begin{figure}[!ht]  % fig:BE
\centerline{\includegraphics[width=0.9\linewidth]{BE.pdf}}
\caption{
The Backward Euler method for decreasing time step values. \label{fig:BE}
}
\end{figure}
%\clearpage % flush figures fig:BE

\index{Crank-Nicolson method}

\begin{figure}[!ht]  % fig:CN
\centerline{\includegraphics[width=0.9\linewidth]{CN.pdf}}
\caption{
The Crank-Nicolson method for decreasing time step values. \label{fig:CN}
}
\end{figure}
%\clearpage % flush figures fig:CN

\index{Forward Euler method}

\begin{figure}[!ht]  % fig:FE
\centerline{\includegraphics[width=0.9\linewidth]{FE.pdf}}
\caption{
The Forward Euler method for decreasing time step values. \label{fig:FE}
}
\end{figure}
%\clearpage % flush figures fig:FE

\section{Error vs $\Delta t$}

\index{error vs time step}

How the error

$E^n = \left(\int_0^T (Ie^{-at} - u^n)^2dt\right)^{\frac{1}{2}}$
varies with $\Delta t$ for the three numerical methods
is shown in Figure~\ref{fig:error}.

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]  % fig:error
\centerline{\includegraphics[width=0.9\linewidth]{error.pdf}}
\caption{
Variation of the error with the time step. \label{fig:error}
}
\end{figure}
%\clearpage % flush figures fig:error

The $E$ numbers corresponding to Figure~\ref{fig:error}
are given in the table below.

\begin{quote}
\begin{tabular}{rrrr}
\hline
\multicolumn{1}{c}{ $\Delta t$ } & \multicolumn{1}{c}{ $\theta=0$ } & \multicolumn{1}{c}{ $\theta=0.5$ } & \multicolumn{1}{c}{ $\theta=1$ } \\
\hline
1.25       & 7.4630     & 0.2161       & 0.2440     \\
0.75       & 0.6632     & 0.0744       & 0.1875     \\
0.50       & 0.2797     & 0.0315       & 0.1397     \\
0.10       & 0.0377     & 0.0012       & 0.0335     \\
\hline
\end{tabular}
\end{quote}

\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}

\bibliographystyle{plain}
\bibliography{.publish_references}

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