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\begin{document}

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\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\
Lost at Sea: Introduction to \\Numerical Methods through Navigation}




\markboth{Lost at Sea}{Navigating Numerical Methods}


\author{
Richard Corban Harwood\\
Department of Mathematics and Applied Science\\ 
George Fox University\\
Newberg OR 97132 USA\\
rharwood@georgefox.edu
}


\makeStitlePDFLaTex


\begin{abstract}
This technique narrative guides a discovery-based approach to learning the basics of numerical methods for first order differential equations, by following the graphical and analytical perspectives of the forward Euler method and second order Taylor method. These methods are motivated by velocity field and dead reckoning navigation over two dimensions to locate a ship lost at sea. 
\end{abstract}

\teacher{
\section*{INTRODUCTION FOR THE TEACHER}
The use of numerical methods to study differential equations is as old as the subject itself, starting with the Euler method published by Leonhard Euler in his calculus textbook in 1768 \cite{Euler1768}. There are many modern benefits to using such methods to give students visual intuition and motivate student learning through investigations and discovery-based activities \cite{Borelli2010}. Modeling-based teaching of ordinary differential equations closely followed the explosion of numerical method implementations in the 1970's and the pedagogical shift in mathematics courses through the availability of mathematical modeling modules developed through NSF grants and the formation of the Consortium for Mathematics and its Applications (COMAP) \cite{West2012}.


This teacher's version provides a guide for planning, facilitating, and assessing the following activities. The student version only includes the STATEMENT section and can be used printed out as a worksheet or distilled into activity components as the teacher decides. No additional supplies are required.

The activities in Section \ref{sec:GraphicalApproach} and \ref{sec:AnalyticApproach} are designed to be done simultaneously, each by half of a group of two or more. Each activity is anticipated to be completed in the classroom within one 50-minute class, but part or all of it can be done outside of class. Some components encourage the use of a calculator or computer for spreadsheet manipulations, but most of the arithmetic has been manipulated to be easy to do by hand, so no technology is required.

Both of these activities are centered around finding a ship, El Perdido (``the Lost" in Spanish), which has been damaged during a storm and is adrift. In Activity \ref{sec:GraphicalApproach}: Graphical Approach of the Rescuers, students take on the role of the Coast Guard Search and Rescue and use slope fields to estimate the position of El Perdido. Using the differing speeds of an airplane, helicopter, and ship, all tasked in the rescue operation, students compare Euler method approximations on the slope fields with varying time steps (faster craft corresponds to higher time step as the search covers greater ground before adjusting). In the process, students develop a graphical understanding of the Euler method and the visual effect of decreasing the time step. Simultaneously, in Activity \ref{sec:AnalyticApproach}: Analytical Approach of El Perdido's Captain, students take on the role of the captain of the damaged vessel who works out dead reckoning approximations of his position based upon the printouts of his ship's Inertial Navigation System. In the process, these students derive the Euler and second order Taylor methods and compare the resultant path estimates. Together, the combined group shares their results and uses two-dimensional error comparisons to search radii to see if any of the rescue craft is successful in finding El Perdido. The search radii were carefully selected to be near two of the position estimates, so finding El Perdido depends upon carefully following the slope fields as well as correctly estimating the numerical approximations.
}


\section*{INTRODUCTION}

The ship, El Perdido, was damaged during a storm which knocked out its main and backup power generators. Before the backup generator failed, Captain Miguel Gomez sent a distress call and the crew have been able to keep El Perdido afloat, but the ship is adrift in the Pacific Ocean off the coast of California. Thankfully, a US Coast Guard rescue operation is underway after receiving the distress call. The Coast Guard has El Perdido's last known position and has mapped out the surface water velocities in this area as slope fields for longitude ($x$) and latitude ($y$), which they have updated using historical data and estimated predictions. Since the search grid is small enough, this curved region on the surface of the earth is relatively flat. 

After days of failed searches, the length of time expected for El Perdido's crew to survive has expired (calculated by the standard Cold Exposure Survivability Model). Accordingly, the Coast Guard mission coordinator downgrades the rescue operation to ``Active Search Suspended, Pending Further Developments." Because the El Perdido crew had kept the ship afloat and had salvaged some of the food reserves, they are able to survive for several more days but their time is still running out.
%, pegged at $(x(0),y(0))=(-1,0)$ on their search grid,

On the fourth day adrift, El Perdido's crew is able to get a short spurt of power, in which they send another distress call and print out logs of ship velocity and acceleration from their Inertial Navigation System (INS). We join the Coast Guard rescue operation, reactivated after receiving the distress call, as they update their slope field maps and launch another search. Simultaneously, Captain Gomez now uses dead reckoning from the INS logs to estimate his own position knowing only El Perdido's previous velocities and accelerations, in hopes that he can assist the rescuers if the crew can get the power back on.

Split your group with half helping the rescuers in Section \ref{sec:GraphicalApproach} and half helping El Perdido's captain in Section \ref{sec:AnalyticApproach} and then compare your results. Based upon the search radius of each rescue craft and Captain Gomez's own dead reckoning, determine if El Perdido is found on this rescue attempt. 

%\newpage

\section{Graphical Approach of the Rescuers}\label{sec:GraphicalApproach}
There are three Coast Guard rescue craft searching for El Perdido: a long-range land-based plane called the Ocean Sentry (S1), a ship-based helicopter called the Eurocopter Dolphin (D2), and a ship with a helipad referred to as a high endurance cutter (C3). All three search craft use the same surface water velocity estimates in the search grid, but each uses a different time step (in units of estimated days spent by the lost vessel as it drifted) since a greater distance is covered by a search craft with a higher speed: $\dt=1$ for S1, $\dt=0.4$ for D2, and $\dt=0.2$ for C3. They each have El Perdido's last known position pegged at $(x(0),y(0))=(-1,0)$ on their search grid, but the velocity field is separated into a longitudinal slope field shown in Figure \ref{fig:ActA} with initial position $(t,x)=(0,-1)$ in Section \ref{sec:Longitude} and a latitudinal slope field shown in Figure \ref{fig:ActB} with initial position $(t,y)=(0,0)$ in Section \ref{sec:Latitude}. Note that the slope fields in Figures \ref{fig:ActA} and \ref{fig:ActB} are snapshots of the overall estimated velocity field and have been graphed with slightly different viewing windows.

%%%%%%%%%%%% Check for source file for dfield_x2_t2_solution, if not open _clean version in GIMP and adding paths with thick, dash, thin %%%%%%%%%



\subsection{Estimating Longitude}\label{sec:Longitude} %%%%%%%%%%%%%%%%%%%%


Traveling the fastest, the S1 plane searched a path that followed the Euler method with time step $\dt=1$ on the slope fields, essentially tracking El Perdido's path using one day of estimated velocities at a time. It's estimate of the longitude has been computed for you in Table \ref{tb:example}. Follow the steps below to compare S1's search path to the slope field, as well as plot the search paths of the D2 helicopter and C3 cutter. Along the way, answer the reflection questions (labeled Q) to check your understanding.
%For a graphical perspective on the Euler method, consider Table \ref{tb:example}. The approximation points were generated with the Euler method using $\dt=1$ to approximate the solution to the differential equation $x'=x^2-t^2$ with $x(-1)=0$. The slope field of this differential equation is shown in Figure \ref{fig:ActA}. Using Table \ref{tb:example} and Figure \ref{fig:ActA}, do the following steps. 

\begin{table}[h!]
 \begin{center}
  \begin{tabular}{c|c||c|c|}%{c|c||c|c|c||c|c}%{c|c|c|c|c|c|cc|c}
$k$ & $t_k$ & $x_k$ & $x_k'= x_k^2 - t_k^2$ \\%& $\qquad\qquad y_k\qquad\qquad$ & $\qquad y_k'= t_k - y_k^2\qquad$ \\%& $\to$Copy for Plotting$\to$&$x_k$ & $y_k$\\ %=f\left(t_k,x_k\right), =g\left(t_k,x_k\right)
\hline \hline
0&0&-1&1 \\  \hline%&0&0\\%&&\teacher{{\bf -1}}&\teacher{{\bf 0}}\\ \hline 
1&1&0&-1 \\ \hline%&\teacher{{\bf 0}}&\teacher{{\bf 1}}\\%&&\teacher{{\bf 0}}&\teacher{{\bf 0}}\\ \hline
2&2&-1&-3 \\ \hline%&\teacher{{\bf 1}}&\teacher{{\bf 1}}\\%&&\teacher{{\bf -1}}&\teacher{{\bf 1}}\\ \hline
3&3&-4&7 \\ \hline%&\teacher{{\bf 2}}&\teacher{{\bf -1}}\\%&&\teacher{{\bf -4}}&\teacher{{\bf 2}}\\ \hline 
4&4&3&-7 %&\teacher{{\bf 1}}&\teacher{{\bf 3}} %&&\teacher{{\bf 3}}&\teacher{{\bf 1}}
  \end{tabular}
 \end{center}
 \caption{\teacher{{\bf (Teacher Example)}} Estimates of El Perdido's path by rescuer S1: longitude ($x$) and latitude ($y$) using the Euler method with $\dt=1$.}
 \label{tb:example}
\end{table}




\begin{enumerate}
\item Plot the points $(t_k,x_k)$ from Table \ref{tb:example} on Figure \ref{fig:ActA} and connect neighbor points with straight line segments. 

Q: Does the slope of each line segment better match the slope field on the left or right end? Does this match your intuition that error accumulates as time progresses?

\teacher{{\bf 
 A: (Teacher Example) These line segments are from individual tangent lines which match the solution's slope on the left edge of each subinterval, and would not match the solution's slope on the right edge of the subinterval unless by coincidence or when the slope vectors were all the same. Thus, starting with the approximate solution value on the left edge, the line segment further approximates the value of the solution on that right edge, accumulating error as time progresses. Yes, this matches our intuition.
}}%\hrule  
\end{enumerate}


\begin{figure}[h!]
\ifnum\TeacherEdition=1 
\centerline{\includegraphics[width=.75\textwidth]{dfield_x2_t2_solution.png}} 
\else 
 \centerline{\includegraphics[width=.75\textwidth]{dfield_x2_t2_clean.png}} 
\fi
\caption{Display of longitudinal velocity field for rescue craft search paths. 
 \teacher{{\bf
  (Teacher Example) Solution of longitudinal search path component for rescue craft S1 (blue, $\dt=1$), D2 (green, $\dt=0.4$), and C3 (magenta, $\dt=0.2$) with points for initial position (black) and final position (red) of El Perdido. From these sketches, the true value of -3.5 for $x(4)$ is estimated as 3 for S1, -3.8 for D2, and -3.75 for C3. To test graph-reading skills, make sure student approximations are only drawn for $0\leq t\leq 4$, leaving the left portion blank.
} }
}
\label{fig:ActA}
\end{figure}

\begin{enumerate}
\setcounter{enumi}{1}
\item Using each line as a hypotenuse of a right triangle, compute the slope (the amount of rise over run) for each and label each line with it.

Q: What values do these slopes match?

\teacher{{\bf 
 A:  (Teacher Example) These are the values for $x'_k$ in Table \ref{tb:example}, which is the data for the derivative on left edge of each subinterval where the slope of the line matches the slope field.
}}%\hrule 


\item On the right edge of Figure \ref{fig:ActA}, circle S1's estimate of El Perdido's longitude on the fourth day as `S1=3'.
 
Q: Based upon the path of S1, do you trust this estimate?

\teacher{{\bf 
 A:  (Teacher Example) Based on the erratic path of S1, however, we should not trust S1's estimate as the path displays unstable behavior.
}}%\hrule 


 
%Now we will improve the approximation by reducing the time step.
Now we will follow the second rescue craft, the D2 helicopter, to recreate El Perdido's path following the slope field every $0.4$ days.
\item Starting again at the initial position $x(0)=-1$ on Figure \ref{fig:ActA}, follow the initial slope for $\dt=0.4$ and mark the right endpoint.

Note: this will stop partway along S1's initial step.
\item Using the arrow direction nearest this point, follow the slope field for another $\dt=0.4$. Then, plot the rest of D2's search path by repeating this process until you reach $t=4$. Estimate this $x(4)$ value and label with `D2='. \label{StepRepeat}

Now we will follow the third rescue craft, the C3 high endurance cutter, to recreate El Perdido's path following the slope field every $0.2$ days.
\item Plot C3's search path, again starting at the initial position $x(0)=-1$ on Figure \ref{fig:ActA} and follow the initial slope for $\dt=0.2$ and mark the right endpoint.   Note: this will stop partway along D2's initial step.
%\item Using the arrow direction nearest this point, follow the slope field another $\dt=0.2$. Then repeat this process until you reach $t=4$.
%Returning to the leftmost point in Figure \ref{fig:ActA}, draw the same tangent line halfway over $\dt=0.2$ and mark the right endpoint. This is your next approximation.
%\item Using the most recent approximation point, follow the slope field over $\dt=0.4$ to get the next approximation point. Then repeat this process until you reach $t=3$.\label{StepRepeat}
\item Repeat step \ref{StepRepeat} with $\dt=0.2$ until you reach $t=4$. Estimate this $x(4)$ value and label with `C3='. 

Q: Compare the search paths you plotted for S1 ($\dt=1$), D2 ($\dt=0.4$), and C3 ($\dt=0.2$) in Figure \ref{fig:ActA}. Do they appear to be converging to a solution function?

\teacher{{\bf 
 A:  (Teacher Example) Yes, as demonstrated in Figure \ref{fig:ActA}, the variation between the search paths of D2 and C3 appears to be much smaller than that of S1 and D2. Note, the exact solution function is converging into a stable separatrix, so these approximations must converge into that same separatrix curve.
}}%\hrule 


If so, highlight that curve in a different color in Figure \ref{fig:ActA}. This is the exact solution. If not, check that your line segments match the arrows on the left ends and plot the paths of S1, D2, and C3 again.
%\item From Figure \ref{fig:ActA} at time $t=4$, estimate the longitude ($x_4$) values for D2 and C3, and record them in column ___ of Table \ref{tb:position} for comparison with Activity \ref{sec:AnalyticApproach}. % This is done at the end of section 1.2
%Estimate the longitude (x) values for D2 and C3 at times $t = 0, 1, 2, 3, 4$, and record them in Table \ref{tb:position??} in Section \ref{sec:AnalyticApproach} for comparison with Activity \ref{sec:AnalyticApproach}.
\end{enumerate}


%\newpage

\subsection{Estimating Latitude}\label{sec:Latitude} %%%%%%%%%%%%%%%%%%%%%%%

The slope field representing the latitudinal velocities can be approximated by the differential equation $y'=t-y^2,\ y(0)=0$. Follow the steps below to compute S1's longitudinal coordinates and then follow the slope field for the D2 helicopter and C3 cutter. Along the way, answer the reflection questions (labeled Q) to check your understanding.


\begin{table}[h!]
 \begin{center}
  \begin{tabular}{c|c||c|c|}
$k$ & $t_k$ & $y_k$ & $\qquad y_k'=t_k - y_k^2\qquad$\\
\hline \hline
0&0&0&0\\ \hline
1&1&\teacher{{\bf 0}}&\teacher{{\bf 1}}\\ \hline
2&2&\teacher{{\bf 1}}&\teacher{{\bf 1}}\\ \hline
3&3&\teacher{{\bf 2}}&\teacher{{\bf -1}}\\ \hline
4&4&\teacher{{\bf 1}}&\teacher{{\bf 3}}
  \end{tabular}
 \end{center}
 \caption{\teacher{{\bf (Teacher Example)}} Latitude ($y$) estimates of El Perdido's path by rescuer S1 using the Euler method with $\dt=1$.}
 \label{tb:fill-in}
\end{table}


\begin{enumerate}
 \item Complete Table \ref{tb:fill-in} using the Euler method approximation, $$y_{k+1}=y_k+\dt\ y_k',$$ using $\dt=1$ for the S1 plane, to update each $y_k$ latitudinal value in the next row and the differential equation to update the slope $y_k'$ in that next row. For example, $$y_1=y_0+\dt\ y_0'=0+(1)(0)=0,\ \textrm{and}\ y_1'=t_1-y_1^2=1-0^2=1.$$ 
% Fill in each subsequent row by updating the time and unknown from the previous row, then computing the new slope $y_{k+1}$.\\

Note that the Euler method is really just the tangent line at the point $\left(t_k,y_k\right),$ which has been rewritten from point-slope form to standard form: $$y_{k+1}=y_k+(t_{k+1}-t_k)\ y_k'=y_k+\dt\ y_k'.$$
Once you have Table \ref{tb:fill-in} filled in, you will repeat the steps you took in plotting and following the slope field for the longitude in Figure \ref{fig:ActA}, only now using the slope field for the latitude in Figure \ref{fig:ActB}.
 \item Plot the latitude estimates  $\left(t_k,y_k\right)$ for S1 on the slope field in Figure \ref{fig:ActB}, draw lines to connect the points, estimate the value of $y(4)$, and label it with `S1='. 
 
Q: Does it look like your approximate solution fits the slope field well? Explain.

  {\bf \teacher{A: (Teacher Example) No, as shown by the oscillations of the approximation segments increasing in amplitude in Figure \ref{fig:ActB}. This is because the time step used by S1 is in the unstable range for the Euler method. This matches the unstable behavior in the longitudinal coordinates of S1's path demonstrated on Figure \ref{fig:ActA}.}}


\begin{figure}[h!]
\ifnum\TeacherEdition=1 
\centerline{\includegraphics[width=.75\textwidth]{dfield_t_x2_solution.png}} 
\else 
\centerline{\includegraphics[width=.75\textwidth]{dfield_t_x2_clean.png}} 
\fi
\caption{Display of longitudinal velocity field for rescue craft search paths. 
\teacher{\bf (Teacher Example) Example solution of latitudinal search path component for rescue craft S1 (blue, $\dt=1$), D2 (green, $\dt=0.4$), and C3 (magenta, $\dt=0.2$) with points for initial position (black) and final position (red) of El Perdido. From these sketches, the true value of 2 for $y(4)$ is estimated as 1 for S1, 1.8 for D2, and 1.9 for C3. To test graph-reading skills, make sure student approximations only go for $0\leq t\leq 4$, leaving the outer portions on the left and right blank.}}
\label{fig:ActB}
\end{figure}
  
  
 \item Again on Figure  \ref{fig:ActB}, graph the search path for D2 helicopter, starting at $y(0)=0$ and stepping with $\dt=0.4$ until $t=4$. Estimate this value of $y(4)$ and label it with `D2='.
\item  Again on Figure  \ref{fig:ActB}, graph the search path for C3 cutter, starting at $y(0)=0$ and stepping with $\dt=0.2$ until $t=4$. Estimate this value of $y(4)$ and label it with `C3='.
 \item Using the slope field, draw your personal best approximation to the solution.\\
Q: What things were you considering when you drew your best approximation?\\
  \teacher{{\bf
   A: (Teacher Example) Example responses could include using a very short time step, balancing the arrows above and below, or being consistent with the arrows at all points (not just the left ends).
  }}%\hrule 





\begin{table}[h!]
 \begin{center}
  \begin{tabular}{c|c|c|c|c|c}
Source & $x(4)$ & $y(4)$ & Search Radius & Distance from El Perdido & Did it See El Perdido?\\
\hline \hline
S1&\teacher{{\bf 3}}&\teacher{{\bf 1}}&1&\teacher{{\bf $\sqrt{6.5^2+1^2}=$6.58}}&\teacher{{\bf No, not $<1$}}\\ \hline
D2&\teacher{{\bf -3.8}}&\teacher{{\bf 1.8}}&.5&\teacher{{\bf $\sqrt{0.3^2+0.2^2}=$0.36}}&\teacher{{\bf Yes, $<0.5$}}\\ \hline
C3&\teacher{{\bf -3.75}}&\teacher{{\bf 1.9}}&.25&\teacher{{\bf $\sqrt{0.25^2+0.1^2}=$0.27}}&\teacher{{\bf No, not $<0.25$}}\\ \hline
El Perdido&\teacher{{\bf -3.5}}&\teacher{{\bf 2}}&N/A&N/A&N/A
  \end{tabular}
 \end{center}
 \caption{\teacher{{\bf (Teacher Example)}} Position of El Perdido at $t=4$ days estimated by rescue craft and dead reckoning.}
 \label{tb:position}
\end{table}

%\subsection{Graphing Rescuers}
Now, you will summarize your work in Table \ref{tb:position} by recording the $t=4$ position of El Perdido estimated by the three rescue craft and comparing it with El Perdido's actual position to see if any of the rescue craft found the ship on this attempt.
\item On Table \ref{tb:position}, record the longitude ($x(4)$) values you labeled for each rescue craft (S1, D2, C3) on Figure \ref{fig:ActA} and the final latitude ($y(4)$) values you labeled for each on Figure \ref{fig:ActB}. 

\item Check in with the rest of your group, sharing how you replicated the search paths of the rescue craft and asking about how they replicated the captain's hand calculations.

\item Get their best estimate of El Perdido's position $(x(4),y(4))$ at the end of Table \ref{tb:INSDataTDR}. Record this point in the El Perdido row of Table \ref{tb:position}.

\item Plot the four $(x(4),y(4))$ positions from Table \ref{tb:position} onto Figure \ref{fig:grid}, labeling each, to visually compare the three estimates of El Perdido's position at $t=4$ days. 

\begin{figure}[h!]
\ifnum\TeacherEdition=1 
\centerline{\includegraphics[width=.75\textwidth]{xy-grid_solutionA.png}} 
\else 
 \centerline{\includegraphics[width=.75\textwidth]{xy-grid_fixed.png}}
 \fi 
 \caption{Longitude-Latitude grid for plotting search paths of El Perdido. \teacher{\bf (Teacher Example) Example solution of longitude-latitude plots for rescue craft S1 (blue, $\dt=1$), D2 (green, $\dt=0.4$), and C3 (magenta, $\dt=0.2$) estimates from Figures \ref{fig:ActA} and \ref{fig:ActB}, and Taylor dead reckoning position (TDR, red) of El Perdido from Table \ref{tb:INSData}.}
}
\label{fig:grid}
\end{figure}

\item Calculate the distance from each search craft estimate to El Perdido's position and check if less than it's search radius to determine if this search craft found the lost ship.
\end{enumerate}																							
%\item Get their best estimate of El Perdido's position $(x_4,y_4)$ at the end of Table \ref{tb:INSDataTDR}. Record this point in the El Perdido row of Table \ref{tb:position} and plot this point on Figure \ref{fig:grid}, labeling it El Perdido and enter the values in Table \ref{tb:position} in the TDR row.
																										
																																																																								
%\subsubsection{Instructions for Activity B2}
%Fill out the first row of table \ref{tab:graphA} using the initial condition $x(0)$ and computing the initial slope, $x'$ by plugging in $x=x(0),t=0$.\\
%Slope can be seen as the ratio of Rise / Run, the opposite and adjacent sides of a right triangle.
%Fill out the other rows by repeating the follow items for each row.
%\begin{enumerate}
%\item Compute the slope $x'=f(t(now), x(now))$
%\item Let Run$=\dt$ and compute the Rise = Slope*Run. 
%\item Update the variables by computing t(next)=t(now)+Run, and x(next)= x(now)+Rise.
%\item Graph the new point $(t(next),x(next))$ and draw a line connecting it to the previous point.
%\end{enumerate}



%\subsubsection{Instructions for activity C?}
%In this activity, you will compare local information to global infomation using the slope field.


\section{Analytic Approach of El Perdido's Captain}\label{sec:AnalyticApproach} %%%%%%%%%%

Miguel Gomez, Captain of El Perdido, was able to extract velocity and acceleration information from the ship's INS (Inertial Navigation System) when his crew momentarily brought backup power online. To aid possible rescuers, he sought to estimate his own position from this data by hand. Captain Gomez selected data from the four days since the day of the storm damage, i.e. times $t=0,1,2,3,4$, and stored them in Table \ref{tb:INSData}.


\subsection{Euler Dead Reckoning}\label{sec:EDR}%%%%%%%%%%%%%%
Seeing that he only had one initial position, but the velocity and acceleration at every time, Captain Gomez knew he needed an equation that would allow him to compute the next position based on the current position and derivatives at either position. Starting with the definition of the derivative of some function $x(t)$ at time $t_k$,
$$
x'(t_k) = \lim_{\dt\to 0} \frac{x(t_k +\dt)-x(t_k)}{\dt},
$$
he used the shorthand notation $x_k=x(t_k),x'(t_k)=x_k',t_{k+1}=t_k+\dt$ to rewrite the derivative definition in the simpler form,
$$ 
x_k'=\lim_{\dt\to 0} \frac{x_{k+1}-x_k}{\dt}.
$$
Captain Gomez realized that if he started near the current position $x_k$ and chose a small enough $\dt$, he could approximate the derivative without taking the limit, leaving him with only one unknown quantity.
 
Q: What equation did the captain get when he solved for this unknown ($x_{k+1}$, the next longitude value)?
$$  x_{k+1}=$$
\teacher{
$$\textbf{A: (Teacher Example)}\qquad x_{k+1} = x_k + \dt x_k'$$
}%\hrule 
 
Q: What similar equation did he obtain for the next unknown latitude?
$$  \teacher{\textbf{A: (Teacher Example)}\qquad } y_{k+1}=\teacher{ y_k + \dt y_k'}$$%\hrule 
 
%\teacher{
%$$\textbf{A: (Teacher Example)}\qquad y_{k+1} = y_k + \dt y_k'$$
%}
These equations are called the {\it (forward) Euler method}. The captain used them to iteratively solve for $x_1,x_2,x_3,x_4$ and $y_1,y_2,y_3,y_4$. Dead Reckoning is a process of estimating the position of a vessel based upon its current movement and past estimated position. When we use the Euler method to estimate such a position, we will refer to it as Euler Dead Reckoning (EDR). 

Q: Compute these values (a calculator or spreadsheet is helpful), and record your values under the heading EDR  in Table \ref{tb:INSData} and then do the following.

\teacher{{\bf 
 A: (Teacher Example) Positions estimated by the Euler method are} 
 
 $(x_k,y_k)=(-1,0),(0,0),(-.5,1),(-1.5,2),(-3,2)$ } 

%record these values in Table \ref{tb:INSData} in the EDR column and then do the following

\begin{table}[h!] 
 \begin{center}
  \begin{tabular}{cc|c|cc||c|cc||rl}
Index &Time &EDR &Data && EDR & Data && Copy Points&for Graphing\\%Data &&&&&&EDR&&TDR&\\
$k$ & $t_k$ & $x_k$ & $x_k'$ & $x_k''$ & $y_k$ & $y_k'$ & $y_k''$ & $x_k$ & $y_k$ \\
%$k$ & $t_k$ & $x_k'$ & $y_k'$ &  $x_k''$ &  $y_k''$ & $x_k$ & $y_k$ & $x_k$ & $y_k$ \\  %%%%% Reorder the values below accordingly
\hline \hline
0&0&-1&1&-2&0&0&1&-1&0\\ \hline
1&1&\teacher{{\bf -1}}&-0.5&-1&\teacher{{\bf 0.5}}&1&0&\teacher{{\bf -1}}&\teacher{{\bf 0.5}}\\ \hline
2&2&\teacher{{\bf -2}}&-1&0&\teacher{{\bf 1.5}}&1&-2&\teacher{{\bf -2}}&\teacher{{\bf 1.5}}\\ \hline
3&3&\teacher{{\bf -3}}&-1.5&2&\teacher{{\bf 1.5}}&0&1&\teacher{{\bf -3}}&\teacher{{\bf 1.5}}\\ \hline
4&4&\teacher{{\bf -3.5}}&-1&-1&\teacher{{\bf 2}}&1&-3&\teacher{{\bf -3.5}}&\teacher{{\bf 2}}
  \end{tabular}
 \end{center}
 \caption{Velocity, acceleration, and initial position data for El Perdido extracted from the Inertial Navigation System (INS). Fill in the dead reckoning positions $x_k,y_k$ using the Euler method (EDR).}\label{tb:INSData}
\end{table}



\begin{figure}[b!]
\ifnum\TeacherEdition=1 
\centerline{\includegraphics[width=.5\textwidth]{xy-grid_solutionB_points.png} \includegraphics[width=.5\textwidth]{xy-grid_solutionB_curves.png}} \caption{Longitude-Latitude grid for plotting the Euler method dead reckoning positions of El Perdido. \bf (Teacher Example) Example solution points and curves for dead reckoning longitude-latitude positions using the Euler method (EDR, orange). For overall comparison, the graphs of the Taylor method (TDR, red) and example positions for rescue craft S1 (blue), D2 (green), and C3 (magenta) are included.}
\else 
 \centerline{\includegraphics[width=.55\textwidth]{xy-grid_fixed.png}} \caption{Longitude-Latitude grid for plotting the Euler method dead reckoning positions of El Perdido.}
\fi
\label{fig:gridB}
\end{figure}

\begin{enumerate}
\item Graph each $(x_k,y_k)$ for $k=0,1,2,3,4$ on Figure \ref{fig:gridB}.
\item To represent the tangent slope at each position $(x_k,y_k)$, graph a short arrow with the direction of the corresponding velocity vector $(x_k',y_k')$ from Table \ref{tb:INSData}.
\item Sketch a curve going through each position $(x_k,y_k)$ and tangent to each slope vector $(x_k',y_k')$. Label it `EDR' for Euler Dead Reckoning.
\end{enumerate}

Q: Does the EDR curve seem like a feasible path for a ship adrift? Why or why not?

\teacher{{\bf
 A: (Teacher Example) For the most part, the EDR curve seems feasible, but there seems to be an incongruity between the first two points and their slopes, requiring an unusual curve to connect them with the correct slopes.
}}%\hrule 



Q: What else could we do to improve our estimate of the path?

\teacher{{\bf
 A: (Teacher Example) It seems like the acceleration data should help smooth out this discrepancy. Also having data closer together would make it more reliable.
}}%\hrule 




\subsection{Taylor Dead Reckoning}\label{sec:TDR}%%%%%%%%%%%%%%
In all the excitement of finding out where he was, the captain had not noticed that the acceleration data had been ignored. Looking again at the equation for the Euler method with $x_{k+1}=x(t_k+\dt)$,
$$ x(t_k+\dt) = x(t_k) + \dt x'(t_k),$$
Captain Gomez remembered another pattern that started out in the same way
$$ x(t_k+\dt) = x(t_k) + \dt x'(t_k) + \frac{\dt^2}{2!}x''(t_k) + \frac{\dt^3}{3!}x'''(t_k)+\ldots +\frac{\dt^n}{n!}x^{(n)}(t_k)+\ldots ,$$
called the {\it Taylor series}. Since he didn't have data for any higher derivatives than two, Captain Gomez truncated the Taylor series and rewrote $x(t_k)=x_k$ to obtain what is called the {\it (second order) Taylor method}.

Q: Write out the (second order) Taylor method for longitude (x) and latitude (y).

$$ \teacher{\textbf{A: (Teacher Example)}\qquad} x_{k+1} = \teacher{x_k + \dt x_k' + \frac{\dt^2}{2!}x_k''}$$%\hrule 


$$ \teacher{\textbf{A: (Teacher Example)}\qquad} y_{k+1} = \teacher{y_k + \dt y_k' + \frac{\dt^2}{2!}y_k''}$$%\hrule 



Captain Gomez now uses the Taylor method to iterate solve again for each of the longitude, $x_k$, and latitude, $y_k$ values. We will refer to these approximations as Taylor Dead Reckoning (TDR). Do the following steps to replicate his work.
%$x_1,x_2,x_3,x_4$ and $y_1,y_2,y_3,y_4$. \\
%Q: Compute these values (a calculator or spreadsheet is helpful),\\
%\teacher{{\bf
% A: (Teacher Example) Positions estimated by the Taylor method are $(-1,0),(-1,.5),(-2,1.5),(-3,1.5),(-3.5,2)$.
%}}%\hrule 
%  \vspace{.3in}
%record these values in Table \ref{tb:INSData} in the TDR column, and then 

\begin{table}[h!] 
 \begin{center}
  \begin{tabular}{cc|c|cc||c|cc||rl}
Index &Time &TDR &Data && TDR & Data && Copy Points&for Graphing\\%Data &&&&&&EDR&&TDR&\\
$k$ & $t_k$ & $x_k$ & $x_k'$ & $x_k''$ & $y_k$ & $y_k'$ & $y_k''$ & $x_k$ & $y_k$ \\
%$k$ & $t_k$ & $x_k'$ & $y_k'$ &  $x_k''$ &  $y_k''$ & $x_k$ & $y_k$ & $x_k$ & $y_k$ \\  %%%%% Reorder the values below accordingly
\hline \hline
0&0&-1&1&-2&0&0&1&-1&0\\ \hline
1&1&\teacher{{\bf -1}}&-0.5&-1&\teacher{{\bf 0.5}}&1&0&\teacher{{\bf -1}}&\teacher{{\bf 0.5}}\\ \hline
2&2&\teacher{{\bf -2}}&-1&0&\teacher{{\bf 1.5}}&1&-2&\teacher{{\bf -2}}&\teacher{{\bf 1.5}}\\ \hline
3&3&\teacher{{\bf -3}}&-1.5&2&\teacher{{\bf 1.5}}&0&1&\teacher{{\bf -3}}&\teacher{{\bf 1.5}}\\ \hline
4&4&\teacher{{\bf -3.5}}&-1&-1&\teacher{{\bf 2}}&1&-3&\teacher{{\bf -3.5}}&\teacher{{\bf 2}}
  \end{tabular}
 \end{center}
 \caption{\teacher{\bf Teacher Example} Velocity, acceleration, and initial position data for El Perdido extracted from the Inertial Navigation System (INS). Fill in the dead reckoning positions $x_k,y_k$ using the Taylor method (TDR).}\label{tb:INSDataTDR}
\end{table}



\begin{figure}[h!]
\ifnum\TeacherEdition=1 
\centerline{\includegraphics[width=.5\textwidth]{xy-grid_solutionB_points.png} \includegraphics[width=.5\textwidth]{xy-grid_solutionB_curves.png}} \caption{Longitude-Latitude grid for plotting the Taylor method dead reckoning positions of El Perdido. \bf (Teacher Example) Example solution points and curves for dead reckoning longitude-latitude positions using the Taylor method (TDR, red). For overall comparison, the graphs of the Euler method (EDR, orange) and example positions for rescue craft S1 (blue), D2 (green), and C3 (magenta) are included.}
\else 
 \centerline{\includegraphics[width=.55\textwidth]{xy-grid_fixed.png}} \caption{Longitude-Latitude grid for plotting the Taylor method dead reckoning positions of El Perdido.}
\fi
\label{fig:gridB2}
\end{figure}


\begin{enumerate}
\item Use the Taylor method to compute the values for $x_1,x_2,x_3,x_4$ and record them in Table \ref{tb:INSDataTDR} in the TDR column for $x_k$.
\item Similarly, compute $y_1,y_2,y_3,y_4$ with the Taylor method and record them in Table \ref{tb:INSDataTDR} in the TDR column for $y_k$.
\item Graph each $(x_k,y_k)$ for $k=0,1,2,3,4$ on Figure \ref{fig:gridB2}.
\item To represent the tangent slope at each position $(x_k,y_k)$, graph a short arrow with the direction of the corresponding velocity vector $(x_k',y_k')$ from Table \ref{tb:INSDataTDR}.
\item Sketch a curve in Figure \ref{fig:gridB2} going through each position $(x_k,y_k)$ and tangent to each slope vector $(x_k',y_k')$. Label it `TDR'.
\end{enumerate}


Q: Does the TDR curve seem like a feasible path for a ship adrift? Why or why not?
 
\teacher{{\bf
 A: (Teacher Example) The TDR curve seems more feasible than the EDR curve, especially between the first two positions and their slopes.
}}%\hrule 


  
Q: What else could we do to improve our estimate of the path?

\teacher{{\bf
 A: (Teacher Example) Again, having data over shorter time intervals would improve it's reliability.
}}%\hrule 



Now, check in with the rest of your group to find their rescuers' estimates on where El Perdido is on day four ($t=4$). 
\begin{enumerate}
\item Ask them how they replicated the search paths of the rescue craft and show them what you did replicating the captain's hand calculations.
\item Give them the final pair of points you computed in Table \ref{tb:INSDataTDR} for them to use in Table \ref{tb:position}.
\item From them, obtain the positions of the 3 rescue craft S1, D2, and C3, and plot these positions on Table \ref{fig:gridB2} to compare with your TDR path of El Perdido.
\item Check back with the rest of your group after they have compared the distance of each rescue craft with their respective search radius.

Q: Did any of the rescue craft find El Perdido? Which were the closest?

 \teacher{{\bf
   A: (Teacher Example) Rescue craft D2 (and possibly C3) should have found El Perdido, but due to not following the slopes as closely, students may not have any of them find the lost ship. Of the estimates, C3 should be closest with D2 nearby, while S1's unstable estimate is very far off.
  }}%\hrule 


\end{enumerate}
%\newpage

\section{Background}
European explorers in the 15th and 16th century sailed throughout the world using a simple navigation technique called dead reckoning. They used a sand glass to precisely measure time, threw flotsam overboard at regular intervals to measure speed, and measured their direction with a magnetic compass. Today, ships, aircraft, and spacecraft continue to use dead reckoning using an Inertial Navigation System (INS) and multiple redundancies such as GPS and radar to verify their position. A modern INS tracks time digitally, measures velocity with rate gyroscopes, and measures acceleration with accelerometers.

%Often thought of as a backup techniques when no exact solution can be found, 
Numerical methods provide unique perspectives to a problem (as well as quick approximations). %Instead of finding an algebraic function to solve a given differential equation, numerical methods construct a sequence of discrete solution values $x_k$ or $y_k$ corresponding to discrete input values $t_k$. Each pair of corresponding sequences, $(t_k,x_k)$ or $(t_k,y_k)$, defines a   discrete function, and a solution to a difference equation which approximates the differential equation. 
Imagine sketching curves along all the arrows in a slope field like Figure \ref{fig:ActA} or \ref{fig:ActB}. This visualizes many member functions of the family of solutions, which are each exact solutions to different initial conditions. In constrast, a numerical solution shifts between member functions of the same family at each approximation. This is demonstrated by your plot of S1's latitudinal path in Figure \ref{fig:ActB}, which you computed using the Euler method in Table \ref{tb:example}. Seeing that the slope field only matches the line segments on each left side demonstrates that the numerical solution visibly slips from one member function to another in the family of solutions with each iteration. It may jump back and forth across member functions, representing oscillations about the exact solution, and may even slip farther and farther away with each jump. These behaviors, demonstrated by S1's path in Figure \ref{fig:ActA} and Figure \ref{fig:ActB}, are related to the stability of a numerical method and can be analyzed with further study.

%\section{Follow up Questions}
%\begin{enumerate}
%\item
%\item
%\end{enumerate}

%\newpage
%\section{Discussion}\label{sec:Discussion}

%\begin{theorem}
%Suppose that the domain $D=\{(t,y)| a\leq t \leq b \textrm{ and } -\infty<y<\infty \}$ and that $f(t,y)$ is continuous on $D$. If $f$ satisfies a Lipschitz condition on $D$ in the variable $y$, then the initial-value problem $$y'(t)=f(t,y),\quad a\leq t\leq b,\quad y(a)=\alpha,$$ has a unique solution $y(t)$ for $a\leq t\leq b$.
%\end{theorem}

%A function $f(t,y)$ defined on a convex domain D satisfies a Lipschitz condition in the variable y if the y-partial of f is bounded within D ($\exists L, |f_y(t,y)|\leq L \forall (t,y)\in D$). %\cite{Burden}.

%An initial-value problem is defined to be well-posed if a unique solution to the problem exists and the associated perturbed initial value problem with a small change in the slope and initial conditions has a unique solution whose difference from the true solution is small as well. (small changes in the statement of the problem introduce correspondingly small changes in the solution) 

%\begin{theorem}
%Suppose that the domain $D=\{(t,y)| a\leq t \leq b \textrm{ and } -\infty<y<\infty \}$ and that $f(t,y)$ is continuous on $D$. If $f$ satisfies a Lipschitz condition on $D$ in the variable $y$, then the initial-value problem $$y'(t)=f(t,y),\quad a\leq t\leq b,\quad y(a)=\alpha,$$ is well-posed.
%\end{theorem}


\teacher{
\section*{NOTES FOR THE TEACHERS}

\section{Implementation Guide}


The examples used in the two activities are significant in a number of ways beyond the scope of these activities. First, both the longitudinal and latitudinal velocity fields have similar pairs of separatrices, but with their stability swapped. These separatrices provide an intriguing extension from the analysis of equilibrium points which students are more familiar with.
Second, students discover and derive the fundamental numerical method for differential equations, originally published by Leonhard Euler in his calculus textbook in 1768 \cite{Euler1768}.
%using the discrete function $x_k$ to approximate the unknown, $x_k\approx x(t_k)$,

In table \ref{tb:example}, The approximation points were generated with the Euler method using $\dt=1$ to approximate the solution to the differential equation $x'=x^2-t^2$ with $x(-1)=0$. The slope field of this differential equation is shown in Figure \ref{fig:ActA}.

\subsection{Example Solutions}
Example solutions provide a helpful comparison with student work. They are included in the statement of the activities, but are not available to students. Look for short answer responses underlined, table values filled in, and figures with example graphs plotted on top.

\subsection{Prerequisites}
No specific differential equations course skills are required before students begin these activities, but it would be helpful if students were familiar with %Before starting this activity, students should be able to
\begin{itemize}
\item Slope fields/direction fields or vector fields.
\item Spreadsheets: relative and fixed referencing of cells and dragging down formulas
\item Equilibrium points or separatrices to provide another layer of insight and connection with the slope fields. 
\end{itemize}

\subsection{Objectives}
Upon successful completion of the Graphical Approach of the Rescuers activity, students will be able to
\begin{enumerate}
\item Explain how the approximation process works graphically for the (forward) Euler method
\item Compute the Euler method by hand for simple values
\item Compute vector error (distance) and compare to a given tolerance
\item Given the slope field and initial condition, 
\begin{enumerate}
\item accurately sketch the unknown exact solution
\item demonstrate the Euler method approximation for a fixed $\dt$
\item use approximate solutions to estimate future positions
\end{enumerate}
\end{enumerate}

Upon successful completion of the Analytical Approach of El Perdido's Captain activity, students will be able to
\begin{enumerate}
\item Derive the (forward) Euler method from the limit definition of the derivative or from a Taylor series
\item Derive the (second order) Taylor method from a Taylor series
\item Implement the Euler and Taylor methods using a calculator, spreadsheet, or extensive (but reasonable) calculations by hand
\item Given 2D discrete position and velocity data, students should be able to  
\begin{enumerate}
\item sketch a limited slope field along the estimated path
\item accurately sketch the solution through the points, tangent to the slope field
\item evaluate approximation smoothness and agreement with the slope field
\end{enumerate}
\end{enumerate}

\subsection{Rubric}
This rubric has been helpful for grading group/individual work during student presentations.
\begin{itemize}
\item Completion of activity components (0 1 2 3 4 5)
\item Mastery of objectives (0 1 2 3 4 5)
\item Depth of mathematical analysis/insight (0 1 2 3 4 5)
\item Individual contribution to presentation/work (0 1 2 3 4 5)
\end{itemize}

\section{Discussion}
These activities can lead into related SIMIODE activities, such as investigating numerical instabilities \cite{Fuselier2016}, found using a quick search for numerical or similar keyword. 

Further investigation of dead reckoning and numerical method comparison can be found in \cite{Banachowitz2012}. For a historic perspective of dead reckoning by Columbus, please see \cite{Pickering1996}. These are two of the articles which inspired the preceding activities.


\begin{thebibliography}{98}
%\bibitem{Hutchinson1978a} Hutchinson, G. Evelyn. 1978. {\it Introduction to Population Ecology\/}. New Haven CT:  Yale University Press.
%\bibitem{Harwood2008}V. S. Manoranjan, M. A. O. Gomez, and  R. C. Harwood, ``Modeling Algae Self-Replenishment,'' {\it Journal of Interdisciplinary Mathematics}, Vol. 11, pp. 681-694, 2008.
% Add ODE textbook and journal articles/books found
\bibitem{West2012}
West, Beverly, Robert Borrelli,  and Lawrence Moore. 2012. Evolution of the Modern ODE Course. {\it CODEE Journal\/}. Vol.~9, Article 9.
\bibitem{Borelli2010}
Borrelli, Robert and Courtney Coleman. 2010. Pitfalls and Pluses in Using Numerical Software to Solve Differential Equations.
{\it CODEE Journal\/}.  Vol.~7, Article 5.
%Available at: http://scholarship.claremont.edu/codee/vol7/iss1/5
%\bibitem{Lal2014} Lal, Nishu and Towse, Christopher (2014) ``Teaching Differential Equations with Graphics and without Linear Algebra," CODEE
%Journal: Vol. 10, Article 3.
%Available at: http://scholarship.claremont.edu/codee/vol10/iss1/3
\bibitem{Euler1768}
Euler, Leonhard. 1768. Institutionum Calculi Integralis, Vol.~1. {\it Opera Omnia\/}. Series 1, Volume 11.

\bibitem{Banachowitz2012} 
Banachowitz, A. and A.~ Wolski. 2012. A Comparison of Numerical Solutions of Dead Reckoning  Navigation. {\it Reports on Geodesy\/}.  93(2): 49-55.

\bibitem{Pickering1996} Pickering, Keith A. 1996. Columbus's Method of Determining Longitude: An Analytical View. {\it The Journal of Navigation\/}. 49:1, 99-113.

\bibitem{Fuselier2016} 
Fuselier, E. 2016. A Comparison of Numerical Techniques through the Study of the Nonlinear Pendulum. 6.2-T: 1-9. Available at \url{https://simiode.org/resources/1657}.

\end{thebibliography}
}





\end{document}

%% Original Table
\begin{table}[h!] %%%%%%%%%%%%%%%%%%%%%  Consider splitting into two tables or at least separating them horizontally
 \begin{center}
  \begin{tabular}{c|c|c|c|c|c||c|c||c|c}
Data &&&&&&EDR&&TDR&\\
$k$ & $t_k$ & $x_k'$ & $y_k'$ &  $x_k''$ &  $y_k''$ & $x_k$ & $y_k$ & $x_k$ & $y_k$ \\  %%%%% Reorder the values below accordingly
\hline \hline
0&0&1&0&-2&1&-1&0&-1&0\\ \hline
1&1&-.5&1&-1&0&\teacher{{\bf 0}}&\teacher{{\bf 0}}&\teacher{{\bf -1}}&\teacher{{\bf 0.5}}\\ \hline
2&2&-1&1&0&-2&\teacher{{\bf -0.5}}&\teacher{{\bf 1}}&\teacher{{\bf -2}}&\teacher{{\bf 1.5}}\\ \hline
3&3&-1.5&0&2&1&\teacher{{\bf -1.5}}&\teacher{{\bf 2}}&\teacher{{\bf -3}}&\teacher{{\bf 1.5}}\\ \hline
4&4&-1&1&-1&-3&\teacher{{\bf -3}}&\teacher{{\bf 2}}&\teacher{{\bf -3.5}}&\teacher{{\bf 2}}
  \end{tabular}
 \end{center}
 \caption{Velocity, acceleration, and initial position data for El Perdido extracted from the Inertial Navigation System (INS). Fill in the dead reckoning positions $x_k,y_k$ using the Euler method (EDR) then the Taylor method (TDR).}\label{tb:INSData}
\end{table}

%\begin{table}[t!] %%%%%%%%%%%%%%%%%%%%%  Previous version of table
% \begin{center}
%  \begin{tabular}{c|c|c|c|c|c|c|c|c|c}
%&&EDR&&TDR&&&&&\\
%$k$ & $t_k$ & $x_k$ & $y_k$ & $x_k$ & $y_k$ & $x_k'$ & $y_k'$ &  $x_k''$ &  $y_k''$\\
%\hline \hline
%0&0&-1&0&-1&0&1&0&-2&1\\ \hline
%1&1&\teacher{{\bf 0}}&\teacher{{\bf 0}}&\teacher{{\bf -1}}&\teacher{{\bf 0.5}}&-.5&1&-1&0\\ \hline
%2&2&\teacher{{\bf -0.5}}&\teacher{{\bf 1}}&\teacher{{\bf -2}}&\teacher{{\bf 1.5}}&-1&1&0&-2\\ \hline
%3&3&\teacher{{\bf -1.5}}&\teacher{{\bf 2}}&\teacher{{\bf -3}}&\teacher{{\bf 1.5}}&-1.5&0&2&1\\ \hline
%4&4&\teacher{{\bf -3}}&\teacher{{\bf 2}}&\teacher{{\bf -3.5}}&\teacher{{\bf 2}}&-1&1&-1&-3
%  \end{tabular}
% \end{center}
% \caption{Velocity, acceleration, and initial position data for El Perdido extracted from the Inertial Navigation System (INS). Fill in the dead reckoning positions $x_k,y_k$ using the Euler method (EDR) then the Taylor method (TDR).}\label{tb:INSData}
%\end{table}
