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--- gibson:teaching:spring-2016:math445:lecture:timestepping [2016/04/14 10:00]
gibson [Problem 3]
+++ gibson:teaching:spring-2016:math445:lecture:timestepping [2016/04/14 18:23] (current)
gibson [Problem 4]
@@ Line 81: / Line 81: @@
  {{ :gibson:teaching:spring-2016:math445:lecture:cylinderpath1.png?direct&400 }}
-Note that the computed trajectory is not very accurate, since we chose quite a large time step $\Delta t = 0.4$, and forward-Euler is only 1st-order accurate (error scales as $\Delta t$).
+Note that the trajectory computed here is not very accurate. The particle shouldexit the box at the same $y$ value it had when it entered. The problem is we chose quite a large time step $\Delta t = 0.4$, and forward-Euler is only 1st-order accurate (error scales as $\Delta t$). In the next problem, we'll reduce the time step to $\Delta t = 0.01$ and get a more accurate solution --though still not as good as the 4th-order accurate ''ode45'' function.
 ----
 ====Problem 2====
-Write Matlab code that plots the //path// of the particle as a red curved line. To do this wee need to save the sequence of $\vec{x}$ values in a matrix, and then plot the rows of that matrix as a line.
+Write Matlab code that plots the //path// of the particle as a red curved line. To do this we need to save the sequence of $\vec{x}$ values in a matrix, and then plot the rows of that matrix as a line.
 <code matlab>
@@ Line 108: / Line 110: @@
 </code>
+ {{ :gibson:teaching:spring-2016:math445:lecture:cylinderpath3.png?400 |}}
+----
 ====Problem 3====
@@ Line 126: / Line 129: @@
 axis equal
 axis tight
-xlim([-xmax,xmax])
+xlim([-3,3])
-ylim([-ymax,ymax])
+ylim([-2,2])
 </code>
@@ Line 150: / Line 153: @@
 </file>
+This produces a plot very like the one for Problem 3.
+----
+==== Problem 4 ====
+Draw a number of particle paths starting with a number of $y$ values and $x=-2.8$.
+<code matlab>
+% use Matlab's ode45 function to do the time integration of
+% dx/dt = v(t, x)
+T = 10;                           % integrate from t=0 to T=10
+for y = -2:0.4:2                  % loop over different y values
+  x0 = [-2.8; y];                 % set initial position of particle
+  [t, x] = ode45(@v, [0 T], x0);  % compute x(t) over range 0 <= t <= T
+  plot(x(:,1), x(:,2), 'r-');     % plot the path
+end
+</code>
+{{ :gibson:teaching:spring-2016:math445:lecture:cylinderpath2.png?direct&400 |}}

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