gibson:teaching:spring-2016:math445:lecture:timestepping
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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] |
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| {{ :gibson:teaching:spring-2016:math445:lecture:cylinderpath1.png?direct&400 }} | {{ :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==== | ====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> | <code matlab> | ||
| Line 108: | Line 110: | ||
| </code> | </code> | ||
| + | {{ :gibson:teaching:spring-2016:math445:lecture:cylinderpath3.png?400 |}} | ||
| + | ---- | ||
| ====Problem 3==== | ====Problem 3==== | ||
| Line 126: | Line 129: | ||
| axis equal | axis equal | ||
| axis tight | axis tight | ||
| - | xlim([-xmax,xmax]) | + | xlim([-3,3]) |
| - | ylim([-ymax,ymax]) | + | ylim([-2,2]) |
| </code> | </code> | ||
| Line 150: | Line 153: | ||
| </file> | </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 |}} | ||
| + | |||
| + | |||
gibson/teaching/spring-2016/math445/lecture/timestepping.1460653220.txt.gz · Last modified: 2016/04/14 10:00 by gibson
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