gibson:teaching:spring-2015:math445:lab11
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gibson:teaching:spring-2015:math445:lab11 [2015/04/13 13:32] gibson |
gibson:teaching:spring-2015:math445:lab11 [2015/04/15 19:36] (current) gibson |
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| **Problem 3.** Superimpose on your previous plot the path of a raindrop that falls at | **Problem 3.** Superimpose on your previous plot the path of a raindrop that falls at | ||
| - | $\vec{x} = (x,y) = (0.9,0.9)$. We're assuming the velocity of the raindrop as it drips down is proportional to the negative of the gradient of the height. That is, $\vec{v} = -c \vec{\nabla} h$, | + | $\vec{x} = (x,y) = (0.9,0.9)$. We're assuming the velocity of the raindrop as it drips down is proportional to the negative of the gradient of the height. That is, $\vec{v} = -c \vec{\nabla} h $, |
| - | where $\vec{v} = (dx/dt, dy/dt)$ is the $x,y$ velocity of the raindrop. For convenience set $c=1$ (this won't change the path). Then compute the path of the raindrop numerically using //forward Euler time-stepping//, | + | where $\vec{v} = d\vec{x}/dt$ is the $x,y$ velocity of the raindrop. For convenience set $c=1$ (this won't change the path). Then compute the path of the raindrop numerically using //forward Euler time-stepping//, |
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| - | \vec{x}(t+\Delta t) = \vec{x}(t) + \Delta t \vec{v} | + | \vec{x}(t+\Delta t) = \vec{x}(t) + \Delta t \, \vec{v}(\vec{x}) |
| - | \end{eqnarry*} | + | \end{eqnarray*} |
| {{:gibson:teaching:spring-2015:math445:montesol_path.png?direct&400|}} | {{:gibson:teaching:spring-2015:math445:montesol_path.png?direct&400|}} | ||
| - | **Problem 4.** Then show the path of the raindrop in 3d, superimposed on the surface plot. | + | **Problem 4.** Then show the path of the raindrop in 3d, superimposed on the surface plot. You will need to use Matlab's ''plot3'' function. |
| {{:gibson:teaching:spring-2015:math445:montesol_path3d.png?direct&400|}} | {{:gibson:teaching:spring-2015:math445:montesol_path3d.png?direct&400|}} | ||
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| **Problem 5.** Recompute the plot of problem 3 using Matlab's ''ode45'' numerical integration function instead of forward Euler. | **Problem 5.** Recompute the plot of problem 3 using Matlab's ''ode45'' numerical integration function instead of forward Euler. | ||
| - | **Problem 6.** Plot the paths of 100 raindrops that fall randomly on the mountains. | + | **Problem 6.** Plot the paths of 100 raindrops that fall randomly on the mountains. Show the hundred raindrop paths on both the contour plot and the 3d surface plot. |
gibson/teaching/spring-2015/math445/lab11.1428957140.txt.gz · Last modified: 2015/04/13 13:32 by gibson
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