gibson:teaching:spring-2012:iam95:hw1
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gibson:teaching:spring-2012:iam95:hw1 [2012/02/29 09:20] gibson [Problem 2] |
gibson:teaching:spring-2012:iam95:hw1 [2012/02/29 10:25] (current) gibson [Problem 2] |
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| long-term stable equilibrium state predicted by the reduced-order model? | long-term stable equilibrium state predicted by the reduced-order model? | ||
| - | **(d)** Use a numerical ODE integration routine to integrate your ODE models from (a), (b), %%(c)%% | + | **(d)** Use a numerical ODE integration routine to integrate your ODE models from (b) and %%(c)%% |
| and the time-integration code from problem 1 for the PDE, for $r=1/8$. For each model and the PDE simulation, | and the time-integration code from problem 1 for the PDE, for $r=1/8$. For each model and the PDE simulation, | ||
| produce phase plots of $a_3(t)$ versus $a_1(t)$ for a handful of initial conditions scattered in the | produce phase plots of $a_3(t)$ versus $a_1(t)$ for a handful of initial conditions scattered in the | ||
| Line 55: | Line 55: | ||
| as well. You should see rapid approach to the center manifold followed by slow evolution on it. | as well. You should see rapid approach to the center manifold followed by slow evolution on it. | ||
| - | **(e)** How well do the ODE models and the reduced-order equilibrium match the asymptotic behavior of | + | **(e)** How accurate are the ODE models and the reduced-order equilibrium in the long term, as a function |
| - | the "real" system as a function of $r$? Take the PDE simulation approximate the real" system | + | of $r$? Assume that the PDE simulation gives an accurate numerical solution of the Swift-Hohenberg |
| - | + | equation. Using the fixed initial condition $w(x,0) = 0.1 cos x + 0.1 cos 3x$, produce a log-log plot | |
| - | Use a fixed initial condition $w(x,0) = 0.1 cos x + 0.1 cos 3x$ | + | of asymptotic error |
| - | + | ||
| - | Compare the long-term behavior of the ODE models and the equilibrium state predicted by center | + | |
| - | manifold theory to the long-term PDE simulation for a variety of vale of $r$. | + | |
| - | + | ||
| - | the equilibrium state | + | |
| - | predicted by center manifold theory, using a fixed initial condition $w(x,0) = 0.1 cos x + 0.1 cos 3x$ | + | |
| - | but changing the value of $r$. | + | |
| - | + | ||
| - | + | ||
| - | and PDE integrations approach the equilibrium | + | |
| - | state predicted by the reduced-order model? Why or why not? Use the L2-norm long-term error | + | |
| <latex> | <latex> | ||
| - | err = \lim_{t \rightarrow \infty} \sqrt{ \int_0^{2\pi} |\hat{w}(x,t) - w_{PDE}(x,t)|^2 dx} | + | err = \lim_{t \rightarrow \infty} \sqrt{ \int_0^{2\pi} |\hat{w}(x,t) - w(x)|^2 dx} |
| </latex> | </latex> | ||
| - | to measure the accuracy of the models, where $\hat{w}$ is the model and $w_{EQB}$ is the | + | versus $r$ where $w(x)$ is the asymptotic state of the PDE simulation and $\hat{w}$ is first the |
| - | equilibrium solution | + | the ODE model from (b), second the reduced-order model from %%(c)%%, and third the reduced-order |
| - | Produce a log-log plot of this error measure versus $r$ for each of the three ODE models. | + | equilibrium. Plot these as three lines in log-log plot of error versus $r$. I suggest |
| - | I suggest using $r = 1/16, 1/8, 1/4, 1/2,$ and $1$. | + | using $r = 1/16, 1/8, 1/4, 1/2,$ and $1$. |
| - | Note that the ODE systems for (a) and (b) will be stiff, in that the high-order coefficients evolve | + | Note that the ODE systems for (b) and %%(c)%% will be stiff, in that the high-order coefficients evolve |
| very rapidly until the system equilibrates to and moves slowly on the center manifold. You might need to | very rapidly until the system equilibrates to and moves slowly on the center manifold. You might need to | ||
| use a stiff ODE integrator instead of the classic explicit schemes like RK4. | use a stiff ODE integrator instead of the classic explicit schemes like RK4. | ||
gibson/teaching/spring-2012/iam95/hw1.1330536047.txt.gz · Last modified: 2012/02/29 09:20 by gibson
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