channelflow.org

====== Differences ====== This shows you the differences between two versions of the page.

--- gibson:teaching:spring-2012:iam95:hw1 [2012/02/08 10:45]
gibson
+++ gibson:teaching:spring-2012:iam95:hw1 [2012/02/29 10:25] (current)
gibson [Problem 2]
@@ Line 1: / Line 1: @@
 ====== IAM 950 HW1 ======
-**Problem 1:** In class we derived via Taylor expansion the following approximation
+=====Problem 1=====
+ In class we derived via Taylor expansion the following approximation
 for the exponential growth rate $\sigma$ of a sinusoidal perturbation of wavenumber
 $q$ for a Type I-s instability, near the critical wavenumber ($q \approx q_c$), and close
@@ Line 18: / Line 19: @@
 and length scales govern?
-**(c)** Adapt the time-integration code for the Kuramoto-Sivashisky equation to Swift-Hohenberg,
+**%%(c)%%** Adapt the time-integration code for the Kuramoto-Sivashisky equation to Swift-Hohenberg,
 and use it to verify your answers to (b) with numerics.
+=====Problem 2=====
+Derive the reduced-order ODE model for the Swift-Hohenberg equation just
+above threshhold and at critical wavenumber and compare its behavior to numerical simulations
+of the PDE, in the following steps:
+**(a)** Starting from the Swift-Hohenberg PDE on the periodic domain $[0, 2\pi]$ and with $0<r<<1$,
+expand the unknown field $w(x,t)$ in Fourier modes
+<latex>
+w(x,t) = \sum_k a_k(t) e^{ikx}
+</latex>
+Substitute into the PDE, exploit orthogonality of the Fourier basis, and truncate to four modes $a_0, a_1, a_2, a_3$
+to obtain a system of four ODEs in the four coefficients (class notes 2012-02-08). You can fix the phase
+to be even in $x$ and use a cosine Fourier expansion, as we did in class, or use a complex Fourier basis
+as written above to represent $w(x,t)$ at arbitrary phase. In the latter case you will need to include the
+complex conjugates of $a_0, a_1, a_2, a_3$ in the expansion.
+**(b)** Show that the equations for $a_0$ and $a_2$ decouple, leaving a 2d system
+of ODEs in just $a_1$ and $a_3$.
+**%%(c)%%** Use Center Manifold Reduction to derive an algebraic model for $a_3$ in terms of $a_1$, and
+then use that result to form a reduced-order nonlinear evolution equation for $a_1$ alone. What is the
+long-term stable equilibrium state predicted by the reduced-order model?
+**(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,
+produce phase plots of $a_3(t)$ versus $a_1(t)$ for a handful of initial conditions scattered in the
+$a_1, a_3$ plane. (If you used the complex Fourier representation, plot $Re a_3$ versus $Re a_1$ and
+choose real-valued initial conditions.) Plot the approximation of the center manifold on the phase plane
+as well. You should see rapid approach to the center manifold followed by slow evolution on it.
+**(e)** How accurate are the ODE models and the reduced-order equilibrium in the long term, as a function
+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
+of asymptotic error
+<latex>
+err = \lim_{t \rightarrow \infty} \sqrt{ \int_0^{2\pi} |\hat{w}(x,t) - w(x)|^2 dx}
+</latex>
+versus $r$ where $w(x)$ is the asymptotic state of the PDE simulation and $\hat{w}$ is first the
+the ODE model from (b), second the reduced-order model from %%(c)%%, and third the reduced-order
+equilibrium. Plot these as three lines in log-log plot of error versus $r$. I suggest
+using $r = 1/16, 1/8, 1/4, 1/2,$ and $1$.
+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
+use a stiff ODE integrator instead of the classic explicit schemes like RK4.

channelflow.org

User Tools

Site Tools

Page Tools