HW 5 for CS 4220
You may (and probably should) talk about problems with the each other, with the TAs, and with me, providing attribution for any good ideas you might get. Your final write-up should be your own.
The equilibrium discrete Klein-Gordon equation has the form
Two common variants are
discrete_kg (generic function with 1 method)
F
J
dFdλ
Warm-up
Characterize all possible solutions to the
Answer
Playing with the problem
Consider a solution at
The following slider controls the coupling constant
Answer
Convergence starts to break down around just past
Continuation in
Write a code that simply runs (unguarded) Newton for a range of increasing
Answer
Pseudo-arclength continuation
Use the pseudo-arclength continuation techniques from the April 22 lecture to produce a bifurcation diagram beyond what can be achieved by just controlling
Answer
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Plotting
As a for-free, we include a visualization tool for all the states visited along the continuation curve. The slider moves the current point along the curve.
Bifurcation computation
We can characterize the turning point where
where
Using an initial guess computed from the continuation code in the previous part, apply Newton iteration to compute the turning point to full accuracy.