Complexity Explorer Santa Fe Institute

Ordinary Differential Equations

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10.1 Unit 3 Homework » Homework Part 3

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Instructions: (The following questions are refered to in the solutions video as Part A, Part B, Part C, Part D, and Part E of Question 3.)

As you observed in the previous two problems, forward and backward Euler both make errors. Combining the two of them, however, can be surprisingly effective. Implement a solver that averages a forward and backward Euler step:

$\vec{x}(t+\Delta t)=\vec{x}(t)+\frac{\Delta t}{2}\bigg[f(\vec{x}(t))+f(\vec{x}{__F}{__E}(t+\Delta t))\bigg]$

Here $\vec{x}{__F}{__E}(t+\Delta t)$ is the result of applying one step of the forward Euler algorithm starting from $\vec{x}$ with time-step $\Delta t$ . This method averages the slope at that point with the slope at the originial point and uses that averaged slope to move forward. This is sometimes called the trapezoidal method in the literature.
Question 1: (a) Using the solver made from the above instructions section, generate a trajectory of the simple harmoic oscillator ODEs with , , and . Start from the initial condition withe a time-step of . What is $[x(t=0.5), v(t=0.5)]^T$ ?
Answer 1
Answer 2
Answer 3
Question 2: (b) Generate a 500-point trajectory of the same ODE system from with a time-step of $\Delta t = 0.01$ . Which of the following describes this trajectory?
Answer 1: A. Ellipses
Answer 2: B. Outward spiral
Answer 3: C. Inward spiral
Answer 4: D. Chaos
Question 3: (c) Remember that the simple harmonic oscillator has no damping (friction). Keeping this in mind, does the trajectory that you generated in part (b) of this problem seem more accurate than the trajectory computer by either of your forward or backward Euler code from Problems 1 and 2?
Answer 1: Yes
Answer 2: No
Question 4: (d) Now generate a 5000-point trajectory of the same ODE system from with a timestep of . How is this plot qualitatively different than the plot you generated that was only a 500-point trajectory?
Answer 1: A. The trajectory has "fattened", taking up more space
Answer 2: B. No change
Answer 3: C. The spiral has changed direction
Answer 4: D. The trajectory is now chaotic
Question 5: (e) What could be the cause of this difference?
Answer 1: A. No change, so no effect.
Answer 2: B. Numerical error is violating the conservation of energy property of a simple harmoic oscillator with. Since this system is a conservative system, a symplectic integrator should be used.
Answer 3: C. The change in trajectory length allow for a bifurcation to chaos.
Answer 4: D. The pendulum arm has reversed direction.

10.1 Unit 3 Homework » Homework Part 3

Quiz scores are NOT recorded.

Instructions

Question 1

Question 2

Question 3

Question 4

Question 5