03_Get_Oscillations - Problem 3

[mom15102]

At the end of Problem 3, it asks "How many time steps does each method need to converge to the same results?"
Is this question asking when they converge to each other, or when they converge to zero?

[evg16108]

The question refers to when they converge to each other. In this case, you know some information about the rk-2 method and implicit method like the fact that they have order 2 convergence while euler step has order 1 convergence. I don't think we are expected to calculate and plot the relative error because you would need the analytical solution but for example you can visually see how they converge by using the above facts. Let me know if this helps!

[mtf16101]

The question refers to when they converge to each other. In this case, you know some information about the rk-2 method and implicit method like the fact that they have order 2 convergence while euler step has order 1 convergence. I don't think we are expected to calculate and plot the relative error because you would need the analytical solution but for example you can visually see how they converge by using the above facts. Let me know if this helps!

We can find when they converge to each other by change the value of N (or more specifically, dt) until all the lines look the same. However, the question seems to want multiple step values so this is why I am unsure.

[evg16108]

dt is different for the different methods based on their convergence rate.

[rcc02007]

All the integration methods should converge to the function:

x= lambda t: exp(-w*z*t)*(cos(sqrt(1-z**2)*w*t)+sin(sqrt(1-z**2)*w*t))

where z=damping ratio and w=natural frequency

spring-mass-damper equation