SciCode / 78.3 / Now write a function to analyze any damped, driven pendulum system to understand…

Background

The Step-Doubling Method

The step-doubling method is a practical and effective way to estimate the truncation error in numerical methods like the Runge-Kutta 4th order (RK4). This method provides an estimate of the error without needing the exact solution, which is often unavailable. Below, I'll explain how to use step-doubling to estimate the global truncation error (GTE) for RK4.

**Perform the Integration with Step Size h and 2h:**
   - **Single Step 2h:** Compute the solution y(t+2h) from y(t) using a single RK4 step with step size 2h.
   - **Two Steps h:** Compute the solution y(t+2h) from y(t) by taking two consecutive RK4 steps with step size h.

2. **Calculate the Difference:**
   - Let y_(2h) denote the result of the single step with step size 2h.
   - Let y_(h,h) denote the result of the two steps with step size h.
   - Compute the difference Δy=y_(h,h)−y_(2h).

3. **Estimate the Error:**
   - The error in the single 2h step can be approximated using the difference Δy scaled by a factor dependent on the order of the method n, which is 4 for RK4. The scaling factor is 2^(n)−1, so for RK4, it is 2⁴−1=15.
   - The estimated error after a single step of 2h is:
     E_(2h)≈(Δy)/(15)

4. **Global Truncation Error (GTE) Estimation:**
   - To estimate the global truncation error over an interval [a,b], perform the step-doubling procedure at several points along the interval, or use it adaptively based on how E_(2h) changes with each segment.
   - Sum these errors to get an estimate of the GTE for the whole interval.

Background

The Step-Doubling Method

The step-doubling method is a practical and effective way to estimate the truncation error in numerical methods like the Runge-Kutta 4th order (RK4). This method provides an estimate of the error without needing the exact solution, which is often unavailable. Below, I'll explain how to use step-doubling to estimate the global truncation error (GTE) for RK4.

**Perform the Integration with Step Size $h$ and $2h$:**
   - **Single Step $2h$:** Compute the solution $y(t + 2h)$ from $y(t)$ using a single RK4 step with step size $2h$.
   - **Two Steps $h$:** Compute the solution $y(t + 2h)$ from $y(t)$ by taking two consecutive RK4 steps with step size $h$.

2. **Calculate the Difference:**
   - Let $y_{2h}$ denote the result of the single step with step size $2h$.
   - Let $y_{h,h}$ denote the result of the two steps with step size $h$.
   - Compute the difference $\Delta y = y_{h,h} - y_{2h}$.

3. **Estimate the Error:**
   - The error in the single $2h$ step can be approximated using the difference $\Delta y$ scaled by a factor dependent on the order of the method $n$, which is 4 for RK4. The scaling factor is $2^n - 1$, so for RK4, it is $2^4 - 1 = 15$.
   - The estimated error after a single step of $2h$ is:
     $$
     E_{2h} \approx \frac{\Delta y}{15}
     $$

4. **Global Truncation Error (GTE) Estimation:**
   - To estimate the global truncation error over an interval $[a, b]$, perform the step-doubling procedure at several points along the interval, or use it adaptively based on how $E_{2h}$ changes with each segment.
   - Sum these errors to get an estimate of the GTE for the whole interval.

Now write a function to analyze any damped, driven pendulum system to understand its dynamic behavior under various conditions. Your function should:

Sweep different timesteps to find the optimized timestep that balances accuracy and time efficiency.
2. Output the trajectory

The combined metric for finding the optimized time should penalize higher computational times while rewarding lower errors.

To better reflect this, we can use a combined metric such as:

Metric=GTE×√(Time)

Global Truncation Error (GTE) is estimated using the step-doubling method. This way, longer computation times will have a more significant penalty, while still prioritizing low errors.

Now write a function to analyze any damped, driven pendulum system to understand its dynamic behavior under various conditions. Your function should:

Sweep different timesteps to find the optimized timestep that balances accuracy and time efficiency.
2. Output the trajectory

The combined metric for finding the optimized time should penalize higher computational times while rewarding lower errors.

To better reflect this, we can use a combined metric such as:

$$
\text{Metric} = \text{GTE} \times \sqrt{\text{Time}}
$$

Global Truncation Error (GTE) is estimated using the step-doubling method. This way, longer computation times will have a more significant penalty, while still prioritizing low errors.