13.2. Runge–Kutta Second-Order Method (RK2)

Also known as the Improved Euler Method or Heun’s Method, RK2 provides a better approximation than Euler’s method by considering the slope at both the beginning and an intermediate point.

The RK2 method, or the Improved Euler Method, improves upon the basic Euler's method. It does this by taking two slope evaluations instead of one: the first at the beginning of the interval and the second at a midpoint. This averaging leads to a more accurate estimate of the function's value after each step.

Imagine you are driving a car and want to estimate how far you will travel in a short time. If you only check your speed at the start (Euler's method), you might get an inaccurate distance. But if you check your speed at both the start and halfway through (RK2), you will have a better estimate of how far you've actually traveled.

Algorithm Given: • Differential equation: = 𝑓(𝑥,𝑦) • Initial condition: 𝑦(𝑥 ) = 𝑦 • Step size: ℎ Steps:

1. Compute intermediate slope: 𝑘₁ = ℎ𝑓(𝑥ₙ,𝑦ₙ)
2. Compute slope at mid-point using 𝑘₁: 𝑘₂ = ℎ𝑓(𝑥ₙ + ℎ/2, 𝑦ₙ + 𝑘₁/2)
3. Update solution: 𝑦ₙ₊₁ = 𝑦ₙ + 𝑘₂

To use the RK2 method, follow these steps: First, calculate the slope at the initial point, which gives you the initial change in y (this is k1). Then, use k1 to find the slope at the midpoint of the interval, providing a better estimate of the slope at that point (this is k2). Finally, update your value of y using k2, leading to the next step in your approximation.

Think of it like planning a trip. First, you note the distance at the start. Then, while driving, you make a note of how your speed changes halfway. By using both notes to calculate your progress, you will have a more accurate estimate of how far you've come rather than relying on just the initial speed.

Example (RK2): Solve 𝑑𝑦/𝑑𝑥 = 𝑥 + 𝑦, with 𝑦(0) = 1, step size ℎ = 0.1, find 𝑦(0.1). Step 1: 𝑥 = 0, 𝑦 = 1, ℎ = 0.1 Calculate: • 𝑘₁ = ℎ𝑓(𝑥ₙ,𝑦ₙ) = 0.1(0 + 1) = 0.1 • 𝑘₂ = 0.1𝑓(0.05,1 + 0.05) = 0.1(0.05 + 1.05) = 0.1 × 1.10 = 0.11 • 𝑦₁ = 1 + 0.11 = 1.11

In this example, we are asked to solve the ordinary differential equation where we know the initial condition and the step size. We first calculate k1, which represents the slope at the start. Then we calculate k2, which takes into account the change at the midpoint. Finally, we update our estimate of y to find its value at the next step.

Let's say you're baking and checking the temperature of the cake. The first reading (k1) tells you if it’s on track to bake well, and the second reading (k2) gives you a better understanding of how it’s cooking through the middle. By taking both into account, you make adjustments to ensure a perfectly baked cake!

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