In scientific computing and engineering, many real-world problems are modeled using ordinary differential equations (ODEs). However, not all ODEs can be solved analytically. To address this, numerical methods such as Euler's method, Runge-Kutta methods, and Heun’s method (also known as the improved Euler's method or the explicit trapezoidal rule) are used.

Consider a first-order initial value problem:
$$\frac{dy}{dx} = f(x,y), \; y(x_0) = y_0$$
We aim to approximate the value of y at discrete points using Heun’s method.

Given the initial condition $(x_0, y_0)$, and a step size $h$, Heun’s method computes the next value \(y_{n+1}\) as follows:

Step 1: Predictor (Euler's estimate)
$$y^* = y_n + h f(x_n, y_n)$$

Step 2: Corrector
$$y_{n+1} = y_n + \frac{h}{2} \left[f(x_n, y_n) + f(x_n + h, y^* )\right]$$

Let us solve over the interval [x0, xn] with step size h:
1. Initialize: Set x0, y0, h, and number of steps n.
2. Loop for each step:
- Compute the predictor: \(y^ = y_n + h \cdot f(x_n, y_n)\)
- Compute the corrector: \(y_{n+1} = y_n + \frac{h}{2} [f(x_n, y_n) + f(x_n + h,y^)]\)
- Update: \(x_{n+1} = x_n + h\)

Given:
$$\frac{dy}{dx} = x + y, \; y(0) = 1$$
Find: y at \(x = 0.1\) using Heun’s Method with \(h = 0.1\)

Step 1:
$$f(x_0, y_0) = f(0,1) = 0 + 1 = 1$$
$$y^* = 1 + 0.1 \times 1 = 1.1$$

Step 2:
$$f(x_1, y^*) = f(0.1,1.1) = 0.1 + 1.1 = 1.2$$
$$y_{1} = 1 + \frac{1}{2} \cdot (1 + 1.2) = 1 + 0.05 \times 2.2 = 1.11$$

Result: y(0.1) ≈ 1.11

Heun’s method can be viewed as applying the trapezoidal rule for numerical integration. Instead of using a single tangent (like Euler's method), it averages the slope at the beginning and the predicted endpoint of the interval. This generally reduces the error significantly.

Advantages of Heun’s Method:
- Better accuracy than Euler's method with the same step size.
- Still relatively simple and easy to implement.
- Reduces local truncation error.

Limitations:
- Requires two evaluations of the function f(x,y) per step.
- May still not be accurate enough for stiff equations or highly nonlinear systems.
- For even better accuracy, higher-order Runge-Kutta methods may be preferred.

Applications:
- Engineering simulations (e.g., heat transfer, fluid dynamics)
- Control systems
- Electrical circuits (RLC models)
- Population dynamics
- Any domain where ODEs govern system behavior.

• Heun’s Method is a second-order improvement over Euler’s method for numerically solving ODEs.
• It uses a predictor-corrector approach: Euler's step to estimate the next value, and a corrected average slope to refine the result.
• It is a simple yet powerful technique, often serving as a stepping stone toward more sophisticated methods like the classical Runge-Kutta.
• The method balances efficiency and accuracy, making it a practical choice for many engineering problems.