Euler-Heun Method

Definition

It is Numerical Method used to find the approximate solution of an 221106 Ordinary Differential Equations an explicit predictor-corrector method that combines Euler’s method (Predictor) with the trapezoidal rule (Corrector) to achieve higher accuracy.

Formula

$$y_{i+1}=\underset{\text{Trapezoidal}}{\underbrace{y_i+\frac{h}{2}[f(t_i,y_i)+f(t_{i+1},\stackrel{\text{Euler}}{\overbrace{y_i+hf(t_i,y_i)}})]}}$$

Derivation

Starting point: The exact solution satisfies

$$y(t_{i+1})=y(t_i)+{\int }_{t_i}^{t_{i+1}}f(t,y(t)),dt$$

Step 1: Apply trapezoid rule to approximate the integral:

$${\int }_{t_i}^{t_{i+1}}f(t,y(t)),dt\approx \frac{h}{2}[f(t_i,y(t_i))+f(t_{i+1},y(t_{i+1}))]$$

where $h=t_{i+1}-t_i$.

Step 2: Replace exact values with approximations:

• $y(t_i)\approx y_i$ (known)
• $y(t_{i+1})\approx y_{i+1}$ (unknown)

This gives:

$$y_{i+1}=y_i+\frac{h}{2}[f(t_i,y_i)+f(t_{i+1},y_{i+1})]$$

Step 3: Make explicit by approximating $y_{i+1}$ on the right side using Euler’s method:

$${\tilde{y}}_{i+1}=y_i+hf(t_i,y_i)$$

Step 4: Final Euler-Heun formula:

$$\boxed{y_{i+1}=y_i+\frac{h}{2}[f(t_i,y_i)+f(t_{i+1},y_i+hf(t_i,y_i))]}$$

Properties of Euler-Heun:

• Explicit method (no implicit equation to solve)
• One-step method (uses only $y_i$ to compute $y_{i+1}$)
• Consistency order 2 (local error $O(h^3)$, global error $O(h^2)$)
• 2 stages: requires 2 evaluations of $f$ per step
• Runge-Kutta method of order 2
• Stability region larger than Euler’s method
• Also called improved Euler or modified Euler method