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

Derivation

Starting point: The exact solution satisfies

Step 1: Apply trapezoid rule to approximate the integral:

where .

Step 2: Replace exact values with approximations:

  • (known)
  • (unknown)

This gives:

Step 3: Make explicit by approximating on the right side using Euler’s method:

Step 4: Final Euler-Heun formula:

Properties of Euler-Heun:

  • Explicit method (no implicit equation to solve)
  • One-step method (uses only to compute )
  • Consistency order 2 (local error , global error )
  • 2 stages: requires 2 evaluations of per step
  • Runge-Kutta method of order 2
  • Stability region larger than Euler’s method
  • Also called improved Euler or modified Euler method