UNIT 4 • INITIAL VALUE PROBLEMS (ODE)

RK2 Method
Heun & Midpoint

The second-order Runge-Kutta method solves $\frac{dy}{dx} = f(x, y)$ by sampling the slope twice per step. Heun's version: $y_{n+1} = y_n + \frac{h}{2}(k_1 + k_2)$ , with $k_1$ at the start and $k_2$ at the predicted end.

HEUN'S RK2ERROR ~ h²

\begin{gathered} k_1 = f(x_n, y_n), \qquad k_2 = f(x_n + h,\ y_n + h k_1) \\ y_{n+1} = y_n + \frac{h}{2}(k_1 + k_2) \end{gathered}

1. Slope at the Start

k_1 = f(x_n, y_n)

2. Slope at the EndTake an Euler step to guess the end point, and find the slope there:

k_2

3. AverageStep forward with the average slope

\frac{k_1 + k_2}{2}

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Solve $\frac{dy}{dx} = x + y$ , $y(0) = 1$ with $h = 0.1$ by Heun's method, and find $y(0.2)$ .

STEP 01 • STEP 1: x = 0 → 0.1

k_1 = 0 + 1 = 1, \qquad k_2 = f(0.1,\ 1 + 0.1 \times 1) = 0.1 + 1.1 = 1.2

y_1 = 1 + \frac{0.1}{2}(1 + 1.2) = 1.11

STEP 02 • STEP 2: x = 0.1 → 0.2

k_1 = 0.1 + 1.11 = 1.21, \qquad k_2 = f(0.2,\ 1.11 + 0.121) = 0.2 + 1.231 = 1.431

y_2 = 1.11 + \frac{0.1}{2}(1.21 + 1.431) = 1.24205

The exact solution is $y = 2e^x - x - 1$ , so $y(0.2) = 1.242806$ : RK2 is off by about 0.0008.

FINAL CONCLUSION

y(0.2) ≈ 1.24205

y(0.2) \approx 1.24205

The calculator below opens on this problem. Switch to the midpoint version to compare.

LABORATORY

Interactive RK2 Calculator

LABORATORY

RK2 Engine

dy/dx = f(x, y)x₀y₀ = y(x₀)

Step hFind y at x =

Version

Common Questions

What is the second-order Runge-Kutta method?

A one-step method that samples the slope twice per step and combines the two. Its error per step shrinks like

h^3

, so it is far more accurate than Euler's method for the same h.

Is Heun's method the same as the modified Euler method?

Yes: Heun's method averages the slope at the start and at an Euler-predicted end point, which is the modified (improved) Euler method with a single correction.

What is the difference between Heun and midpoint?

Heun averages the slopes at the two ends of the step. Midpoint takes a half step, measures the slope there, and uses only that. Both are second order; for many problems they give similar accuracy.

PreviousRunge-Kutta Methods

Unit 4•35 of 63

NextAdams-Bashforth-Moulton

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 4 • INITIAL VALUE PROBLEMS (ODE)

RK2 Method
Heun & Midpoint

HEUN'S RK2ERROR ~ h²

\begin{gathered} k_1 = f(x_n, y_n), \qquad k_2 = f(x_n + h,\ y_n + h k_1) \\ y_{n+1} = y_n + \frac{h}{2}(k_1 + k_2) \end{gathered}

1. Slope at the Start

k_1 = f(x_n, y_n)

2. Slope at the EndTake an Euler step to guess the end point, and find the slope there:

k_2

3. AverageStep forward with the average slope

\frac{k_1 + k_2}{2}

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Solve $\frac{dy}{dx} = x + y$ , $y(0) = 1$ with $h = 0.1$ by Heun's method, and find $y(0.2)$ .

STEP 01 • STEP 1: x = 0 → 0.1

k_1 = 0 + 1 = 1, \qquad k_2 = f(0.1,\ 1 + 0.1 \times 1) = 0.1 + 1.1 = 1.2

y_1 = 1 + \frac{0.1}{2}(1 + 1.2) = 1.11

STEP 02 • STEP 2: x = 0.1 → 0.2

k_1 = 0.1 + 1.11 = 1.21, \qquad k_2 = f(0.2,\ 1.11 + 0.121) = 0.2 + 1.231 = 1.431

y_2 = 1.11 + \frac{0.1}{2}(1.21 + 1.431) = 1.24205

The exact solution is $y = 2e^x - x - 1$ , so $y(0.2) = 1.242806$ : RK2 is off by about 0.0008.

FINAL CONCLUSION

y(0.2) ≈ 1.24205

y(0.2) \approx 1.24205

The calculator below opens on this problem. Switch to the midpoint version to compare.

LABORATORY

Interactive RK2 Calculator

LABORATORY

RK2 Engine

dy/dx = f(x, y)x₀y₀ = y(x₀)

Step hFind y at x =

Version

Common Questions

What is the second-order Runge-Kutta method?

A one-step method that samples the slope twice per step and combines the two. Its error per step shrinks like

h^3

, so it is far more accurate than Euler's method for the same h.

Is Heun's method the same as the modified Euler method?

Yes: Heun's method averages the slope at the start and at an Euler-predicted end point, which is the modified (improved) Euler method with a single correction.

What is the difference between Heun and midpoint?

Heun averages the slopes at the two ends of the step. Midpoint takes a half step, measures the slope there, and uses only that. Both are second order; for many problems they give similar accuracy.

PreviousRunge-Kutta Methods

Unit 4•35 of 63

NextAdams-Bashforth-Moulton

RK2 Method Heun & Midpoint

Step-by-Step Worked Example

y(0.2) ≈ 1.24205

Interactive RK2 Calculator

RK2 Engine

Common Questions

NETZ Engine

RK2 Method Heun & Midpoint

Step-by-Step Worked Example

y(0.2) ≈ 1.24205

Interactive RK2 Calculator

RK2 Engine

Common Questions

RK2 Method
Heun & Midpoint

RK2 Method
Heun & Midpoint