UNIT 4 • INITIAL VALUE PROBLEMS (ODE)

Runge-Kutta 4th Order
RK4 Classical Method

The gold standard of ODE solvers, Runge-Kutta 4th Order (RK4) samples 4 trial slopes across each step interval to achieve 4th-order $\mathcal{O}(h^4)$ accuracy—matching Taylor series precision without computing high-order analytical derivatives!

RK4 FOUR-STAGE STENCIL4TH-ORDER ACCURACY

k_1 = h \cdot f(x_n, y_n)

k_2 = h \cdot f\left(x_n + \frac{h}{2}, y_n + \frac{k_1}{2}\right)

k_3 = h \cdot f\left(x_n + \frac{h}{2}, y_n + \frac{k_2}{2}\right)

k_4 = h \cdot f\left(x_n + h, y_n + k_3\right)

y_{n+1} = y_n + \frac{1}{6} \left( k_1 + 2k_2 + 2k_3 + k_4 \right)

1. Start & End Slopes

k_1

at start

x_n

and

k_4

at full step

x_n + h

2. Midpoint Slopes (2×)

k_2

and

k_3

evaluated at midpoint

x_n + \frac{h}{2}

with double weight.

3. Simpson-Like WeightingWeights

1:2:2:1

divided by 6 yield optimal 4th-order cancelation.

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Solve the initial value problem:

\frac{dy}{dx} = x + y, \quad \text{with initial condition } y(0) = 1

Find the value of $y(0.1)$ using single RK4 step $h = 0.1$ !

STEP 01 • EVALUATE 4 INTERMEDIATE SLOPES (k₁, k₂, k₃, k₄)

Compute each RK4 stage starting from $x_0 = 0, y_0 = 1$ :

k_1 = (0.1)(0 + 1) = 0.10000

k_2 = (0.1)(0.05 + 1.05) = 0.11000

k_3 = (0.1)(0.05 + 1.055) = 0.11050

k_4 = (0.1)(0.1 + 1.1105) = 0.12105

STEP 02 • WEIGHTED AVERAGE INCREMENT

Combine the 4 slopes into the RK4 update equation:

\Delta y = \frac{1}{6} \left[ k_1 + 2k_2 + 2k_3 + k_4 \right]

\Delta y = \frac{1}{6} \left[ 0.10000 + 2(0.11000) + 2(0.11050) + 0.12105 \right]

\Delta y = \frac{1}{6} \left[ 0.10000 + 0.22000 + 0.22100 + 0.12105 \right] = \frac{0.66205}{6} = 0.110342

y(0.1) = y_0 + \Delta y = 1.00000 + 0.110342 = 1.11034

FINAL RESULT

RK4 Value: y(0.1) ≈ 1.11034

\boxed{y(0.1) \approx 1.11034} \quad (\text{Exact: } 2e^{0.1} - 1.1 \approx 1.1103418)

With a single step of size $h = 0.1$ , RK4 computes 1.11034, matching the exact analytical solution to 6 decimal places!

LABORATORY

Interactive Live Calculator & Visualizer

HIGH-PRECISION ODE LABORATORY

Runge-Kutta 4th Order Engine

PreviousModified Euler's Method

Unit 4•21 of 28

NextFitting Straight Lines

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 4 • INITIAL VALUE PROBLEMS (ODE)

Runge-Kutta 4th Order
RK4 Classical Method

RK4 FOUR-STAGE STENCIL4TH-ORDER ACCURACY

k_1 = h \cdot f(x_n, y_n)

k_2 = h \cdot f\left(x_n + \frac{h}{2}, y_n + \frac{k_1}{2}\right)

k_3 = h \cdot f\left(x_n + \frac{h}{2}, y_n + \frac{k_2}{2}\right)

k_4 = h \cdot f\left(x_n + h, y_n + k_3\right)

y_{n+1} = y_n + \frac{1}{6} \left( k_1 + 2k_2 + 2k_3 + k_4 \right)

1. Start & End Slopes

k_1

at start

x_n

and

k_4

at full step

x_n + h

2. Midpoint Slopes (2×)

k_2

and

k_3

evaluated at midpoint

x_n + \frac{h}{2}

with double weight.

3. Simpson-Like WeightingWeights

1:2:2:1

divided by 6 yield optimal 4th-order cancelation.

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Solve the initial value problem:

\frac{dy}{dx} = x + y, \quad \text{with initial condition } y(0) = 1

Find the value of $y(0.1)$ using single RK4 step $h = 0.1$ !

STEP 01 • EVALUATE 4 INTERMEDIATE SLOPES (k₁, k₂, k₃, k₄)

Compute each RK4 stage starting from $x_0 = 0, y_0 = 1$ :

k_1 = (0.1)(0 + 1) = 0.10000

k_2 = (0.1)(0.05 + 1.05) = 0.11000

k_3 = (0.1)(0.05 + 1.055) = 0.11050

k_4 = (0.1)(0.1 + 1.1105) = 0.12105

STEP 02 • WEIGHTED AVERAGE INCREMENT

Combine the 4 slopes into the RK4 update equation:

\Delta y = \frac{1}{6} \left[ k_1 + 2k_2 + 2k_3 + k_4 \right]

\Delta y = \frac{1}{6} \left[ 0.10000 + 2(0.11000) + 2(0.11050) + 0.12105 \right]

\Delta y = \frac{1}{6} \left[ 0.10000 + 0.22000 + 0.22100 + 0.12105 \right] = \frac{0.66205}{6} = 0.110342

y(0.1) = y_0 + \Delta y = 1.00000 + 0.110342 = 1.11034

FINAL RESULT

RK4 Value: y(0.1) ≈ 1.11034

\boxed{y(0.1) \approx 1.11034} \quad (\text{Exact: } 2e^{0.1} - 1.1 \approx 1.1103418)

With a single step of size $h = 0.1$ , RK4 computes 1.11034, matching the exact analytical solution to 6 decimal places!

LABORATORY

Interactive Live Calculator & Visualizer

HIGH-PRECISION ODE LABORATORY

Runge-Kutta 4th Order Engine

PreviousModified Euler's Method

Unit 4•21 of 28

NextFitting Straight Lines

Runge-Kutta 4th Order RK4 Classical Method

Step-by-Step Worked Example

RK4 Value: y(0.1) ≈ 1.11034

Interactive Live Calculator & Visualizer

Runge-Kutta 4th Order Engine

NETZ Engine

Runge-Kutta 4th Order RK4 Classical Method

Step-by-Step Worked Example

RK4 Value: y(0.1) ≈ 1.11034

Interactive Live Calculator & Visualizer

Runge-Kutta 4th Order Engine

Runge-Kutta 4th Order
RK4 Classical Method

Runge-Kutta 4th Order
RK4 Classical Method