UNIT 4 • INITIAL VALUE PROBLEMS (ODE)

Modified Euler's
Predictor-Corrector Method

Standard Euler's method suffers from slope drift. Modified Euler's Method (Heun's Method) predicts a trial step, calculates the destination slope, and averages both slopes for 2nd-order $\mathcal{O}(h^2)$ accuracy!

PREDICTOR-CORRECTOR SCHEMESLOPE AVERAGING

Predictor Step (Euler Trial)

y_{n+1}^{(0)} = y_n + h \cdot f(x_n, y_n)

Corrector Step (Average Slope)

y_{n+1}^{(k+1)} = y_n + \frac{h}{2} \left[ f(x_n, y_n) + f(x_{n+1}, y_{n+1}^{(k)}) \right]

1. Predictor TrialUses starting slope to estimate endpoint

y_{n+1}^{(0)}

2. Slope AverageAverages initial slope

f(x_n,y_n)

and trial slope

f(x_{n+1},y_{n+1})

3. Iterative RefinementRepeat corrector loop until successive estimates converge.

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Given the differential equation:

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

Find the value of $y(0.1)$ using step size $h = 0.1$ and 2 corrector iterations!

STEP 01 • PREDICTOR STEP (TRIAL EULER ESTIMATE)

Evaluate starting slope $f(0, 1) = 0 + 1 = 1.0$ and project trial endpoint:

y_1^{(0)} = y_0 + h \cdot f(x_0, y_0) = 1.0 + (0.1)(1.0) = 1.10000

STEP 02 • CORRECTOR ITERATIONS (AVERAGE SLOPES)

1st Corrector: Evaluate slope at trial point $(0.1, 1.10000)$ :

f(0.1, 1.10000) = 0.1 + 1.1 = 1.20000

y_1^{(1)} = 1.0 + \frac{0.1}{2} [1.0 + 1.20000] = 1.0 + (0.05)(2.2) = 1.11000

2nd Corrector: Re-evaluate slope at updated point $(0.1, 1.11000)$ :

f(0.1, 1.11000) = 0.1 + 1.11000 = 1.21000

y_1^{(2)} = 1.0 + \frac{0.1}{2} [1.0 + 1.21000] = 1.0 + (0.05)(2.21) = 1.11050

FINAL RESULT

Corrected Value: y(0.1) = 1.11050

\boxed{y(0.1) \approx 1.11050} \quad (\text{Exact: } 1.11034)

With 2 corrector iterations, Modified Euler's method improves accuracy from 1.10000 (standard Euler) to 1.11050 (exact: $1.11034$ )!

LABORATORY

Interactive Live Calculator & Visualizer

PREDICTOR-CORRECTOR ODE LABORATORY

Modified Euler (Heun's) Engine

PreviousEuler's Method

Unit 4•20 of 28

NextRunge-Kutta Method

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 4 • INITIAL VALUE PROBLEMS (ODE)

Modified Euler's
Predictor-Corrector Method

PREDICTOR-CORRECTOR SCHEMESLOPE AVERAGING

Predictor Step (Euler Trial)

y_{n+1}^{(0)} = y_n + h \cdot f(x_n, y_n)

Corrector Step (Average Slope)

y_{n+1}^{(k+1)} = y_n + \frac{h}{2} \left[ f(x_n, y_n) + f(x_{n+1}, y_{n+1}^{(k)}) \right]

1. Predictor TrialUses starting slope to estimate endpoint

y_{n+1}^{(0)}

2. Slope AverageAverages initial slope

f(x_n,y_n)

and trial slope

f(x_{n+1},y_{n+1})

3. Iterative RefinementRepeat corrector loop until successive estimates converge.

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Given the differential equation:

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

Find the value of $y(0.1)$ using step size $h = 0.1$ and 2 corrector iterations!

STEP 01 • PREDICTOR STEP (TRIAL EULER ESTIMATE)

Evaluate starting slope $f(0, 1) = 0 + 1 = 1.0$ and project trial endpoint:

y_1^{(0)} = y_0 + h \cdot f(x_0, y_0) = 1.0 + (0.1)(1.0) = 1.10000

STEP 02 • CORRECTOR ITERATIONS (AVERAGE SLOPES)

1st Corrector: Evaluate slope at trial point $(0.1, 1.10000)$ :

f(0.1, 1.10000) = 0.1 + 1.1 = 1.20000

y_1^{(1)} = 1.0 + \frac{0.1}{2} [1.0 + 1.20000] = 1.0 + (0.05)(2.2) = 1.11000

2nd Corrector: Re-evaluate slope at updated point $(0.1, 1.11000)$ :

f(0.1, 1.11000) = 0.1 + 1.11000 = 1.21000

y_1^{(2)} = 1.0 + \frac{0.1}{2} [1.0 + 1.21000] = 1.0 + (0.05)(2.21) = 1.11050

FINAL RESULT

Corrected Value: y(0.1) = 1.11050

\boxed{y(0.1) \approx 1.11050} \quad (\text{Exact: } 1.11034)

With 2 corrector iterations, Modified Euler's method improves accuracy from 1.10000 (standard Euler) to 1.11050 (exact: $1.11034$ )!

LABORATORY

Interactive Live Calculator & Visualizer

PREDICTOR-CORRECTOR ODE LABORATORY

Modified Euler (Heun's) Engine

PreviousEuler's Method

Unit 4•20 of 28

NextRunge-Kutta Method

Modified Euler's Predictor-Corrector Method

Step-by-Step Worked Example

Corrected Value: y(0.1) = 1.11050

Interactive Live Calculator & Visualizer

Modified Euler (Heun's) Engine

NETZ Engine

Modified Euler's Predictor-Corrector Method

Step-by-Step Worked Example

Corrected Value: y(0.1) = 1.11050

Interactive Live Calculator & Visualizer

Modified Euler (Heun's) Engine

Modified Euler's
Predictor-Corrector Method

Modified Euler's
Predictor-Corrector Method