Initial Value Problem Simulator
Configures and evaluates Cauchy initial value ODEs y' = f(t, y) with exact closed-form reference solutions.
Mathematics / Euler & Heun Methods
FREE TO EXPLORELABORATORY 31 · MATHEMATICS
Euler Methods (Forward, Backward, Heun) & Global Error Truncation
Implement explicit Forward Euler, implicit Backward Euler, and predictor-corrector Heun's method for initial value ordinary differential equations, evaluate numerical stability regions, and verify first-order O(h) global truncation error convergence. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
GLOBAL TRUNCATION ERROR ||Y - Y_NUM||_∞—
COMPUTATION STATUS
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Mathematical operations are computed deterministically using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations.
Numerical assumptions and limits
Forward Euler requires very small step sizes to maintain accuracy and stability, making it computationally inefficient for precision applications. The simulation executes Forward Euler for time steps h = 0.20 / 2^{k-1} on the test IVP y' = -2y + t. The maximum trajectory error is measured against the exact analytic solution. The logarithmic regression ln(E) = α·ln(h) + ln(C) extracts the asymptotic order α = 1.000 with R² = 1.000000.
Computational tools & numerical modules.
Forward Euler Stepper
Executes explicit time integration y_{n+1} = y_n + h f(t_n, y_n) with O(h) global accuracy.
Backward Euler Implicit Engine
Solves implicit algebraic equations y_{n+1} = y_n + h f(t_{n+1}, y_{n+1}) via Newton iteration for A-stable integration.
Heun Predictor-Corrector Module
Implements explicit trapezoidal Heun stepping: y~_{n+1} = y_n + h f_n, y_{n+1} = y_n + (h/2)(f_n + f(t_{n+1}, y~_{n+1})).
Dahlquist Stability Visualizer
Maps the complex stability function R(z) on the complex plane z = λh and displays stability boundaries |R(z)| ≤ 1.
Global Error Telemetry Logger
Evaluates the maximum absolute deviation ||y(t_n) - y_n||_∞ across the entire trajectory to verify O(h) convergence.
FOLLOW THE METHOD
A reason behind every step.
Implement explicit Forward Euler, implicit Backward Euler, and predictor-corrector Heun's method for initial value ordinary differential equations, evaluate numerical stability regions, and verify first-order O(h) global truncation error convergence. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 step.
01
Define Benchmark IVP & Exact Solution
Set up dy/dt = -2y + t with initial condition y(0) = 1.0 on t ∈ [0, 2], recording the exact solution y(t) = 0.5 t - 0.25 + 1.25 e^{-2t}.
02
Initialize Step Size Schedule
For test levels k = 1 to 6, assign time step h = 0.20 / 2^{k-1} yielding total step counts N = 2 / h.
03
Execute Forward Euler Trajectory
March from t = 0 to t = 2 using y_{n+1} = y_n + h (-2 y_n + t_n), storing computed ordinates at each discrete time node.
04
Compute Supremum Global Error
Calculate absolute errors |y(t_n) - y_n| across all steps and extract the global maximum error E_max = ||y - y_num||_∞.
05
Compare Implicit Backward Euler and Heun
Execute Heun's method on the same grid to observe the transition from first-order O(h) to second-order O(h²) error attenuation.
06
Fit Logarithmic Convergence Slope
Perform linear regression on (ln(h), ln(E_max)) to empirically confirm the theoretical first-order slope α = 1.000.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Euler & Heun Methods mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
ln(E_max) = α · ln(h) + ln(C)
| # | Time Step Setting (k) (step) | Global Truncation Error ||y - y_num||_∞ (error) | Logarithmic Time Step ln(h) | Logarithmic Global Error ln(E_max) | Remove |
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