History Buffer Register
Maintains sliding window of previous solution points (y_n, y_{n-1}, y_{n-2}, y_{n-3}) and derivative vectors.
Mathematics / Adams-Bashforth-Moulton Multistep
FREE TO EXPLORELABORATORY 33 · MATHEMATICS
Adams-Bashforth-Moulton Linear Multistep Predictor-Corrector Methods
Implement the four-step explicit Adams-Bashforth predictor and implicit Adams-Moulton corrector algorithm (PECE), utilize self-starting RK4 initialization, analyze the Dahlquist root condition, and verify fourth-order O(h⁴) global convergence with only two function evaluations per step. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
ABM GLOBAL 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
Linear multistep methods are not self-starting, requiring auxiliary starter routines whenever integration begins or restarts. The simulation executes the 4-step Adams-Bashforth-Moulton predictor-corrector method for time steps h = 0.20 / 2^{k-1}. Global trajectory error is measured against the exact analytic solution of y' = -y + t². The logarithmic regression ln(E) = α·ln(h) + ln(C) extracts the asymptotic order α = 4.000 with R² = 1.000000.
Computational tools & numerical modules.
RK4 Bootstrap Starter
Generates the initial 3 history steps with 4th-order accuracy to jumpstart the multistep pipeline.
Adams-Bashforth Predictor
Extrapolates polynomial through 4 historical slopes to predict y*_{n+1} with O(h⁴) local error.
Adams-Moulton Corrector
Interpolates polynomial including predicted slope f(t_{n+1}, y*_{n+1}) to refine y_{n+1}.
Milne Local Error Estimator
Computes Milne's estimate ε ≈ (19/270) |y*_{n+1} - y_{n+1}| using difference between predictor and corrector.
Log-Log Regression Slope Processor
Performs linear regression on (ln(h), ln(E_max)) to verify the theoretical fourth-order slope α = 4.000.
FOLLOW THE METHOD
A reason behind every step.
Implement the four-step explicit Adams-Bashforth predictor and implicit Adams-Moulton corrector algorithm (PECE), utilize self-starting RK4 initialization, analyze the Dahlquist root condition, and verify fourth-order O(h⁴) global convergence with only two function evaluations per step. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 step.
01
Initialize IVP & Benchmark Manifold
Set up dy/dt = -y + t² on t ∈ [0, 2] with y(0) = 1.0 and exact solution y(t) = t² - 2t + 2 - e^{-t}.
02
Assign Step Size Schedule
For test settings k = 1 to 6, assign time step h = 0.20 / 2^{k-1} across refinement levels.
03
Bootstrap History Buffer via RK4
Execute 3 steps of RK4 to populate initial history registers (t₁, y₁, f₁), (t₂, y₂, f₂), and (t₃, y₃, f₃).
04
Execute PECE Multistep Cycle
For n ≥ 3, compute AB4 predictor y*_{n+1}, evaluate f*_{n+1}, apply AM4 corrector to obtain y_{n+1}, and update history buffer.
05
Evaluate Maximum Global Trajectory Error
Record the supremum norm discrepancy ||y(t_n) - y_n||_∞ against the exact analytic reference solution.
06
Verify Asymptotic Fourth-Order Slope
Plot ln(E_max) versus ln(h) and compute the linear regression slope to verify α = 4.000 with R² = 1.000000.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Adams-Bashforth-Moulton Multistep mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
ln(E_max) = α · ln(h) + ln(C)
| # | Time Step Setting (k) (step) | ABM Global Error ||y - y_num||_∞ (error) | Logarithmic Time Step ln(h) | Logarithmic Global Error ln(E_max) | Remove |
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