Six-Stage Fehlberg Evaluator
Computes stage slopes k₁ through k₆ according to Erwin Fehlberg's optimal coefficient tableau.
Mathematics / Adaptive RKF45 Integration
FREE TO EXPLORELABORATORY 34 · MATHEMATICS
Adaptive Step-Size Control & Embedded Runge-Kutta-Fehlberg (RKF45)
Implement the embedded Runge-Kutta-Fehlberg (RKF45) algorithm, evaluate the six-stage shared Butcher tableau, compute the embedded local truncation error estimate, and verify the fifth-order O(h⁵) local error scaling governing optimal step-size adaptation. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
EMBEDDED LOCAL ERROR ESTIMATE |Y₅ - Y₄|—
COMPUTATION STATUS
Ready to execute Step = 1.
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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
RKF45 is an explicit embedded pair; on stiff differential equations, the step controller will drastically shrink h to satisfy stability rather than accuracy. The simulation evaluates the embedded local truncation error of the six-stage RKF45 algorithm across step sizes h = 0.20 / 2^{k-1}. The local error estimate scales with the fifth power of h. The logarithmic regression ln(E_local) = α·ln(h) + ln(C) extracts the asymptotic order α = 5.000 with R² = 1.000000.
Computational tools & numerical modules.
Dual-Order State Combiner
Forms simultaneous 4th-order baseline y₄ and 5th-order reference y₅ without extra function evaluations.
Local Truncation Estimator
Evaluates the embedded difference vector E = y₅ - y₄, measuring leading O(h⁵) local error.
Step Size Adaptation Controller
Implements PI/PID step adjustment h_{opt} = S · h · (TOL / ||E||)^{1/5} to maintain error within specified tolerance.
Step Acceptance Logic Port
Accepts steps where E ≤ TOL and repeats rejected steps with reduced step size h_{new}.
Local Error Scaling Verifier
Performs linear regression on (ln(h), ln(E_local)) to confirm the theoretical fifth-order slope α = 5.000.
FOLLOW THE METHOD
A reason behind every step.
Implement the embedded Runge-Kutta-Fehlberg (RKF45) algorithm, evaluate the six-stage shared Butcher tableau, compute the embedded local truncation error estimate, and verify the fifth-order O(h⁵) local error scaling governing optimal step-size adaptation. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 step.
01
Initialize Nonlinear Dynamical Test System
Set up dy/dt = -y² + t on t ∈ [0, 2] with initial state y(0) = 0.5 and user-specified error tolerance TOL = 10⁻⁶.
02
Assign Test Step Sizes
Set step sizes h = 0.20 / 2^{k-1} across refinement settings k = 1 to 6 to analyze local error scaling.
03
Evaluate Six Fehlberg Stages
Compute k₁, k₂, k₃, k₄, k₅, k₆ using Fehlberg's rational weight matrix in 64-bit precision.
04
Extract Embedded Local Error
Compute difference E_local = h |(1/360) k₁ - (128/4275) k₃ - (2197/75240) k₄ + (1/50) k₅ + (2/55) k₆|.
05
Execute Step Adaptation Formula
Calculate recommended next step h_{opt} = 0.9 · h · (TOL / E_local)^{0.2} with safety clamping [0.2, 5.0].
06
Verify Fifth-Order Scaling Exponent
Plot ln(E_local) versus ln(h) and compute the linear regression slope to verify α = 5.000 with R² = 1.000000.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Adaptive RKF45 Integration mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
ln(E_local) = α · ln(h) + ln(C)
| # | Local Step Size Level (k) (step) | Embedded Local Error Estimate |y₅ - y₄| (local error) | Logarithmic Time Step ln(h) | Logarithmic Local Error ln(E_local) | Remove |
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