Overview
Runge-Kutta-Fehlberg computes two estimates of different orders from shared stages, using their difference to adjust the next time step.
ExtendedSource checked
Runge-Kutta-Fehlberg
Solves ODEs with paired Runge-Kutta estimates used for adaptive step sizing.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Numerical Methods
- Difficulty
- Advanced
- Family
- ODE Solver
- Tier
- Extended
- Status
- Verified
- Review
- Source Checked
- Sources
- 2
- Formulas
- 1
- Relationships
- 3
- Code example
- Ready
- Visualization
- Ready
Core Concepts
adaptive step sizeembedded methodlocal error estimateode integration
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(stages)
Average
O(steps * stages)
Worst
O(steps * stages)
Space
O(state)
When to Use
- Use when solving equations needs adaptive step size.
- Use when the input matches the assumptions behind oDE Solver.
When Not to Use
- Avoid when the problem is ill-conditioned and no scaling or stability checks are available.
- Avoid when a simpler algorithm gives the required result with less setup.
What It Does
- Solves ODEs with paired Runge-Kutta estimates used for adaptive step sizing
- Uses adaptive step size and embedded method to guide the work.
- Produces approximate numeric result.
Use Cases
- Solving equations
- Approximating integrals
- Working with matrices and differential equations
Inputs
- Numeric function, matrix, or sampled data
- Initial estimate or discretization
- Tolerance
Outputs
- Approximate numeric result
- Error estimate or residual
Key Formulas (1)
Step error
error = ||yhigh order − ylow order||
error = ||y_high_order - y_low_order||Estimates local truncation error from two embedded solutions.
Pseudocode
- 1.Prepare the inputs and state needed for Runge-Kutta-Fehlberg.
- 2.Apply the adaptive step size step while the stopping condition has not been met.
- 3.Update intermediate state and discard invalid candidates.
- 4.Return approximate numeric result.
Canonical Example
- Input
- Runge-Kutta-Fehlberg input with a small representative case
- Output
- Approximate numeric result
- Explanation
- Runge-Kutta-Fehlberg applies adaptive step size to produce approximate numeric result.
Implementation Notes
- Check convergence with residuals or estimated error, not iteration count alone.
- Keep the adaptive step size step separate enough to test independently.
Common Pitfalls
- Ignoring conditioning and residuals can produce plausible-looking but wrong numeric answers.
- Forgetting to check edge cases can make Runge-Kutta-Fehlberg look correct on only ideal inputs.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow the numeric approximation improving against the target function, matrix, or state.
Step 1 of 4: Prepare the inputs and state needed for Runge-Kutta-Fehlberg.
numerical-methods-mechanismNotes
Runge-Kutta-Fehlberg estimates local error from embedded formulas and adapts step size; poor tolerances can still miss important dynamics.
Sources (2)
What each reference supports for this entry.
NASA TR R-315
PaperFehlberg's NASA technical report on low-order Runge-Kutta formulas with step-size control.
Open sourceNumerical Recipes
BookReference text and implementation-oriented treatment for root finding, interpolation, integration, and ODE methods.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Relationships
View full relationship pageAlternative toAdams-Bashforth Method
Both solve ODEs with different step-control strategies.
Variant ofRunge-Kutta Method
Runge-Kutta-Fehlberg is an adaptive Runge-Kutta method.
Related Algorithms
Tags
Numerical MethodsOde SolverAdaptiveNumerical Integration