Overview
Runge-Kutta methods combine multiple derivative estimates to produce a more accurate state update.
FoundationSource checked
Runge-Kutta Method
Approximates differential equations by sampling several slopes per step.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Simulation
- Difficulty
- Intermediate
- Family
- Numerical Integration
- Tier
- Foundation
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 4
- Code example
- Not ready
- Visualization
- Ready
- Aliases
- RK4
Core Concepts
slope samplingweighted averagetime stepintegration accuracysimulated stateintegral approximationnumeric approximationnumerical integration
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(t)
Average
O(t)
Worst
O(t)
Space
O(1)
When to Use
- Use it when ordinary differential equations need more accuracy than Euler integration.
- Use fourth-order Runge-Kutta for a common balance of accuracy and cost.
- Use adaptive variants when step size should respond to local error.
When Not to Use
- Avoid fixed-step methods when the system has rapidly changing error behavior.
- Avoid it when derivative evaluations are too expensive for the chosen order.
- Avoid assuming higher order fixes a poor model or unstable time step.
What It Does
- Samples slopes within a time step
- Combines slope estimates into one update
Use Cases
- Accurate physics simulation
- Solving ordinary differential equations
- Engineering models
Inputs
- Current state
- Derivative function
- Time step
Outputs
- Next state
Key Formulas (1)
RK4 update
ynext = y + dt · (k1 + 2·k2 + 2·k3 + k4) / 6
y_next = y + dt * (k1 + 2*k2 + 2*k3 + k4) / 6Combines four slope estimates into one higher-accuracy step.
Pseudocode
- 1.Evaluate the derivative at the start of the step
- 2.Estimate intermediate states
- 3.Evaluate derivatives at those intermediate states
- 4.Combine the derivative estimates with weights
- 5.Update the state
Canonical Example
- Input
- dy/dt = y, y(0) = 1, step from t = 0 to t = 0.1
- Output
- an approximation near e^0.1
- Explanation
- A fourth-order Runge-Kutta step combines several slope estimates across the interval.
Implementation Notes
- Keep each intermediate state separate from the original state until the final weighted update.
- Document the specific Runge-Kutta tableau or formula being implemented.
- Test on differential equations with known analytical solutions.
Common Pitfalls
- Combining intermediate slopes with the wrong weights.
- Updating the state before all intermediate derivative estimates are computed.
- Using a fixed step size without checking error on difficult systems.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow several slope samples combining into a curved step.
Step 1 of 5: Evaluate the derivative at the start of the step
multi-slope-stepNotes
Runge-Kutta names a family of methods; order, step-size policy, and coefficients define the specific algorithm.
Sources (1)
What each reference supports for this entry.
Numerical Recipes
BookNumerical methods reference covering Runge-Kutta methods for ordinary differential equations.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Alternative toEuler Integration
Simulation · Foundation
Runge-Kutta methods use multiple derivative estimates for higher accuracy.
Compare with thisRelationships
View full relationship pageAlternative toEuler Integration
Runge-Kutta methods use multiple derivative estimates for higher accuracy.
GeneralizesRunge-Kutta-Fehlberg
Runge-Kutta-Fehlberg is an adaptive Runge-Kutta method.
Related Algorithms
Tags
SimulationNumerical IntegrationNumerical Methods