Overview
Euler Integration updates a value by adding the current derivative multiplied by a time step.
FoundationSource checked
Euler Integration
Approximates motion by stepping forward using the current rate of change.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Simulation
- Difficulty
- Beginner
- Family
- Numerical Integration
- Tier
- Foundation
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 17
- Code example
- Not ready
- Visualization
- Ready
- Aliases
- Forward Euler
Core Concepts
state derivativetime stepexplicit updateintegration errorsimulated statephysical stateintegral approximationiterative update
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 for simple simulations, prototypes, and teaching numerical integration.
- Use it when small time steps make its error acceptable.
- Use semi-implicit variants for some physical systems when stability is more important.
When Not to Use
- Avoid it for stiff systems or long simulations where error accumulation dominates.
- Avoid large time steps that make the update unstable.
- Avoid it when energy preservation or high accuracy is required.
What It Does
- Advances state in fixed steps
- Uses current velocity or derivative
Use Cases
- Simple physics demos
- Quick differential equation approximations
- Teaching numerical integration
Inputs
- Current state
- Derivative or velocity
- Time step
Outputs
- Next state
Key Formulas (1)
Euler update
statenext = state + dt · derivative(state)
state_next = state + dt * derivative(state)Advances a state using the derivative at the start of the time step.
Pseudocode
- 1.Read the current state
- 2.Compute the current derivative
- 3.Multiply the derivative by the time step
- 4.Add the result to the state
- 5.Repeat for each time step
Canonical Example
- Input
- position = 0, velocity = 2, dt = 0.5
- Output
- position = 1
- Explanation
- The update adds velocity * dt, so 0 + 2 * 0.5 = 1.
Implementation Notes
- Use new_state = state + derivative(state, time) * dt.
- Keep units consistent across state, derivative, and time step.
- Compare against a smaller time step to estimate integration error.
Common Pitfalls
- Using a time step that is too large for the dynamics.
- Confusing explicit Euler with semi-implicit Euler.
- Assuming small local error will not accumulate over many steps.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow a moving point advancing with straight step estimates.
Step 1 of 5: Read the current state
time-step-motionNotes
Euler integration is simple and cheap, but its error and stability limits should be visible in tests.
Sources (1)
What each reference supports for this entry.
Numerical Recipes
BookNumerical methods reference for scalar optimization, interpolation, integration, and practical algorithm tradeoffs.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Has variantsEuler-Maruyama Method
Numerical Methods · Extended
Euler-Maruyama adapts Euler-style updates to stochastic differential equations.
Compare with thisAlternativesRunge-Kutta Method
Simulation · Foundation
Runge-Kutta methods use multiple derivative estimates for higher accuracy.
Compare with thisAlternative toVerlet Integration
Simulation · Core
Euler and Verlet are simple time integration schemes with different stability behavior.
Compare with thisOften paired withFLIP Fluid Simulation
Simulation · Core
FLIP Fluid Simulation can be combined with Euler Integration in simulation pipelines that need compatible state updates.
Compare with thisRelationships
View full relationship pageGeneralizesSemi-Implicit Euler
Semi-implicit Euler modifies the ordering of velocity and position updates.
Alternative toVerlet Integration
Euler and Verlet are simple time integration schemes with different stability behavior.
Related Algorithms
Tags
SimulationPhysicsNumerical IntegrationIterative