Overview
Stable Fluids is a simulation method that advances physical or agent state using forces, constraints, collisions, or transport. It is useful when stability, plausibility, or physical behavior needs an explicit update rule.
CoreSource checked
Stable Fluids
Stable Fluids advances physical or agent state under forces, constraints, or collisions.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Simulation
- Difficulty
- Intermediate
- Family
- Simulation
- Tier
- Core
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 1
- Code example
- Ready
- Visualization
- Ready
Core Concepts
time stepstate updateconstraint or forceconstraints
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(n t)
Average
O(n t)
Worst
O(n t)
Space
O(n)
When to Use
- Use Stable Fluids when the simulated system matches its force, collision, or constraint model.
- Use it when time-step limits and stability checks can be enforced.
When Not to Use
- Avoid when the time step, collision model, or constraints make the method unstable.
- Avoid when visual plausibility is enough and a cheaper approximation is acceptable.
What It Does
- Computes simulation within Simulation workloads (O(n t) time, O(n) space).
- Works by It is useful when stability, plausibility, or physical behavior needs an explicit update rule.
- Maintains key invariants across time step, state update, constraint or force.
Use Cases
- Physics simulation
- Animation
- Fluid or particle systems
Inputs
- Physical state
- Forces, constraints, or collision geometry
- Time step
Outputs
- Updated state
- Collision response
- Simulated motion or field
Key Formulas (1)
Semi-Lagrangian advection
qnext(x) = qcurrent(x − δt · velocity(x))
q_next(x) = q_current(x - delta_t * velocity(x))Samples backward along velocity to advect quantities stably.
Pseudocode
- 1.Initialize positions, velocities, fields, constraints, or agents
- 2.Compute forces, collisions, constraints, or transport
- 3.Advance state by the time step
- 4.Resolve constraints or collisions
- 5.Return the updated simulation state
Canonical Example
- Input
- state at time t with delta_t = 0.016
- Output
- state at time t + delta_t
- Explanation
- Stable Fluids applies its update rule to advance the simulation by one step.
Implementation Notes
- Clamp or adapt time steps when stability depends on them.
- Keep collision, force, and constraint phases independently testable.
Common Pitfalls
- Large time steps can create unstable motion even when equations are correct.
- Resolving constraints in the wrong order can inject energy or jitter.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow particles, bodies, or fields evolving as each simulation step is applied.
Step 1 of 5: Initialize positions, velocities, fields, constraints, or agents
simulationNotes
Stable Fluids operates in O(n t) time and O(n) space within the Simulation family. Key governing relation: q_next(x) = q_current(x - delta_t * velocity(x)). Often compared with Euler Integration and Verlet Integration.
Sources (1)
What each reference supports for this entry.
Fluid Simulation for Computer Graphics
BookFluid and simulation reference for state updates or fluid behavior in Stable Fluids.
Explore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Relationships
View full relationship pageOften paired withEuler Integration
Stable Fluids can be combined with Euler Integration in simulation pipelines that need compatible state updates.
Related Algorithms
Tags
Simulation