Stable fluids

Jos Stam

article1999SIGGRAPH1,913 citations

Introduces an unconditionally stable numerical method for solving the Navier-Stokes equations via semi-Lagrangian advection and implicit diffusion, enabling fast, real-time, and blowup-free fluid simulation in computer graphics.

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Simulating realistic fluid behaviors such as smoke, water, fire, and liquid paint is a vital requirement across visual effects, digital art, and interactive graphics. Historically, physics-based fluid animations relied on explicit numerical methods that become unstable and numerically blow up if large simulation time steps or grid resolutions are used. These numerical breakdowns forced animators to use tiny time steps, severely limiting simulation speed, increasing computational costs, and preventing real-time, interactive manipulation of fluid motion.

The article demonstrates an unconditionally stable method for solving the full, three-dimensional Navier-Stokes equations governing fluid flow. The primary objective is to enable real-time, interactive fluid modeling for computer graphics without risking numerical failure, regardless of the time step size chosen by the user.

To achieve this, the author replaced traditional explicit schemes with a four-step modular framework: adding external forces, transporting velocity by tracing particles backward along flow paths via the method of characteristics, resolving internal viscosity through an implicit diffusion solver, and applying a projection step to enforce mass conservation. The resulting software was implemented in a compact C library of roughly 500 lines of code and tested within an interactive system on an SGI Octane workstation, simulating both two-dimensional and three-dimensional fluid flows and advecting scalar properties such as density, temperature, and visual texture coordinates across grid sizes ranging from 16-cubed to 36-cubed.

The findings confirm three key breakthroughs. First, the algorithm achieves unconditional numerical stability, allowing users to take significantly larger time steps without simulation blowups. Second, the solver operates with linear computational complexity, or near-linear complexity when using Fast Fourier Transforms for periodic boundaries, enabling real-time interactive performance during 3D simulations. Third, by advecting texture coordinates alongside dynamic density fields, the system successfully produces highly detailed, wispy gaseous effects even on coarse, low-resolution computational grids.

These results demonstrate that computer graphics tools can prioritize visual realism and user interactivity over strict physical exactness. By eliminating catastrophic numerical instabilities, production teams and software developers can drastically reduce simulation turnaround times, avoid expensive re-runs, and allow artists to manipulate fluid phenomena directly in real time.

Software teams and tool developers should adopt this stable semi-Lagrangian framework for interactive fluid and gas effects, and combine the fluid solver with dynamic solid textures to maximize visual detail at minimal computational expense. When implementing periodic boundaries, teams should leverage Fourier transform methods for simple and efficient pipelines.

Decision-makers should note that the primary trade-off of this solver is high numerical dissipation, meaning fluids lose energy and dampen more rapidly than in real-world physics, making the method unsuitable for engineering or aerodynamic validation. Additionally, the baseline implementation does not address dynamically moving free boundaries, such as splashing water surfaces. Nonetheless, for computer graphics and interactive artistic tools, confidence in the algorithm's stability, interactivity, and visual quality is exceptionally high.

Alias|Wavefront fluid models non-newtonian fluids stable model Stable fluids Navier-Stokes equations fluid solvers

Stam (1999).pdf

Paper: Large steps in cloth simulation, D. Baraff et al. (1998). This seminal work pioneered the use of implicit integration to achieve stable, large-time-step physical simulation in computer graphics, establishing the numerical paradigm that Stam adapted for fluid flow.

Paper: Physics-informed neural networks (PINNs) for fluid mechanics: a review, Shengze Cai et al. (2021). This review surveys modern physics-informed neural network approaches that build beyond classical numerical grid solvers for the Navier-Stokes equations.

Abstract

Building animation tools for fluid-like motions is an important and challenging problem with many applications in computer graphics. The use of physics-based models for fluid flow can greatly assist in creating such tools. Physical models, unlike key frame or procedural based techniques, permit an animator to almost effortlessly create interesting, swirling fluid-like behaviors. Also, the interaction of flows with objects and virtual forces is handled elegantly. Until recently, it was believed that physical fluid models were too expensive to allow real-time interaction. This was largely due to the fact that previous models used unstable schemes to solve the physical equations governing a fluid. In this paper, for the first time, we propose an unconditionally stable model which still produces complex fluid-like flows. As well, our method is very easy to implement. The stability of our model allows us to take larger time steps and therefore achieve faster simulations. We have used our model in conjuction with advecting solid textures to create many fluid-like animations interactively in two- and three-dimensions.

Topics

fluid models non-newtonian fluids stable model key frames differential equations

Keywords

Stable fluids Navier-Stokes equations fluid solvers fluid dynamics numerical dissipation divergent free fields Poisson equation velocity fields boundary conditions diffusion operator

1 Introduction
2 Stable Navier-Stokes
2.1 Basic Equations
2.2 Method of Solution
2.3 Periodic Boundaries and the FFT
2.4 Moving Substances through the Fluid
3 Our Solver
3.1 Setup
3.2 The Simulator
4 Results
5 Conclusions
Acknowledgments
A Method of Characteristics
B FISHPAK Routines
References

Knowls

Knowl 1 — Projected Incompressible Navier-Stokes Evolution Equation
equation
The motion of an incompressible fluid with constant density $\rho$ and kinematic viscosity $\nu$ , subject to an external force $\mathbf{f}(\mathbf{x}, t)$ , is governed by the incompressible Navier-Stokes equations:
$\nabla \cdot \mathbf{u} = 0$ $\frac{\partial \mathbf{u}}{\partial t} = -(\mathbf{u} \cdot \nabla)\mathbf{u} - \frac{1}{\rho}\nabla p + \nu \nabla^2 \mathbf{u} + \mathbf{f}$
where $\mathbf{u}(\mathbf{x}, t)$ is the velocity vector field and $p(\mathbf{x}, t)$ is the scalar pressure field in $\mathbb{R}^n$ ( $n = 2, 3$ ).

By the Helmholtz-Hodge decomposition, any vector field $\mathbf{w}$ uniquely decomposes into a divergence-free component $\mathbf{u}$ and the gradient of a scalar field $q$ :
$\mathbf{w} = \mathbf{u} + \nabla q, \quad \text{where } \nabla \cdot \mathbf{u} = 0$
Taking the divergence of both sides gives a Poisson equation for $q$ :
$\nabla^2 q = \nabla \cdot \mathbf{w}$
subject to homogeneous Neumann boundary conditions $\frac{\partial q}{\partial n} = 0$ on the domain boundary $\partial D$ . An orthogonal projection operator $\mathbf{P}$ is defined as $\mathbf{P}\mathbf{w} = \mathbf{w} - \nabla q$ .

Applying $\mathbf{P}$ to both sides of the momentum equation, and using the identities $\mathbf{P}\mathbf{u} = \mathbf{u}$ and $\mathbf{P}(\nabla p) = \mathbf{0}$ , eliminates the pressure term to yield the single projected evolution equation for velocity:
$\frac{\partial \mathbf{u}}{\partial t} = \mathbf{P}\left( -(\mathbf{u} \cdot \nabla)\mathbf{u} + \nu \nabla^2 \mathbf{u} + \mathbf{f} \right)$
Knowl 2 — Operator-Splitting Time-Stepping for Fluid Velocity
algorithm
Given the velocity field $\mathbf{u}(\mathbf{x}, t) = \mathbf{w}_0(\mathbf{x})$ at time $t$ , the simulation advances the velocity to time $t + \Delta t$ through four sequential steps over time step $\Delta t$ :
1. Force addition: $\mathbf{w}_1(\mathbf{x}) = \mathbf{w}_0(\mathbf{x}) + \Delta t \, \mathbf{f}(\mathbf{x}, t)$ .
2. Advection: $\mathbf{w}_2(\mathbf{x}) = \mathbf{w}_1(\mathbf{p}(\mathbf{x}, -\Delta t))$ , where $\mathbf{p}(\mathbf{x}, -\Delta t)$ is obtained by backtracing the streamline through $\mathbf{w}_1$ over duration $\Delta t$ .
3. Viscous diffusion: implicit solve of $(I - \nu \Delta t \nabla^2)\mathbf{w}_3(\mathbf{x}) = \mathbf{w}_2(\mathbf{x})$ , where $I$ is the identity operator.
4. Divergence-free projection: solve $\nabla^2 q = \nabla \cdot \mathbf{w}_3$ and set $\mathbf{w}_4(\mathbf{x}) = \mathbf{w}_3(\mathbf{x}) - \nabla q(\mathbf{x})$ .
The final divergence-free velocity is $\mathbf{u}(\mathbf{x}, t + \Delta t) = \mathbf{w}_4(\mathbf{x})$ .
Input: Velocity field $\mathbf{u}$ , force field $\mathbf{f}$ , kinematic viscosity $\nu$ , time step $\Delta t$
Output: Divergence-free velocity field $\mathbf{u}$ at time $t + \Delta t$
$\mathbf{w}_0 \leftarrow \mathbf{u}$
$\mathbf{w}_1 \leftarrow \mathbf{w}_0 + \Delta t \cdot \mathbf{f}$
for each grid cell coordinate $\mathbf{x}$ do
$\mathbf{x}_0 \leftarrow \text{TraceParticle}(\mathbf{x}, \mathbf{w}_1, -\Delta t)$
$\mathbf{w}_2(\mathbf{x}) \leftarrow \text{LinInterp}(\mathbf{x}_0, \mathbf{w}_1)$
$\mathbf{w}_3 \leftarrow \text{SolveLinearSystem}((I - \nu \Delta t \nabla^2)\mathbf{w}_3 = \mathbf{w}_2)$
$q \leftarrow \text{SolvePoisson}(\nabla^2 q = \nabla \cdot \mathbf{w}_3)$
$\mathbf{w}_4 \leftarrow \mathbf{w}_3 - \nabla q$
$\mathbf{u} \leftarrow \mathbf{w}_4$
return $\mathbf{u}$
The algorithm has $O(N)$ computational complexity per step when using linear-time Poisson/multigrid solvers for a grid of $N$ cells, and is unconditionally stable for any time step $\Delta t > 0$ .
Knowl 3 — Unconditionally Stable Semi-Lagrangian Advection
model/method
To solve the nonlinear advection equation $\frac{\partial a}{\partial t} = -(\mathbf{u} \cdot \nabla)a$ for any scalar or vector field $a$ over time step $\Delta t$ , the method of characteristics is solved backwards in time.

Along a characteristic path $\mathbf{p}(\mathbf{x}_0, t)$ satisfying $\frac{d}{dt}\mathbf{p}(\mathbf{x}_0, t) = \mathbf{u}(\mathbf{p}(\mathbf{x}_0, t), t)$ with $\mathbf{p}(\mathbf{x}_0, 0) = \mathbf{x}_0$ , the total derivative vanishes:
$\frac{d}{dt} a(\mathbf{p}(\mathbf{x}_0, t), t) = \frac{\partial a}{\partial t} + \mathbf{u} \cdot \nabla a = 0$
Hence, the field value is invariant along the streamline: $a(\mathbf{p}(\mathbf{x}_0, \Delta t), t + \Delta t) = a(\mathbf{x}_0, t)$ .

To compute the updated field value at each grid node $\mathbf{x}$ :
1. Trace a particle backwards from $\mathbf{x}$ through the velocity field $\mathbf{u}$ over time $-\Delta t$ (using a 2nd-order Runge-Kutta integrator or adaptive step refinement) to locate the origin point $\mathbf{x}_0 = \mathbf{p}(\mathbf{x}, -\Delta t)$ .
2. Interpolate the field $a$ at position $\mathbf{x}_0$ from the previous time step using multilinear interpolation: $a(\mathbf{x}, t + \Delta t) = a(\mathbf{x}_0, t)$ .
Because multilinear interpolation computes a convex combination of existing grid node values, the maximum norm is bounded:
$\max_{\mathbf{x}} |a(\mathbf{x}, t + \Delta t)| \le \max_{\mathbf{x}} |a(\mathbf{x}, t)|$
This guarantees unconditional stability regardless of time step size $\Delta t$ , removing the standard Courant-Friedrichs-Lewy (CFL) limit $\Delta t < \Delta x / \|\mathbf{u}\|$ .
Knowl 4 — Implicit Backward Euler Formulation for Viscous Diffusion
model/method
Viscous diffusion in fluid flow is governed by the parabolic partial differential equation:
$\frac{\partial \mathbf{w}}{\partial t} = \nu \nabla^2 \mathbf{w}$
where $\nu$ is the kinematic viscosity.

Applying an implicit backward Euler discretization over time step $\Delta t$ yields:
$(I - \nu \Delta t \nabla^2) \mathbf{w}^{t+\Delta t}(\mathbf{x}) = \mathbf{w}^t(\mathbf{x})$
where $I$ is the identity operator.

On a Cartesian grid with spatial cell spacings $\Delta x, \Delta y, \Delta z$ , central differencing of $\nabla^2$ produces a sparse, diagonally dominant, symmetric positive-definite linear system for each spatial velocity component:
$\left(1 + 2\nu\Delta t\sum_{d \in \{x,y,z\}} \frac{1}{\Delta d^2}\right) w_{i,j,k}^{t+\Delta t} - \nu\Delta t\left(\frac{w_{i+1,j,k}^{t+\Delta t} + w_{i-1,j,k}^{t+\Delta t}}{\Delta x^2} + \frac{w_{i,j+1,k}^{t+\Delta t} + w_{i,j-1,k}^{t+\Delta t}}{\Delta y^2} + \frac{w_{i,j,k+1}^{t+\Delta t} + w_{i,j,k-1}^{t+\Delta t}}{\Delta z^2}\right) = w_{i,j,k}^t$
This implicit system is unconditionally stable for arbitrary $\Delta t > 0$ and any viscosity $\nu \ge 0$ .
Knowl 5 — Fast Fourier Transform Fluid Solver for Periodic Domains
algorithm
For fluids defined on an $n$ -dimensional torus (periodic boundary conditions), spatial differential operators simplify in the Fourier domain: the spatial gradient $\nabla$ becomes multiplication by $i\mathbf{k}$ , where $i = \sqrt{-1}$ and $\mathbf{k}$ is the spatial frequency wave vector with magnitude $k = \|\mathbf{k}\|$ .

In the Fourier domain, diffusion and projection operators take explicit algebraic forms:
- Viscous diffusion acts as an isotropic low-pass filter: $\hat{\mathbf{w}}_3(\mathbf{k}) = \frac{\hat{\mathbf{w}}_2(\mathbf{k})}{1 + \nu \Delta t k^2}$
- Divergence-free projection maps each Fourier vector onto the plane normal to $\mathbf{k}$ : $\hat{\mathbf{w}}_4(\mathbf{k}) = \hat{\mathbf{w}}_3(\mathbf{k}) - \frac{\mathbf{k} \cdot \hat{\mathbf{w}}_3(\mathbf{k})}{k^2}\mathbf{k}$
Input: Velocity field $\mathbf{w}_0$ , external force $\mathbf{f}$ , kinematic viscosity $\nu$ , time step $\Delta t$
Output: Divergence-free velocity field $\mathbf{w}_4$ at time $t + \Delta t$
$\mathbf{w}_1 \leftarrow \mathbf{w}_0 + \Delta t \cdot \mathbf{f}$
for each spatial grid point $\mathbf{x}$ do
     $\mathbf{w}_2(\mathbf{x}) \leftarrow \mathbf{w}_1(\mathbf{p}(\mathbf{x}, -\Delta t))$
$\hat{\mathbf{w}}_2 \leftarrow \text{FFT}(\mathbf{w}_2)$
for each frequency wave vector $\mathbf{k}$ do
     $k^2 \leftarrow \|\mathbf{k}\|^2$
     $\hat{\mathbf{w}}_3(\mathbf{k}) \leftarrow \hat{\mathbf{w}}_2(\mathbf{k}) / (1 + \nu \cdot \Delta t \cdot k^2)$
    if $k^2 > 0$ then
         $\hat{\mathbf{w}}_4(\mathbf{k}) \leftarrow \hat{\mathbf{w}}_3(\mathbf{k}) - \frac{\mathbf{k} \cdot \hat{\mathbf{w}}_3(\mathbf{k})}{k^2} \mathbf{k}$
    else
         $\hat{\mathbf{w}}_4(\mathbf{0}) \leftarrow \hat{\mathbf{w}}_3(\mathbf{0})$
$\mathbf{w}_4 \leftarrow \text{FFT}^{-1}(\hat{\mathbf{w}}_4)$
return $\mathbf{w}_4$
The algorithm exhibits $O(N \log N)$ complexity for an $N$ -voxel grid.
Knowl 6 — Advection, Diffusion, and Dissipation of Scalar Fields
model/method
Non-reactive substances transported by the fluid (such as smoke density, dust, temperature, or texture coordinates) are governed by an advection-diffusion-dissipation equation for a scalar field $a(\mathbf{x}, t)$ :
$\frac{\partial a}{\partial t} = -(\mathbf{u} \cdot \nabla)a + \kappa_a \nabla^2 a - \alpha_a a + S_a$
where $\mathbf{u}$ is the fluid velocity, $\kappa_a \ge 0$ is the diffusion coefficient, $\alpha_a \ge 0$ is the dissipation rate, and $S_a(\mathbf{x}, t)$ is an external source term.

Over time step $\Delta t$ , the four terms are updated sequentially:
1. Source addition: $a_1(\mathbf{x}) = a_0(\mathbf{x}) + \Delta t \, S_a(\mathbf{x}, t)$
2. Semi-Lagrangian transport: $a_2(\mathbf{x}) = a_1(\mathbf{p}(\mathbf{x}, -\Delta t))$
3. Implicit diffusion: solve $(I - \kappa_a \Delta t \nabla^2) a_3(\mathbf{x}) = a_2(\mathbf{x})$
4. Dissipation: $a_4(\mathbf{x}) = \frac{a_3(\mathbf{x})}{1 + \Delta t \, \alpha_a}$
The updated scalar field is $a(\mathbf{x}, t + \Delta t) = a_4(\mathbf{x})$ , maintaining unconditional stability.
Knowl 7 — Flow Detail Synthesis via Multi-Layer Texture Advection and Periodic Blending
model/method
To generate detailed, wispy visual features in gaseous flows without requiring high-resolution simulation grids, solid texture coordinates are advected through the fluid velocity field $\mathbf{u}$ using the scalar transport solver.

To prevent distortions that accumulate as texture coordinates stretch over time:
1. Three independent sets of texture coordinates $\mathbf{T}_1(\mathbf{x}, t), \mathbf{T}_2(\mathbf{x}, t), \mathbf{T}_3(\mathbf{x}, t)$ are advected simultaneously by the velocity field.
2. The coordinate sets have staggered cycle phases and are periodically reset to their unperturbed coordinate values $\mathbf{T}_i(\mathbf{x}) = \mathbf{x}$ .
3. At each render frame, the synthesized texture value is evaluated as a smooth weighted superposition (linear blend) of texture lookups across the three coordinate sets, smoothly fading out highly distorted coordinates while fading in freshly reset coordinates.
This produces high-frequency flowing details on coarse simulation grids (e.g., $16^3$ to $30^3$ ).
Knowl 8 — Cell-Centered Finite Difference Formulation of the Projection Step
algorithm
Physical quantities are discretized on a grid with cell dimensions $D[0], D[1], D[2]$ , where velocity components $U[0], U[1], U[2]$ and scalar values $S$ are located at voxel centers $(i+0.5, j+0.5, k+0.5) \cdot \mathbf{D}$ .

The divergence-free projection $\mathbf{w}_4 = \mathbf{w}_3 - \nabla q$ is solved as follows:
1. The discrete divergence $F[i,j,k] = \nabla \cdot \mathbf{w}_3$ is computed via central differences on the input velocity $\mathbf{U}$ : $F[i,j,k] = \frac{1}{2}\left(\frac{U[0]_{i+1,j,k} - U[0]_{i-1,j,k}}{D[0]} + \frac{U[1]_{i,j+1,k} - U[1]_{i,j-1,k}}{D[1]} + \frac{U[2]_{i,j,k+1} - U[2]_{i,j,k-1}}{D[2]}\right)$
2. The Poisson equation $\nabla^2 S = F$ for potential $S = q$ is solved using a 7-point finite difference Laplacian subject to boundary conditions.
3. The divergence-free velocity $\mathbf{U}_1 = \mathbf{w}_4$ is obtained by subtracting the central-difference gradient of $S$ from the previous field $\mathbf{U}_0 = \mathbf{w}_3$ : $U_1[0]_{i,j,k} = U_0[0]_{i,j,k} - \frac{S_{i+1,j,k} - S_{i-1,j,k}}{2 D[0]}$ $U_1[1]_{i,j,k} = U_0[1]_{i,j,k} - \frac{S_{i,j+1,k} - S_{i,j-1,k}}{2 D[1]}$ $U_1[2]_{i,j,k} = U_0[2]_{i,j,k} - \frac{S_{i,j,k+1} - S_{i,j,k-1}}{2 D[2]}$
Knowl 9 — Numerical Dissipation in Stable Fluid Solvers
limitation
The unconditional stability of the Stable Fluids solver introduces significant numerical dissipation (artificial damping):
1. Linear interpolation during semi-Lagrangian advection smooths out high-frequency spatial gradients and vortical structures at every time step.
2. Backward Euler implicit integration introduces first-order temporal numerical damping.
As a consequence, the simulated fluid dampens faster than real physical flows, making the method unsuitable for engineering simulations requiring precise quantitative bounds on physical quantities. In interactive computer graphics, however, this dissipation prevents numerical blow-up and stabilizes the simulation under large interactive time steps $\Delta t$ .

Coverage note — No substantial contributed material was omitted; specific third-party library routine parameter wrappers (FISHPAK POIS3D) and generic graphical user interface routines were excluded as standard implementation specifics.

References

1.M. B. Abbott. Computational Fluid Dynamics: An Introduction for Engineers. Wiley, New York, 1989.
2.J. X. Chen, N. da Vittoria Lobo, C. E. Hughes, and J. M. Moshell. Real-Time Fluid Simulation in a Dynamic Virtual Environment. IEEE Computer Graphics and Applications, pages 52–61, May-June 1997.
3.A. J. Chorin and J. E. Marsden. A Mathematical Introduction to Fluid Mechanics. Springer-Verlag. Texts in Applied Mathematics 4. Second Edition., New York, 1990.
4.D. Ebert, K. Musgrave, D. Peachy, K. Perlin, and S. Worley. Texturing and Modeling: A Procedural Approach. AP Professional, 1994.
5.D. S. Ebert, W. E. Carlson, and R. E. Parent. Solid Spaces and Inverse Particle Systems for Controlling the Animation of Gases and Fluids. The Visual Computer, 10:471–483, 1994.
6.N. Foster and D. Metaxas. Realistic Animation of Liquids. Graphical Models and Image Processing, 58(5):471–483, 1996.
7.N. Foster and D. Metaxas. Modeling the Motion of a Hot, Turbulent Gas. In Computer Graphics Proceedings, Annual Conference Series, 1997, pages 181–188, August 1997.
8.M. N. Gamito, P. F. Lopes, and M. R. Gomes. Two-dimensional Simulation of Gaseous Phenomena Using Vortex Particles. In Proceedings of the 6th Eurographics Workshop on Computer Animation and Simulation, pages 3–15. Springer-Verlag, 1995.
9.M. Griebel, T. Dornseifer, and T. Neunhoeffer. Numerical Simulation in Fluid Dynamics: A Practical Introduction. SIAM, Philadelphia, 1998.
10.W. Hackbusch. Multi-grid Methods and Applications. Springer Verlag, Berlin, 1985.
11.F. H. Harlow and J. E. Welch. Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface. The Physics of Fluids, 8:2182–2189, December 1965.
12.M. Kass and G. Miller. Rapid, Stable Fluid Dynamics for Computer Graphics. ACM Computer Graphics (SIGGRAPH ’90), 24(4):49–57, August 1990.
13.N. Max, R. Crawfis, and D. Williams. Visualizing Wind Velocities by Advecting Cloud Textures. In Proceedings of Visualization ’92, pages 179–183, Los Alamitos CA, October 1992. IEEE CS Press.
14.W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes in C. The Art of Scientific Computing. Cambridge University Press, Cambridge, 1988.
15.W. T. Reeves. Particle Systems. A Technique for Modeling a Class of Fuzzy Objects. ACM Computer Graphics (SIGGRAPH ’83), 17(3):359–376, July 1983.
16.M. Shinya and A. Fournier. Stochastic Motion - Motion Under the Influence of Wind. In Proceedings of Eurographics ‘92, pages 119–128, September 1992.
17.K. Sims. Particle Animation and Rendering Using Data Parallel Computation. ACM Computer Graphics (SIGGRAPH ’90), 24(4):405–413, August 1990.
18.K. Sims. Choreographed Image Flow. The Journal Of Visualization And Computer Animation, 3:31–43, 1992.
19.J. Stam. A General Animation Framework for Gaseous Phenomena. ERCIM Research Report, R047, January 1997. http://www.ercim.org/publications/technical_reports/047-abstract.html.
20.J. Stam and E. Fiume. Turbulent Wind Fields for Gaseous Phenomena. In Proceedings of SIGGRAPH ’93, pages 369–376. Addison-Wesley Publishing Company, August 1993.
21.J. Stam and E. Fiume. Depicting Fire and Other Gaseous Phenomena Using Diffusion Processes. In Proceedings of SIGGRAPH ’95, pages 129–136. Addison-Wesley Publishing Company, August 1995.
22.P. N. Swarztrauber and R. A. Sweet. Efficient Fortran Subprograms for the Solution of Separable Elliptic Partial Differential Equations. ACM Transactions on Mathematical Software, 5(3):352–364, September 1979.
23.J. Wejchert and D. Haumann. Animation Aerodynamics. ACM Computer Graphics (SIGGRAPH ’91), 25(4):19–22, July 1991.
24.L. Yaeger and C. Upson. Combining Physical and Visual Simulation. Creation of the Planet Jupiter for the Film 2010. ACM Computer Graphics (SIGGRAPH ’86), 20(4):85–93, August 1986.

Citation

MLA

Stam, J. “Stable Fluids”. Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques - SIGGRAPH '99, 1999, pp. 121–28, https://doi.org/10.1145/311535.311548.

APA

Stam, J. (1999). Stable fluids. Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques - SIGGRAPH '99, 121–128. https://doi.org/10.1145/311535.311548

Chicago

Stam, J. 1999. “Stable Fluids”. Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques - SIGGRAPH '99, 121–28. https://doi.org/10.1145/311535.311548.

Harvard

Stam, J. (1999) “Stable fluids”, Proceedings of the 26th annual conference on Computer graphics and interactive techniques - SIGGRAPH '99. ACM Press, pp. 121–128. Available at: https://doi.org/10.1145/311535.311548.

Vancouver

1. Stam J (1999) Stable fluids. In: Proceedings of the 26th annual conference on Computer graphics and interactive techniques - SIGGRAPH '99. ACM Press, pp 121–128

BibTeX

@inproceedings{Stam_1999, series={SIGGRAPH ’99}, title={Stable fluids}, url={http://dx.doi.org/10.1145/311535.311548}, DOI={10.1145/311535.311548}, booktitle={Proceedings of the 26th annual conference on Computer graphics and interactive techniques  - SIGGRAPH ’99}, publisher={ACM Press}, author={Stam, Jos}, year={1999}, pages={121–128}, collection={SIGGRAPH ’99} }

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Stable fluids

Abstract

Topics

Keywords

Table of Contents

Knowls

Knowl 1 — Projected Incompressible Navier-Stokes Evolution Equation

Knowl 2 — Operator-Splitting Time-Stepping for Fluid Velocity

Knowl 3 — Unconditionally Stable Semi-Lagrangian Advection

Knowl 4 — Implicit Backward Euler Formulation for Viscous Diffusion

Knowl 5 — Fast Fourier Transform Fluid Solver for Periodic Domains

Knowl 6 — Advection, Diffusion, and Dissipation of Scalar Fields

Knowl 7 — Flow Detail Synthesis via Multi-Layer Texture Advection and Periodic Blending

Knowl 8 — Cell-Centered Finite Difference Formulation of the Projection Step

Knowl 9 — Numerical Dissipation in Stable Fluid Solvers

References

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