[SIGGRAPH 1999] Stable fluids. Jos Stam.

Modern C++ 23 / CUDA Implementation with C ABI of the paper Stable fluids by Jos Stam.

1. Algorithm Pipeline

Fields

• $\mathbf{w}$: velocity field
• $\mathbf{\rho}$: density field

1.1 Velocity Pipeline

$$ \mathbf{w}^n_0 \xrightarrow{\text{add force}} \mathbf{w}^n_1 \xrightarrow{\text{advect}} \mathbf{w}^n_2 \xrightarrow{\text{diffuse}} \mathbf{w}^n_3 \xrightarrow{\text{project}} \mathbf{w}^n_4 \Rightarrow{} \mathbf{w}^{n+1}_0. $$

add force

$$ \mathbf{w}_1(\mathbf{x}) = \mathbf{w}_0(\mathbf{x}) + \Delta t, f(\mathbf{x}, t) $$

advect (semi-Lagrangian method)

$$ \mathbf{w}_2(\mathbf{x}) = \mathbf{w}_1(\mathbf{p}(\mathbf{x}, -\Delta t)). $$

diffuse

$$ (I - \nu \Delta t \nabla^2) \mathbf{w}_3(\mathbf{x}) = \mathbf{w}_2(\mathbf{x}) $$

where $\nu$ is the viscosity coefficient.

project

$$ \nabla^2 q = \nabla \cdot \mathbf{w}_3 \qquad \mathbf{w}_4 = \mathbf{w}_3 - \nabla q $$

1.2 Density Pipeline

$$ \mathbf{\rho}^n_0 \xrightarrow{\text{add source}} \mathbf{\rho}^n_1 \xrightarrow{\text{advect}} \mathbf{\rho}^n_2 \xrightarrow{\text{diffuse}} \mathbf{\rho}^n_3 \Rightarrow{} \mathbf{\rho}^{n+1}_0. $$

add source

$$ \mathbf{\rho}_1(\mathbf{x}) = \mathbf{\rho}_0(\mathbf{x}) + \Delta t S(\mathbf{x}, t) $$

where $S(\mathbf{x}, t)$ is the scalar source term (e.g., injected smoke or dye).

advect (semi-Lagrangian method)

$$ \mathbf{\rho}_2(\mathbf{x}) = \mathbf{\rho}_1(\mathbf{p}(\mathbf{x}, -\Delta t)) $$

diffuse

$$ (I - \kappa \Delta t \nabla^2) \mathbf{\rho}_3(\mathbf{x}) = \mathbf{\rho}_2(\mathbf{x}) $$

where $\kappa$ is the diffusion coefficient (similar to viscosity for density dissipation).

2. Build Instruction

Build C ABI library

• CMake 4.3.0 or higher
• Ninja build system (for CXX std module support)
• A C++23 compliant compiler (tested on Arch Linux with gcc/g++ 15.2.1, Windows with MSVC 17.14.29)
• NVIDIA CUDA 13.2 or higher

cmake -B build -S . -G Ninja
cmake --build build --parallel

Build with Vulkan visualizer

A built-in Vulkan visualizer is provided. To enable it, you need to install latest Vulkan SDK.

• Vulkan SDK 1.4 or higher.

cmake -B build -S . -G Ninja -DSTABLE_FLUIDS_BUILD_VULKAN_APP=ON
cmake --build build --parallel