Stable Fluids implemented in Python/NumPy

In his 1999 SIGGRAPH Paper Jos Stam introduced a famous algorithm that is still ubiquitous in Computer Graphics and Video Game Physics. It solves the Navier-Stokes equations unconditionally stable and simulates fluid dynamics. Here is the code: https://github.com/Ceyron/machine-learning-and-simulation/blob/main/english/simulation_scripts/stable_fluids_python_simple.py

Unconditionally stable means that the time steps can be chosen arbitrarily large, and the kinematic viscosity can also be selected freely. This is extremely advantageous for computer graphics applications. Surely, this algorithm is unable to compete with state-of-the-art CFD codes in terms of accuracy and modelling capabilities. However, I think it is beautiful and encourages one to dig deeper.

You can find Jos' original paper here: https://d2f99xq7vri1nk.cloudfront.net/legacy_app_files/pdf/ns.pdf

His modified solver which is suitable to run in real-time is described here: http://graphics.cs.cmu.edu/nsp/course/15-464/Fall09/papers/StamFluidforGames.pdf

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? : Check out the GitHub Repository of the channel, where I upload all the handwritten notes and source-code files (contributions are very welcome): https://github.com/Ceyron/machine-learning-and-simulation

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Timestamps:
00:00 Introduction
00:23 About Stable Fluids
00:59 Problem Scenario
01:14 Upwards Forcing
01:31 Algorithm in Detail
04:06 Note on Boundary Conditions
04:17 Imports
05:11 Defining Simulation Parameters (=Constants)
05:53 Some Boilerplate
06:07 Creating a mesh
09:24 Forcing Function Definition
11:39 Vectorizing the Forcing Function
12:33 Time Loop + Initial condition
13:09 Step 1: Apply forces
13:50 Step 2: Nonlinear Convection (Self-Advection)
17:04 Step 3: Diffuse
17:19 Laplace Operator
18:58 Implicit Diffusion Operator
20:46 Step 3: Diffuse (cont.)
22:30 Step 4.1: Compute Pressure
23:00 Divergence + Partial Derivatives
26:00 Poisson Operator
26:54 Step 4.1: Compute Pressure (cont.)
27:53 Step 4.2: Pressure Correction
28:07 Gradient Operator
29:31 Step 4.2: Pressure Correction (cont.)
30:04 Advance in time
30:22 Initial visualization
31:43 First run + debugging
32:47 Curl Operator
34:00 Visualizing the Curl
36:15 Discussing the Plot
36:51 Playing with the Viscosity
37:20 Instability
37:56 Outro