2-D Navier-Stokes Solver in a periodic box #124

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ns5678 wants to merge 10 commits into NVIDIA:main from ns5678:sn/ns-2d-solver

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ns5678 commented Jan 15, 2026

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⚠️ Unsigned Commits Detected

This pull request contains 8 unsigned commit(s):

5d04acd first push of notebook
18e8767 update notebook
92e5247 remove 50MB gif
f352bf8 reduce the size of notebook temporarily
0ba5186 consolidate comments
f186973 update notebook and it's solution
669dcf8 remove python script
7521f26 restore the previous version of ising notebook

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ns5678 added 8 commits

January 14, 2026 17:58


          first push of notebook

43cae31


          update notebook

72cdd87


          remove 50MB gif

93128ff


          reduce the size of notebook temporarily

70e20fa


          consolidate comments

09d5a72


          update notebook and it's solution

03f3e3f


          remove python script

10169b7


          restore the previous version of ising notebook

8b773ca

ns5678 force-pushed the sn/ns-2d-solver branch from 7521f26 to 8b773ca Compare

January 15, 2026 02:00

nv-kriehl assigned shi-eric

shi-eric suggested changes

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shi-eric left a comment

Some initial comments

Accelerated_Python_User_Guide/notebooks/2D_Navier_Stokes_In_Warp.ipynb Outdated

+                      "\n",
+                      "## Overview\n",
+                      "\n",
+                      "In this notebook, we will implement a 2-D Navier-Stokes (N-S) solver in a periodic box using the vorticity-streamfunction formulation. The Poisson equation relating streamfunction to vorticity is solved spectrally using tile-based FFT in Warp.\n",

Collaborator

shi-eric Jan 15, 2026

This intro sentence, while precise, is very technical. We can help novices by at least mentioning that we're building a solver that simulates fluid dynamics.

Collaborator

shi-eric Jan 15, 2026

You should also emphasize the Python aspect, e.g. something like "We will build our high-performance solver entirely in Python using the Warp framework..."

Contributor Author

ns5678 Jan 16, 2026

"In this notebook, we will build a high-performance fluid solver entirely in Python using the Warp framework. Specifically, we will simulate 2-D turbulent flow in a box." -- How does this sound?

Accelerated_Python_User_Guide/notebooks/2D_Navier_Stokes_In_Warp.ipynb Outdated

+                      "## Introduction\n",
+                      "\n",
+                      "This seemingly simple 2-D N-S solver example combines multiple Warp features that can be leveraged to build industrial-grade solvers:\n",
+                      " - CUDA kernels for finite difference operators\n",

Collaborator

shi-eric Jan 15, 2026

CUDA kernels should probably be Accelerated kernels for finite-difference operators. When I see "CUDA kernels", I think of all the nastiness associated with CUDA C++, which our users don't need to worry about.

Contributor Author

ns5678 Jan 16, 2026

Updated to "Accelerated Warp kernels..."

Accelerated_Python_User_Guide/notebooks/2D_Navier_Stokes_In_Warp.ipynb Outdated

Comment on lines 110 to 116

+                      "### 1. Vorticity-stream function formulation\n",
+                      "\n",
+                      "For 2-D incompressible flow, we define:\n",
+                      "- **Vorticity**: $\\omega = \\partial v/\\partial x - \\partial u/\\partial y$\n",
+                      "- **Streamfunction**: $\\psi$ such that $u = \\partial\\psi/\\partial y$, $v = -\\partial\\psi/\\partial x$\n",
+                      "\n",
+                      "This formulation automatically satisfies continuity and eliminates pressure from the Navier-Stokes equations.\n",

Collaborator

shi-eric Jan 15, 2026

You don't explain what the variables u and v are

Contributor Author

ns5678 Jan 16, 2026

Nice catch, fixed.

Accelerated_Python_User_Guide/notebooks/2D_Navier_Stokes_In_Warp.ipynb Outdated

+                      "\n",
+                      "### 3. Spectral Poisson solver\n",
+                      "\n",
+                      "For periodic domains, equation (2) can be directly solved in Fourier space. The wavenumbers are defined as $k_x = \\frac{2\\pi m}{L_x}, \\quad k_y = \\frac{2\\pi n}{L_y}$, where $m$ and $n$ are the Fourier space indices. The vorticity field $\\omega$ is converted to its Fourier representation $\\hat{\\omega}_{m,n}$ using a 2-D FFT. The Poisson equation then becomes algebraic as follows\n",

Collaborator

shi-eric Jan 15, 2026

Maybe you can add some more discussion to help novices. You start off with saying this can be solved in Fourier space, but maybe it's better to acknowledge that it might be tempting to solve the Poisson equation in physical space but we can make use of periodicity to solve the equation in spectral space, which is both efficient and simple to implement.

You can also explain what "Fourier space" is and specifically what the "FFT" means and why it deserves the "Fast" in its name

Accelerated_Python_User_Guide/notebooks/2D_Navier_Stokes_In_Warp.ipynb Outdated

+                      "$\\hat{\\psi}_{m,n}$ is then converted back to physical space using inverse FFT.\n",
+                      "\n",
+                      "### 4. Time integration \n",
+                      "For time stepping, we use a strong-stability preserving third-order Runge-Kutta method.\n",

Collaborator

shi-eric Jan 15, 2026

Maybe give a layperson explanation of what SSP and third-order mean

Accelerated_Python_User_Guide/notebooks/2D_Navier_Stokes_In_Warp.ipynb Outdated

+                      "\n",
+                      "### 5. Initial condition: 2-D decaying turbulence\n",
+                      "\n",
+                      "In this problem, we solve the governing equations on a square box of dimensions $2\\pi \\times 2\\pi$. The initial condition is generated using the energy spectrum from San & Staples CNF (2012)\n",

Collaborator

shi-eric Jan 15, 2026

Explain why you picked $2 \pi \times 2 \pi$ for the box size. Someone might find it a non-intuitive choice and wonder if this solver can handle other domain sizes.

Contributor Author

ns5678 Jan 16, 2026

Done!

Accelerated_Python_User_Guide/notebooks/2D_Navier_Stokes_In_Warp.ipynb

+                      "\n",
+                      "**NOTE**: Other similar time-stepping schemes can be implemented with minimal changes to the code, but we use the scheme above as a demonstration example.\n",
+                      "\n",
+                      "### 5. Initial condition: 2-D decaying turbulence\n",

Collaborator

shi-eric Jan 15, 2026

I think this is too much. I would create an appendix section and put the details about the initialization there, and then in this section you just say we're gonna use this particular E(k) because of <...> and you can find more discussion behind this choice of I.C. in the appendix.

Contributor Author

ns5678 Jan 16, 2026

I removed the construction of \zeta to the appendix A. I still left the definition of E(k) there.

Accelerated_Python_User_Guide/notebooks/2D_Navier_Stokes_In_Warp.ipynb Outdated

+                      "$$\n",
+                      "\\boxed{\n",
+                      "\\begin{array}{c}\n",
+                      "\\textbf{Solver Pipeline} \\\\[10pt]\n",

Collaborator

shi-eric Jan 15, 2026

This "Solver Pipeline" diagram is already pretty good. I would add a "IFFT" in the first arrow that follows "Initialize $\hat{\omega}$"

ns5678 added 2 commits

January 16, 2026 16:25


          resolve comments

e871495


          finish adding some extra details to remaining markdown cells

1f60f30

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