turing-sphere#

Reaction–diffusion systems (Turing patterns) solved live in the browser on the surface of a sphere, using a spectral spherical-harmonic method with the transforms running on the GPU via WebGPU.

Live demo: https://concept-collection.github.io/turing-sphere/

What it does#

It solves the N-species system

d(u_k)/dt = D_k*lap_s(u_k) + f_k(t, x, y, z, u_1, ..., u_N),    k = 1, ..., N

on the unit sphere, where lap_s is the Laplace–Beltrami operator. Diffusion is treated implicitly in spherical-harmonic coefficient space, where lap_s is diagonal with eigenvalues -l(l+1); reaction is treated explicitly on the grid. The two are combined with a first-order IMEX Euler step — the entire time loop is

V_k  = synth(U_k)                          # spectral -> grid
R_k  = analys(f_k(t, x, y, z, V_1..V_N))   # reaction on grid -> spectral
U_k  = (U_k + dt*R_k) / (1 + dt*D_k*l(l+1))

You watch the patterns emerge in real time on orbitable 3D spheres (one per species, cameras synced), with pause/resume, re-seeding, live parameter editing, and colormap selection.

Three presets are included:

• Schnakenberg — Turing spots (unstable band 14 ≤ l ≤ 40, peak l = 24)
• Brusselator — stripes and spots from a stiffer reaction
• Allen–Cahn — a single species whose interfaces form and coarsen

Provenance#

This is the browser port of a MATLAB reference implementation (SphericalReactionDiffusion.m, "websph"), which defines the solver through a four-member porting boundary: coeffs2vals, vals2coeffs, grid.lat, grid.lon. Profiling of the MATLAB version shows the transforms are ~96% of compute, so this port swaps in:

• Transforms: shtns-webgpu — fp32 spherical harmonic transforms in WGSL compute shaders, modeled on SHTNS. Its source is vendored under src/sht/ (CECILL-2.1), including the f64 CPU reference transform used for testing and as a no-WebGPU fallback.
• Rendering: three.js spheres with per-vertex colormaps, adapted from the SphereEmbedding view in figpack's experimental extension package (src/render/).
• Solver: src/solver/simulation.ts, a direct TypeScript port of the MATLAB IMEX loop, in f64 on the coefficients with the transforms in fp32 on the GPU.

Numerics#

• Grid: Gauss–Legendre × equispaced-phi, dealiased for the cubic reactions with the (pdeg+1) rule from the reference implementation: nlat ≥ ((pdeg+1)·lmax+1)/2, nphi ≥ (pdeg+1)·lmax+1 (rounded up to a power of two for the GPU FFT path). At the default lmax 63 that is a 128×256 grid.
• Spectral layout: SHTNS conventions — orthonormal + Condon–Shortley, complex coefficients for m ≥ 0, m-major ordering.
• fp32 transforms introduce ~1e-6 relative error per step (verified against the f64 CPU path); for pattern formation from 1e-2 seeded noise this is inconsequential.

Tests#

• npm run test:node — f64 solver correctness in Node: exact single-mode linear recurrence, exact uniform-state reaction ODE, and the linearized Turing-mode 2×2 IMEX recurrence (all at ~1e-12).
• npm run test:gpu — builds and drives headless Chrome: GPU-vs-CPU transform and solver cross-checks, plus a 100-step stability run.
• node scripts/longrun-node.ts — CPU run to t = 100 confirming pattern saturation.
• node scripts/screenshot.mjs out.png [light|dark] [minSteps] — screenshot the demo after a number of steps.

Development#

npm install
npm run dev       # local dev server
npm run build     # type-check + production build to dist/

Deployed to GitHub Pages by .github/workflows/deploy.yml on push to main.

License#

CECILL-2.1 (inherited from SHTNS via shtns-webgpu, whose sources are vendored).