turing-surface#

Reaction–diffusion systems (Turing patterns) on closed surfaces given by spherical-harmonic embeddings, solved live in the browser with a spectral method whose transforms run on the GPU via WebGPU.

This is the sibling of turing-sphere, which solves the same systems on the round sphere. Everything there is here; what is added is a surface.

The geometry is in the operator: the models evaluate the surface Laplace–Beltrami operator lap_g inside the implicit solve, in a flux form that costs 5 spherical-harmonic transforms per species per iteration (plus one Legendre-free FFT derivative) where the textbook Cartesian-gradient form needs 12. See The geometry in the operator and docs/reduced-transforms.md.

What a surface is here#

A geometry is an embedding of the sphere into R³: three scalar fields x, y, z over the (θ, φ) parametrization, each carried as spherical-harmonic coefficients. The unit sphere is the case where all three are pure degree-1 harmonics.

You write one down as MATLAB, in geometries/:

function [gx, gy, gz] = shape(theta, phi, waist, stretch)
  st = sin(theta);
  r = 1 - waist * (st .^ 2);
  gx = r .* (st .* cos(phi));
  gy = r .* (st .* sin(phi));
  gz = (1 + stretch) * (r .* cos(theta));
end

That is ordinary element-wise MATLAB and goes through the same compiler and the same WGSL backend the models do. It is evaluated once on the solver's grid, and then analysed into coefficients, which is the form everything downstream uses. Two things follow from going through the coefficients rather than keeping the pointwise values:

• It is exactly band-limited at lmax. The surface has as many derivatives as the scheme needs and no aliased content the solver cannot see. What the solver and the renderer both use is the synthesis of the coefficients, so for a shape with sharp features the surface being solved on is not quite the one that was written down — which is the honest thing for a spectral method to do.
• It can be evaluated on any grid. The renderer draws the surface on the (possibly finer) display grid by synthesizing the same coefficients there. That is exact interpolation, not subdivision — the same argument that lets the species fields be oversampled, and it is checked directly in the tests.

Four geometries ship: sphere (the reference case), ellipsoid, peanut — a dumbbell whose waist is a saddle — and bumpy. Each is editable in the page, with its own parameters. Changing a shape does not recompile the solver and does not disturb the run: the geometry is data whose shape in the bindings depends only on the grid, so a swap is sixteen buffer writes and the pattern carries straight on.

A morph slider blends the drawn surface back to the unit sphere. The parametrization is the sphere's either way, so sweeping it shows which point went where.

The scheme, and where the geometry enters#

It solves the N-species system

d(u_k)/dt = D_k*lap_g(u_k) + f_k(t, u_1, ..., u_N),    k = 1, ..., N

where lap_g is the Laplace–Beltrami operator of the surface. On the round sphere lap_g is diagonal in spherical-harmonic space with eigenvalues -l(l+1), which is what makes turing-sphere's implicit diffusion a single divide. On a general surface it is not diagonal, and not even constant- coefficient, so that divide has to become a solve.

The models split the operator:

lap_g = lap_s + dlap

with lap_s the round-sphere one. (I - dt*D*lap_s) is still exactly invertible, so the implicit step

(I - dt*D*lap_g) Unew = B

rearranges into a fixed point that keeps the whole geometry on the right-hand side,

Unew = (B + dt*D*dlap(Unew)) ./ (1 + dt*D*lam)

and the loop iterates it from the round-sphere answer. That is preconditioned Richardson, with the operator we can invert exactly as the preconditioner; it converges while dt*D*dlap stays small against (I - dt*D*lap_s), which is what keeps the cost to a few transforms per step rather than a full elliptic solve (see docs/richardson-iteration.md). One species of models/schnakenberg.m's solve loop:

lamJ = lam ./ jhat;             % mean-J preconditioner eigenvalues (below)
...
for k = 1:niter
  Fu = Un .* filt;              % zero the top 2 degrees before differentiating
  vtu = dthetac(Fu);
  vpu = dphic(Fu);
  [Ftu, Fpu] = synth(vtu, vpu); % sin(theta)*dtheta(u), dphi(u) -- smooth on
                                % the sphere, one batched dispatch
  Pu = p1 .* Ftu + p2 .* Fpu;   % the two fluxes, also smooth: the precomputed
  Qu = p2 .* Ftu + q2 .* Fpu;   % weights carry every 1/sin(theta) there is
  PAu = analys(Pu);
  Pcu = PAu .* filt;
  scu = dthetac(Pcu);           % theta part of the divergence, coefficients
  Lu = synth(scu);              % sin(theta) * dtheta(P) on the grid
  dQu = dphig(Qu);              % d/dphi is diagonal in the Fourier index:
                                % two FFT stages, no Legendre work at all
  lapu = r .* (Lu + dQu);       % = lap_g(u) on the grid
  dLu = (analys(lapu) + lamJ .* Un) .* filt;   % dlap, projected onto the band
  Un = (Bu + (dt * D1) * dLu) ./ (1 + (dt * D1) * lamJ);
end

On the sphere dlap is mathematically zero — p1 = q2 = 1, p2 = 0, r = 1/sin²θ, jhat = 1, and the composition collapses to lap_s — so the sphere case reproduces turing-sphere to fp32 round-off, and the tests assert the state stays put across 0, 1 and 4 iterations.

The preconditioner folds in the symbol of the operator. jhat is the host's minimax scale 2/(μmin + μmax) over the eigenvalues μ(x) of the operator's principal symbol — the inverse squared principal stretches of the embedding, direction included, read straight off the flux-metric arrays (S = (1/J)·[[p1,p2],[p2,q2]]). Preconditioning with lam/jhat then contracts every mode and every direction at rate (μmax − μmin)/(μmax + μmin) < 1 on any surface, where the plain lam diverges wherever μ > 2 — peanut reaches μ = 6.2. A det-based mean of the area factor (μ's geometric mean, exact only for conformal surfaces) is not enough: it under-corrects anisotropic stretching and leaves directional high-degree bands with amplification > 1, which surfaced as patterns going high-frequency and diverging as niter or lmax grew. The answer never depends on jhat — the lamJ term added inside dLu is the term divided back out — only the convergence rate does.

The correction is projected onto the band (.* filt on dLu, matching algos.tex Algorithm 5's zeroing of the top coefficients). Without it the top two degrees iterate toward the undiffused Bu — each solve iteration strips a bit more of their implicit diffusion, at species- dependent rates, which manufactures a spurious Turing band at the band edge: visible on the round sphere as top-degree energy growing ~3%/step at lmax 127, niter 8. With both fixes the whole niter × geometry sweep converges, the spectral centroid of the pattern is resolution-independent (l ≈ 26 at lmax 63 and 127 alike), and jhat: 1 is kept as the divergent control in the tests.

The geometry in the operator#

dlap = lap_g - lap_s is applied to the current iterate at every solve iteration, so its transform count is what the whole step's cost scales with. Two formulations ship:

1. The flux form (above, all three models): lap_g u as the weighted divergence of two weighted fluxes of the sin-scaled derivatives. The weights p1, p2, q2, r are grid arrays precomputed once per surface from the embedding's θ/φ tangents (src/geom/metric.ts), chosen so that every field that gets analysed is a smooth function on the sphere — the property that makes spherical-harmonic analysis meaningful, and the entire difficulty near the poles. Cost: 5 Legendre transforms per species per iteration (3 syntheses + 2 analyses; the phi flux never needs the Legendre basis — dphig differentiates it on the grid with two FFT stages, masking m past the top-degree filter — and dthetac/dphic are O(nlm) coefficient shuffles). The derivation, the smoothness argument and the fp32 error analysis are in docs/reduced-transforms.md.
2. The Cartesian-gradient form (Algorithm 4 of docs/algos.pdf), kept as a live reference in models/schnakenberg_alg4.m and selectable in the app: the surface gradient carried as three ambient components through the inverse metric quantities Vt*/Vp*. Cost: 12 transforms per species per iteration. The tests hold both forms to the same answer on a curved surface, and both metric formulations are precomputed and uploaded for every geometry, so either kind of model runs.

The θ-derivative machinery both forms need — the α± recurrence (sin θ ∂θ Y_l^m = α⁺Y_{l+1}^m + α⁻Y_{l-1}^m) as a coefficient-space shuffle feeding the existing scalar synthesis — lives in src/sht/deriv.ts; no Legendre-derivative tables are required.

for loops, unrolled#

A plan is a fixed list of GPU operations with no branching, which is what makes a timestep pure command recording — one submit, no CPU in the loop. A counted loop still fits: the planner (src/mgpu/plan.ts) unrolls it, planning the body once per iteration.

Nothing else had to change for that, because numbl gives a variable one cName for every assignment to it: the buffer an iteration writes is the buffer the next one reads, which is exactly a loop-carried value. The loop variable gets no buffer at all — it is bound as a derived scalar to that iteration's literal, so a kernel reading k folds the number in.

Two consequences worth stating:

• The bounds must be known when the model compiles. niter is supplied as a fixed scalar rather than a tunable one, so changing it recompiles — unlike a parameter, which is a uniform. A runtime bound is refused at compile time with a source position, not silently mis-compiled, and there is a test for that.
• Fusion survives. numbl's inline pass recurses into loop bodies, so a line inside the loop is still one kernel. It runs there with no protected names, though, which means an assignment whose only visible use is later in the same body can be elided — correct for a body-local temp, wrong if something outside the loop wanted it. src/mgpu/compile.ts snapshots what each loop body assigns before the pass and refuses the ones that escape, so that case is a compile error rather than a stale read.

Unrolling is exactly linear in the trip count: 19 GPU ops per species per iteration (6 transforms, 4 coefficient shuffles, 9 kernels), asserted in the tests.

MATLAB, compiled to WebGPU#

Unchanged from turing-sphere, and it now compiles the geometry files too. numbl parses and lowers each function for the concrete argument types of the current grid; its inline pass folds single-use temps back into their consumer, so one line of MATLAB becomes one expression tree; and this repo emits one WGSL compute kernel per element-wise statement (src/mgpu/wgsl.ts). synth / analys are external operations whose type rules numbl learns from a .mtoc2.js workspace file, and which the backend maps onto the spherical-harmonic pipelines. Anything it cannot express is refused at compile time with a source position.

The Schnakenberg step compiles to 51 GPU operations at one solve iteration: 16 transforms, 8 coefficient-space shuffles, 25 generated kernels, and 2 buffer copies feeding the new state back.

Transforms batch. The expensive part of every Legendre stage is generating the associated Legendre values on the fly by recurrence — work that depends only on the grid, not on the field. synth/analys therefore take multiple fields, and a grouped call runs as one batched dispatch: one walk of the recurrence, one accumulator lane per field —

[Ftu, Fpu, Ftv, Fpv] = synth(vtu, vpu, vtv, vpv);   % one Legendre dispatch

The grouping is a promise of independence, never of a lane width: the planner (src/mgpu/plan.ts, materializeTransforms) chunks each group into whatever the device supports — one ×4 batch under the default WebGPU limits, or scalar dispatches with SHT_BATCH=0 for A/B — so the same source runs anywhere. Ungrouped transforms that happen to sit on consecutive independent lines are batched the same way. Per-lane arithmetic is identical to the scalar kernels', so batched and scalar plans produce bit-identical states, asserted in the tests along with compile-time refusal of a group that drops one of its outputs. All 16 transforms of the step above land in batches, worth ~25% of the whole step (0.88 vs 1.14 ms/step at lmax 127, 2 iterations, on bumpy).

Two consequences carried over:

• The step is synchronous. WebGPU's encode path is synchronous and every pipeline is built once at compile time, so a timestep is pure command recording; the only await in the loop is the single readback per rendered frame.
• Parameters are uniforms, not constants. Moving a slider rewrites a small buffer instead of triggering a recompile. Editing the MATLAB recompiles; changing dt does not. niter is the deliberate exception, above.

Provenance#

• turing-sphere, which this is a fork of: the solver, the transforms backend, the compilation path, the benchmarks and the analytic tests.
• Transforms: shtns-webgpu — fp32 spherical harmonic transforms in WGSL compute shaders, modeled on SHTNS. Vendored under src/sht/ (CECILL-2.1), including the f64 CPU reference transform used for testing.
• Rendering: three.js meshes with per-vertex colormaps, adapted from the SphereEmbedding view in figpack's experimental extension package (src/render/). That view displays a time-varying embedded geometry with fields on it, which is the same picture this draws — including its sphere/surface morph, which turing-sphere had dropped as having nothing to morph to.

turing-sphere additionally carries a comparison against a native build of upstream SHTNS (bench/shtns/). That is not duplicated here: the transforms are the same code, and its C-side transcription of the model would have to be maintained against a step this project intends to change.

Because the algorithm is compiled to compute shaders, WebGPU is required — there is no CPU fallback (the f64 CPU transform remains, for tests).

Numerics#

• Grid: Gauss–Legendre × equispaced-φ, dealiased for the cubic reactions with the (pdeg+1) rule: 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; for pattern formation from 1e-2 seeded noise this is inconsequential. The geometry goes through one analysis/synthesis round trip and picks up the same round-off: the unit sphere comes back with radius 1 to ~2e-5 under Dawn, ~4e-4 under SwiftShader.
• The shipped geometries are all degree ≤ 5, far below any lmax the app offers, so band-limiting removes nothing from them. A shape you write yourself may not be so lucky — see the note in geometries/bumpy.m.

Desktop vs browser#

scripts/bench.ts runs the same thing the app runs — same .m, same generated WGSL, same transforms — from Node on desktop WebGPU (Google Dawn), and the app prints the command line that reproduces whatever it is currently simulating:

npm run bench -- --preset schnak-spots --geometry ellipsoid --lmax 63 --niter 1 \
  --steps 2000 --seed 1 --a 0.1 --b 0.9 --D1 0.0004 --D2 0.008 --dt 0.05 \
  --gax 1.5 --gay 1 --gaz 0.6

Copy it from under the stats line and compare the ms/step it reports with the app's. Both sides go through the one shared src/bench/runSpec.ts — the app formats a run into that command, the benchmark parses it back — so there is no second copy of the defaults for the two runs to drift apart on. Geometry parameters take a g prefix (--gwaist) so a shape parameter can never collide with a model one.

The app reports two numbers and only the first is comparable to the benchmark: solver is the batch of steps alone, waited for but not read back; ms/frame additionally carries a GPU→CPU readback per species, the colormapping, and the vertex upload. Those per-frame costs are fixed and do not shrink when the GPU gets faster, so on a quick GPU a frame can easily cost ten times the steps inside it. That is expected and is not the solver being slower in the browser.

To attribute the gap rather than guess at it:

node scripts/compare-perf.mjs [--lmax 63] [--steps 300]

measures the same solver work in both — batched, nothing read back, no rendering on either side — and reports each with its CPU-encoding share, the Fourier stage, and the adapter. It stops you first if the two are not even the same device, which is a common cause of "the browser is much slower". Both sides resolve the geometry and the iteration count from the same constants, because the iteration count is unrolled into the step and a mismatch would compare two different amounts of work.

The app's Benchmark button runs the same measurement in the page, plus the ramp — the first third of the run against the last. GPUs downclock when idle and an animation-paced loop leaves them idle most of every frame, so a large ramp means the steady-state number is limited by clocks rather than work.

Is it really the same computation?#

node scripts/compare-env.mjs [--lmax 31] [--steps 200] [--preset schnak-spots]

runs one identical spec on the desktop and in a real browser and compares the final spectral state. The pipeline is deterministic given (model source, geometry, parameters, lmax, niter, seed, steps), so the two should agree to fp32 round-off — not bit for bit, since GPUs differ in fused-multiply-add and other latitude fp32 allows. It also reports which Fourier stage each side chose, since FFT and DFT are genuinely different algorithms that round differently.

Desktop WebGPU comes from the webgpu package (prebuilt Dawn, ~70 MB), an optional dependency so that an unsupported platform fails the install of that package alone. Its binaries need glibc 2.29+. Other flags: --steps, --warmup, --batch, --json, --help; DAWN_FLAGS='backend=vulkan' (;-separated) passes Dawn options through.

Tests#

There is no second implementation of the solver to diff against, so the .m path is checked against closed-form answers and against exact structural properties. Five modules, run in both environments:

test/analyticChecks.ts — cases whose evolution is known exactly, run through the whole real pipeline. All three are statements about the round sphere, so all three build on the sphere geometry:

• A — a linear reaction leaves every mode independent, growing by exactly (1 + dt*c) / (1 + dt*D*l(l+1)) per step. Pins the transform round trip, the eigenvalue mapping, the IMEX update and the state feedback at once. ~2e-7 over 20 steps.
• B — a nonlinear reaction on a uniform field stays uniform, so each step is exactly the scalar ODE map. 1.5e-8 over 25 steps.
• C — a 1e-6 perturbation of the Schnakenberg fixed point follows the linearized 2×2 IMEX recurrence, and (l=24, m=7) is confirmed unstable. Looser (~4e-3) because fp32 keeps about four digits of a perturbation that small.

test/geometryChecks.ts — the surface and the loop:

• every geometry compiles and closes; the sphere has radius 1 everywhere and is exactly degree 1 in the harmonics, which is what makes the reference case exact rather than merely accurate;
• the peanut matches its own closed-form radial profile at every grid point, and the same coefficients give the same surface on a 2× grid — the 2× Gauss latitudes share no point with the 1× ones, so agreeing there is agreeing everywhere, which is what "rendered exactly, not subdivided" means;
• unrolling is exactly linear in the trip count, and on the sphere — where the geometric correction is mathematically zero — the state after 20 steps stays within fp32 round-off of the 0-iteration one at 1 and 4 iterations;
• a runtime loop bound is refused at compile time;
• swapping the surface mid-run leaves the spectral state untouched.

test/fluxChecks.ts — the six-transform flux-form Laplace-Beltrami scheme (docs/reduced-transforms.md):

• on the sphere, the precomputed weights match their closed form and the analysed fluxes are exactly band-limited (beyond-band tails at f64 round-off, ~1e-13), while the deliberately non-smooth control Q̃/sin θ keeps a fat tail (~1e-2) — the discrimination the whole scheme rests on;
• on a non-axisymmetric surface, the flux tails match the Cartesian gradient component's, the doc's §7.1 criterion;
• the compiled op sequences add 5 Legendre transforms per species per iteration against Algorithm 4's 12, and a real simulation driven by each stays within fp32 accumulation of the other.

test/modelChecks.ts compiles every model the app offers and asserts how many kernels it compiles to, split into the base step and what one solve iteration adds. That is a fusion guard: if numbl's inline pass stops folding, the results stay correct while every operator becomes its own dispatch, which is invisible in the numbers.

test/transformChecks.ts compares the WGSL transforms against shtns-webgpu's f64 CPU twin, and holds every compiled batch width to the scalar transforms lane by lane; a model run with SHT_BATCH=0 must reproduce the batched run's state exactly.

• npm run test:node — under Dawn on the desktop, via vite-node. Needs a GPU; --skip-without-gpu lets a machine without one say so and move on (which is what CI does, since the browser suite covers the same modules).
• npm run test:gpu — builds and drives headless Chrome, on SwiftShader in CI. Also runs the soak. A few geometry tolerances are set by SwiftShader's fp32, which is about an order of magnitude looser than Dawn's.

Other commands:

• npm run bench -- --help — the desktop benchmark.
• npm run bench:sht -- --help — the transforms alone, no solver.
• npx vite-node scripts/diagnose-sht.ts — when the transform tests fail on a GPU, say which stage is wrong.
• npx vite-node scripts/diagnose-leg.ts [--m 0] — read the Legendre recurrence out of the production shader term by term.
• npx vite-node scripts/longrun-node.ts [lmax] — run to t = 100 and confirm the pattern saturates rather than decaying or diverging.
• node scripts/soak.mjs [steps] [lmax] — drive the demo for many steps, sampling JS heap and catching crashes.
• node scripts/screenshot.mjs out.png [light|dark] [minSteps] — screenshot the demo after a number of steps.
• node scripts/check-live.mjs [url] — smoke-check a deployed URL.
• test.html?soak=<steps>&lmax=<n> — solver-only soak with no rendering.

Development#

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

The numbl dependency#

numbl is a local file:../../numbl dependency, so a sibling checkout of numbl is required. We use its compiler internals — parser, lowerer, IR, inline pass — which its package exports map does not publish, so they are reached through the numbl-src path alias in vite.config.ts.

The exact surface we depend on is written down in src/mgpu/numbl.d.ts and TypeScript checks against that, not against numbl's sources. This keeps this project's compiler settings independent of numbl's, and means a change to one of those shapes upstream breaks the build here with a clear diff rather than deep inside numbl's tree. The For IR node is spelled out there, since the planner now walks it.

CI clones numbl to the sibling path that the file: dependency expects, pinned to a commit, with --ignore-scripts (npm runs a linked package's prepare script, and numbl's is husky). numbl's own node_modules are not needed: the slice we import is self-contained TypeScript.

The scripts/*.ts entry points that touch the compiler go through vite-node, so they resolve imports exactly as the browser build does. Plain node cannot: numbl's sources import each other as ./foo.js while the files are .ts.

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).