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.

Note

The geometry is in the operator. The models solve with the surface's Laplace–Beltrami operator, iterated by a fixed-count preconditioned Richardson solve (solvers/richardson.m, applying lib/dlap.m). The iteration count is fixed at compile time with no residual check, so a shape/timestep/diffusivity combination outside its convergence radius diverges over many steps rather than being caught — the tests pin the known cases. See Where the geometry enters the operator.

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 six 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.

The pieces of that sentence are separate files, because they are separate ideas. The operator — dlap applied to a spectral field — is lib/dlap.m. The solver — the fixed point above, iterated niter times — is solvers/richardson.m:

function X = richardson(B, dtD, lam, filt, Vtx, Vty, Vtz, Vpx, Vpy, Vpz, niter)
  X = B ./ (1 + dtD * lam);
  for k = 1:niter
    dL = dlap(X, filt, Vtx, Vty, Vtz, Vpx, Vpy, Vpz, lam);
    X = (B + dtD * dL) ./ (1 + dtD * lam);
  end
end

And a model is a reaction plus one solve per species — the whole of models/schnakenberg.m's step is:

function [Un, Vn, u, v] = step(U, V, lam, filt, gx, gy, gz, Vtx, Vty, Vtz, Vpx, Vpy, Vpz, a, b, D1, D2, dt, niter)
  u = synth(U);
  v = synth(V);
  uuv = u .* u .* v;

  Bu = U + dt * analys(a - u + uuv);
  Bv = V + dt * analys(b - uuv);

  Un = richardson(Bu, dt * D1, lam, filt, Vtx, Vty, Vtz, Vpx, Vpy, Vpz, niter);
  Vn = richardson(Bv, dt * D2, lam, filt, Vtx, Vty, Vtz, Vpx, Vpy, Vpz, niter);
end

Trying a different solver against the same operator is a change to those two call lines: every solver composes from dlap (the matvec is (1 + dtD.*lam).*x - dtD.*dlap(x), the preconditioner the elementwise divide), and which one a model calls is part of what compiles — swapping recompiles, like changing niter already does. The solver is written as a full re-evaluation rather than an accumulated correction on purpose: where dlap computes to zero there is no correction to mis-round, and the divide is turing-sphere's arithmetic unchanged.

Three solvers ship. solvers/bicgstab.m solves the same system by preconditioned BiCGSTAB — same dlap, same preconditioner, a Krylov recurrence instead of a stationary one, at two dlap evaluations per iteration instead of one. Its scalars (rho, alpha, omega) never touch the CPU: dot is a GPU reduction into a 1-element buffer, the recurrences on its results compile to 1-element kernels, and a single-element value broadcasts into the vector updates. Inner products carry the half-spectrum weight wlm (m > 0 counts twice), making them the real L2 inner products on the sphere. With no residual test, every ratio a/b is written in the guarded form a*b/(b*b + 1e-30), so a converged (or broken-down) iteration goes stationary instead of dividing noise by noise. The difference is not academic: at the app's default lmax, Schnakenberg on the peanut sits outside the Richardson iteration's convergence radius for niter ≥ 2 and diverges, while BiCGSTAB on the identical operator converges monotonically — the tests pin both behaviors, side by side.

solvers/gmres.m is right-preconditioned GMRES(niter) — one Arnoldi sweep, no restart — with the residual minimized over the whole Krylov space. Its bookkeeping is what the other solvers never need: a basis of niter+1 spectral fields, a Hessenberg matrix, Givens rotations, a triangular back-substitution. The basis lives in a bank (getslab / setslab: the k-th 2 × nlm field of a wider array), the small matrices are element-addressed (getat / setat), and both are functional updates the planner compiles to static-offset buffer copies — MATLAB's own H(i,j) = h cannot lower, because numbl must prove an indexed write in bounds before the loop unrolls, and a loop variable has no value yet at that point. Written as calls, the index resolves at planning, where unrolling has made it a literal. The same resolution lets an inner loop bound depend on the outer loop's variable, which is what makes the for i = 1:j orthogonalization sweep compile.

Where the geometry enters the operator#

lib/dlap.m is Algorithm 3 of the evolving-surface notes: the field's θ/φ derivatives (the dtheta/dphi transforms, src/sht/deriv.ts) are contracted through the inverse metric quantities into a tangential gradient; each Cartesian component is re-analysed and differentiated again; the results recombine into the surface divergence, and lam .* F adds back what the round-sphere part already carries. The metric quantities Vt*/Vp* (src/geom/metric.ts) are built once from the embedding's derivatives when the geometry is (re)built — the geometry is static, so per step they are just six more buffers the kernels read. filt zeroes the top two spectral degrees wherever the operator re-differentiates, because the derivative recurrences cannot exactly represent a derivative there.

On the sphere dlap computes to (numerical) zero, so any niter lands within transform round-off of the exact round-sphere answer — asserted in the tests. Off the sphere the correction genuinely moves the answer, and convergence is a real constraint: the fixed-count loop has no residual check, so the tests also pin which shape/niter combinations are known to sit outside the convergence radius and diverge.

Subroutines#

A model file is not limited to init and step: it can define further functions and call them, and every model compiles against the shared library files — lib/ for operators, solvers/ for solvers — with MATLAB's visibility rules (a file's namesake function is public; a model-local function of the same name shadows it). numbl specializes each callee for the argument types at its call sites, and the host then splices the lowered body into the caller, one clone per call site (src/mgpu/inlineCalls.ts): arguments bind by renaming rather than copying, and assignments to a callee output become assignments to the caller's variable, which is what lets a solver iterate its result in place. Expansion runs before the fusion pass, so a call fuses exactly as the same code written inline would — the boundary costs nothing, and describe()'s op listing names the expanded internals (richardson#1.X). Recursion cannot unroll into a fixed op sequence and is refused at compile time, like a runtime loop bound.

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. The loop that matters is solvers/richardson.m's for k = 1:niter, expanded into each model's step at every solve call site.

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 bound may also be an enclosing unrolled loop's variable (for i = 1:j — each unrolled j plans its own inner trip count, which is how GMRES's triangular sweeps compile). A genuinely 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: 26 GPU ops per species per iteration (the operator's twelve transforms and the solve's 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; user-function calls are expanded into the caller, one clone per call site; the 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 (and the derivative pair dtheta / dphi) are external operations whose type rules numbl learns from a .mtoc2.js workspace file, and which the backend maps onto the spherical-harmonic pipelines; dot is one more, mapped onto a single-dispatch reduction (src/mgpu/reduce.ts) whose 1-element result stays on the GPU — scalars computed from it become 1-element kernels, and reading one inside a vector expression broadcasts it. The indexed-access ops (getslab/setslab, getat/setat) compile to static-offset buffer copies, their indices evaluated at planning time where the unrolled loop's variable is a literal. Anything it cannot express is refused at compile time with a source position.

The Schnakenberg step above compiles to 65 GPU operations at one solve iteration: 28 transforms, 35 generated kernels, and 2 buffer copies feeding the new state back.

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. Four 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; on the sphere the geometric correction computes to (numerical) zero, so 0, 1 and 4 iterations agree to transform round-off; on the peanut it measurably moves the answer;
• a niter × geometry sweep stays finite except the combinations known to sit outside the Richardson convergence radius, which are pinned as diverging — and on exactly those combinations bicgstab and gmres keep converging, also pinned;
• at equal niter, bicgstab and gmres land far closer to the converged answer than richardson on the same operator;
• a runtime loop bound is refused at compile time;
• swapping the surface mid-run leaves the spectral state untouched.

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. It also compiles a model built of user-defined subroutines — multi-output, scalar-returning, a solver-like local with its own loop — asserts recursion is refused, and checks the reduction and indexing primitives directly against exact expected values: the dot sum and the wlm-weighted inner product against the CPU, scalar arithmetic on a GPU-resident result bit for bit, the 1-element broadcast, and element/slab round-trips through setat/getat and setslab/getslab.

test/transformChecks.ts compares the WGSL transforms against shtns-webgpu's f64 CPU twin.

• 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).