% Schnakenberg reaction-diffusion on a closed surface.
%
%   du/dt = D1*lap_g(u) + a - u + u^2*v
%   dv/dt = D2*lap_g(v) + b     - u^2*v
%
% Explicit reaction, implicit diffusion (IMEX Euler). The implicit solve
% splits lap_g = lap_s + dlap: the round-sphere part lap_s is diagonal in
% spherical-harmonic space (eigenvalues -lam), and the loop iterates the
% geometric correction dlap from that exact solve. Grid fields are npts x 1;
% spectral fields are real 2 x nlm. See docs/richardson-iteration.md.
%
% The correction evaluates lap_g in flux form -- 7 transforms per species
% per iteration where the Cartesian-gradient form (Algorithm 4 of
% evolving_surface/notes/algos.tex) needs 12. See
% docs/reduced-transforms.md, and models/schnakenberg_alg4.m
% for the original form kept as a live reference.
%
% The flux divergence is split against the round sphere: the sphere's share
% of it is -jinv*lap_s(u), exact in spectral space, and only the geometry
% *deviation* meets r ~ 1/sin^2(theta). Without that split the concentrated
% division amplifies the polar roundoff of the whole flux, and since a
% Turing pattern is seeded by whatever is largest in its unstable band, the
% amplified polar noise -- static, and re-injected every step -- picks the
% nucleation site and grows a spot at the pole. See docs/reduced-transforms.md
% Sec 5.

% The uniform steady state, perturbed by a smooth random field: chebfun's
% randnfun3 on the surface's bounding box, restricted to the surface by
% evaluating it at the grid points -- the way surfacefun seeds a run. lam3
% is its wavelength; the draw is seeded on the host, the sum over its
% Fourier modes runs on the GPU (src/mgpu/randnfun3.ts).
function [U, V, u, v] = init(lam3, gx, gy, gz, a, b)
  f = randnfun3(lam3, gx, gy, gz);
  us = a + b;
  vs = b / (us * us);
  [U, V] = analys(us + 0.01*f, vs * ones(numel(f), 1));
  [u, v] = synth(U, V);
end

function [Un, Vn, u, v] = step(U, V, lam, filt, gx, gy, gz, p2, r, dp1, dq2, jinv, jhat, a, b, D1, D2, dt, niter)
  % Grouped transforms -- [a, b] = synth(x, y) -- are explicit batching:
  % output k is the transform of input k, and the whole group runs as one
  % batched Legendre dispatch, or as many as the device's lane width allows
  % (src/mgpu/plan.ts, materializeTransforms). The grouping is a promise of
  % independence, never of a lane width, so the same source runs anywhere.
  [u, v] = synth(U, V);
  uuv = u .* u .* v;

  % Right-hand side of the implicit solve (I - dt*D*lap_g) Unew = B.
  ru = a - u + uuv;
  rv = b - uuv;
  [Ru, Rv] = analys(ru, rv);
  Bu = U + dt * Ru;
  Bv = V + dt * Rv;

  % Preconditioned solve, then iterate the geometric correction. jhat is
  % the host's minimax scale over the operator's symbol eigenvalues mu(x)
  % -- the inverse squared principal stretches of the embedding, direction
  % included (src/geom/geometry.ts, Jhat): preconditioning with lam/jhat
  % contracts every mode and direction at rate
  % (muMax - muMin)/(muMax + muMin) < 1 on any surface, where the plain
  % lam (jhat = 1) diverges wherever mu > 2 -- docs/reduced-transforms.md
  % Sec 10. The answer never depends on jhat (the lamJ term added inside
  % dLu is the term divided back out); only the convergence rate does. On
  % the sphere mu = 1 and lamJ = lam.
  lamJ = lam ./ jhat;
  Un = Bu ./ (1 + (dt * D1) * lamJ);
  Vn = Bv ./ (1 + (dt * D2) * lamJ);

  for k = 1:niter
    % dlap = lap_g - lap_s at the current iterate, in flux form
    % (docs/reduced-transforms.md Sec 4). The sin-weighted derivatives
    % A = sin(theta)*dtheta(u) and B = dphi(u) -- both smooth on the sphere,
    % synthesized straight from the dthetac/dphic coefficient shuffles --
    % are combined pointwise through the precomputed weights into two
    % fluxes P,Q, also smooth. The theta flux P goes back to
    % coefficients, through the same shuffle again, and is synthesized as
    % sin(theta)*dtheta(P); the phi flux Q never leaves the grid -- d/dphi
    % is diagonal in the Fourier index, so dphig differentiates it with two
    % FFT stages and no Legendre work (masking m past filt's reach). Their
    % sum, scaled by r, is lap_g(u). The only division by sin(theta)
    % anywhere is folded into the weights at precompute time.
    %
    % The weights here are the *sphere-subtracted* ones: p1 = 1 + dp1 and
    % q2 = 1 + dq2 (p2 is zero on the sphere already), so P,Q below are the
    % deviation fluxes P' = P - A, Q' = Q - B. What that leaves out is the
    % round sphere's own divergence, sin(theta)*dtheta(A) + dphi(B) =
    % -sin^2(theta)*lap_s(u), which needs no flux machinery at all: lap_s is
    % diagonal, so it is -lam.*Fu synthesized once (S below, riding along in
    % the gradient's batched synthesis) and scaled by the bounded
    % jinv = 1/J = r*sin^2(theta). r therefore multiplies only the deviation
    % -- the difference between this and multiplying the whole flux is two
    % orders of magnitude of polar roundoff, and it is what keeps a pattern
    % from nucleating at the pole (src/geom/geometry.ts, dp1/dq2/jinv).
    % lamJ.*Un adds back the preconditioner's -lap_s(Un)/jhat, since lam
    % holds +l(l+1). filt zeroes the top two degrees, where the derivative
    % recurrences cannot exactly represent a derivative -- and the correction
    % itself is projected onto the same band (algos.tex Algorithm 5 zeroes
    % the same coefficients): without that, each iteration replaces a bit
    % more of the top degrees' implicit diffusion with nothing (their fixed
    % point is the undiffused Bu), and the two species un-diffuse at
    % different rates -- a spurious Turing band at the band edge.
    %
    % The two species share each grouped call: the six gradient-and-sphere
    % syntheses, the two theta-flux analyses, the two divergence syntheses
    % and the two final analyses each run as one batched dispatch.
    Fu = Un .* filt;
    Fv = Vn .* filt;
    vtu = dthetac(Fu);
    vpu = dphic(Fu);
    vtv = dthetac(Fv);
    vpv = dphic(Fv);
    [Ftu, Fpu, Ftv, Fpv, Su, Sv] = synth(vtu, vpu, vtv, vpv, lam .* Fu, lam .* Fv);
    Pu = dp1 .* Ftu + p2 .* Fpu;
    Qu = p2 .* Ftu + dq2 .* Fpu;
    Pv = dp1 .* Ftv + p2 .* Fpv;
    Qv = p2 .* Ftv + dq2 .* Fpv;
    [PAu, PAv] = analys(Pu, Pv);
    Pcu = PAu .* filt;
    Pcv = PAv .* filt;
    scu = dthetac(Pcu);
    scv = dthetac(Pcv);
    [Lu, Lv] = synth(scu, scv);
    dQu = dphig(Qu);
    dQv = dphig(Qv);
    lapu = r .* (Lu + dQu) - jinv .* Su;
    lapv = r .* (Lv + dQv) - jinv .* Sv;
    [LAu, LAv] = analys(lapu, lapv);
    dLu = (LAu + lamJ .* Un) .* filt;
    dLv = (LAv + lamJ .* Vn) .* filt;

    Un = (Bu + (dt * D1) * dLu) ./ (1 + (dt * D1) * lamJ);
    Vn = (Bv + (dt * D2) * dLv) ./ (1 + (dt * D2) * lamJ);
  end
end