1% Brusselator reaction-diffusion on a closed surface.
2%
3% du/dt = D1*lap_g(u) + A - (B+1)*u + u^2*v
4% dv/dt = D2*lap_g(v) + B*u - u^2*v
5%
6% Same scheme as models/schnakenberg.m, including the grouped transforms:
7% [a, b] = synth(x, y) runs the group as batched Legendre dispatches.
9function [U, V, u, v] = init(noise, A, B)
10 [U, V] = analys(A + noise, (B / A) * ones(numel(noise), 1));
11 [u, v] = synth(U, V);
12end
14function [Un, Vn, u, v] = step(U, V, lam, filt, gx, gy, gz, p1, p2, q2, r, jhat, A, B, D1, D2, dt, niter)
15 [u, v] = synth(U, V);
16 uuv = u .* u .* v;
18 ru = A - (B + 1) * u + uuv;
19 rv = B * u - uuv;
20 [Ru, Rv] = analys(ru, rv);
21 Bu = U + dt * Ru;
22 Bv = V + dt * Rv;
24 % Mean-J preconditioning -- see models/schnakenberg.m.
25 lamJ = lam ./ jhat;
26 Un = Bu ./ (1 + (dt * D1) * lamJ);
27 Vn = Bv ./ (1 + (dt * D2) * lamJ);
29 for k = 1:niter
30 % dlap = lap_g - lap_s, evaluated at the current iterate in flux form
31 % (see models/schnakenberg.m, docs/richardson-iteration.md and
32 % docs/reduced-transforms.md for the derivation and the ordering).
33 Fu = Un .* filt;
34 Fv = Vn .* filt;
35 vtu = dthetac(Fu);
36 vpu = dphic(Fu);
37 vtv = dthetac(Fv);
38 vpv = dphic(Fv);
39 [Ftu, Fpu, Ftv, Fpv] = synth(vtu, vpu, vtv, vpv);
40 Pu = p1 .* Ftu + p2 .* Fpu;
41 Qu = p2 .* Ftu + q2 .* Fpu;
42 Pv = p1 .* Ftv + p2 .* Fpv;
43 Qv = p2 .* Ftv + q2 .* Fpv;
44 [PAu, PAv] = analys(Pu, Pv);
45 Pcu = PAu .* filt;
46 Pcv = PAv .* filt;
47 scu = dthetac(Pcu);
48 scv = dthetac(Pcv);
49 [Lu, Lv] = synth(scu, scv);
50 dQu = dphig(Qu);
51 dQv = dphig(Qv);
52 lapu = r .* (Lu + dQu);
53 lapv = r .* (Lv + dQv);
54 [LAu, LAv] = analys(lapu, lapv);
55 dLu = (LAu + lamJ .* Un) .* filt;
56 dLv = (LAv + lamJ .* Vn) .* filt;
58 Un = (Bu + (dt * D1) * dLu) ./ (1 + (dt * D1) * lamJ);
59 Vn = (Bv + (dt * D2) * dLv) ./ (1 + (dt * D2) * lamJ);
60 end
61end