Interleaved

[mongibellili]

function interleaved(du, u, p, t)
    C=1e4;L=1e4;R=0.1;T=1e-4;DC=0.5;ROn = 1e-5; ROff = 1e5;N=4.0
    du[1] =C*(u[2]+u[3]-R*u[1]+u[4]+u[5])
    for k in 2:5 
        du[k]=(((24.0/p[k+3]) - u[k]) * (p[k+3]*p[k-1]/(p[k+3]+p[k-1])) - u[1])*L;
    end 
end

rel,abs=(-3,-4)
liqss2:
The total simulation steps: 842
The number of events: 177
The number of state steps per Var: [119, 115, 132, 152, 147]
  376.100 μs (82 allocations: 45.84 KiB)

[mongibellili]

algName = :Rosenbrock23
sol.stats = SciMLBase.DEStats(570, 0, 148, 444, 104, 0, 0, 1561, 104, 44, 0.0)
  37.600 μs (218 allocations: 19.11 KiB)
algName = :BS3
sol.stats = SciMLBase.DEStats(527453, 0, 0, 0, 0, 0, 0, 2362091, 132626, 12719, 0.0)
  153.376 ms (398050 allocations: 39.34 MiB)

[mongibellili]

liqss2() is not advantageous because:

• events are not local: each event triggers diff.equation 2 to 5
• states are not loosely coupled
• stats of liqss2 show that all vars are almost equally updated (not advantage of asynchronous updates)

[mongibellili]

Jacobian Matrix

J = [
  -1000    10000    10000    10000    10000
 -10000   -15000        0        0        0
 -10000        0   -21000        0        0
 -10000        0        0   -26666.67     0
 -10000        0        0        0   -32142.86
]

Eigenvalues

λ ≈ { -11270.66 ± 16034.37i,  -17930.20,  -24604.54,  -30733.46 }

Why is it still stiff with ratio ≈ 3?

Stiffness isn’t only about eigenvalue ratio—it’s also about absolute size and time-step restrictions for explicit methods.
Here, eigenvalues are large (up to -30k), so explicit solvers like BS3 need tiny steps for stability, even if the ratio isn’t huge.
Rosenbrock23 is implicit and handles large negative eigenvalues well, so it’s faster.

[mongibellili]

abstol, reltol = 1e-4,1e-3
(typeof(alg)).name.name = :TRBDF2
sol.stats = SciMLBase.DEStats(11009, 0, 29, 10991, 21, 8773, 0, 44227, 2172, 0, 0.0)
Interleaved_err = 3.5217677309489823
(typeof(alg)).name.name = :QNDF2
sol.stats = SciMLBase.DEStats(2604, 0, 690, 2025, 38, 1775, 0, 5973, 473, 371, 0.0)
29.3 µs
Interleaved_err = 0.149302308649211
(typeof(alg)).name.name = :Trapezoid
sol.stats = SciMLBase.DEStats(60827, 0, 3093, 36305, 38, 36079, 0, 252564, 16022, 1215, 0.0)
55.7µs
Interleaved_err = 1.1990524907425624
(typeof(alg)).name.name = :Heun
sol.stats = SciMLBase.DEStats(442972, 0, 0, 0, 0, 0, 0, 2550337, 167782, 9, 0.0)
209.9ms
Interleaved_err = 0.6068817466232987
liqss2
The total simulation steps: 522
The number of events: 110
The number of state steps per Var: [77, 77, 84, 87, 87]
236.600 μs (82 allocations: 13.97 KiB)
Interleaved_err = 0.3608466363108351

qss2
The total simulation steps: 1256902
The number of events: 561398
The number of state steps per Var: [78, 147971, 165835, 182062, 199558]
432.911 ms (134 allocations: 25.73 MiB)
Interleaved_err = 0.3599578136940765