Dimension scaling

State dimension is the second axis (after stiffness) along which solver choice matters. Per-step cost grows differently for explicit and implicit schemes, and the crossover where one beats the other moves with n.

The chart

Wall-clock runtime versus state dimension for DoPri5 (explicit) and
Radau5 (implicit) on a coupled linear ODE. The explicit scheme scales
roughly linearly in n; the implicit scheme pays for a dense linear
solve every Newton iterate, so it scales faster.

What it shows

The benchmark integrates a tridiagonally-coupled linear ODE,

dy_i/dt = -α y_i + β (y_{i-1} + y_{i+1}),     α = 2,  β = 0.5

over t ∈ [0, 5], at n ∈ {2, 10, 50, 200}.

DoPri5 has slope ≈ 1.0 on a log-log plot of runtime versus n — each RHS evaluation does O(n) work, and the number of steps doesn’t depend on n here, so the total scales linearly.
Radau5 has a steeper slope. Each step solves a dense linear system inside Newton, which is O(n³) for an unstructured Jacobian. The full sweep n ∈ {2, 10, 50, 200} runs comfortably inside Criterion’s budget — the n = 200 Radau5 point lands at ~46 ms, dominated by the dense LU factorisation rather than the step count.

When does this matter?

The two regimes:

Non-stiff problems with large n — stay explicit. Even a high-order RK is cheaper than factoring a 200×200 dense matrix every step.
Stiff problems with structured Jacobians — the implicit-solver cost in this chart is worst case (dense Jacobian). For problems where the Jacobian is banded, sparse, or block-structured, the cost-per-step drops dramatically. See the linear algebra chapter for what Numra exposes there.

If you’re unsure: pass Hints::dimension(n) to auto_solve_with_hints and the heuristic will pick an appropriate scheme.

What’s not in the chart

This is a non-stiff problem. For stiff high-dimensional problems, the calculus shifts — the implicit solver may still win by a wide margin because its step size doesn’t shrink with stiffness, even if each step is more expensive. The PDE method-of-lines bench (separate file bench_pde_mol.rs) explores that regime; see the appendix’s solver-reference for the underlying tradeoffs.