CFL Limit: Adaptive Timestep for Wave Simulation

Grid & Wave

Grid resolution N 120 Finer grid → smaller Δx → the CFL limit forces a smaller stable Δt.

Wave speed c 2.0

Timestep Mode

Courant target r 0.60 r = c·Δt/Δx. Stable while r ≤ 1; Δt is recomputed from this target every frame.

Manual Δt (ms) 1.0 Δt stays fixed as you change N — raise resolution far enough and this same Δt goes unstable.

Live Metrics

Stable

Δt (effective)

— ms

Courant r (actual)

—

Sim speed vs real-time (compute budget = 120 steps/frame)

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How it works

The bars are nodes of a 1D wave equation u_tt = c²u_xx, stepped with the explicit central-difference (leapfrog) scheme:

u[i]ⁿ⁺¹ = 2u[i]ⁿ − u[i]ⁿ⁻¹ + r²(u[i+1]ⁿ − 2u[i]ⁿ + u[i−1]ⁿ)
r = c·Δt / Δx   (Courant–Friedrichs–Lewy number)

This explicit scheme is only numerically stable while r ≤ 1 — the CFL condition. Physically, information cannot be allowed to cross more than one grid cell per timestep, or the discretisation overshoots and every step amplifies its own error until the field diverges.

Adaptive mode picks Δt = r·Δx/c every frame from the Courant slider, so raising resolution (N) automatically shrinks Δt to stay stable — exactly the "smaller timestep, more steps needed" trade-off from the article: finer spatial resolution is only affordable if you also pay for more, smaller timesteps.
Manual mode fixes Δt directly. Push N up while Δt stays the same and r = cΔt/Δx quietly climbs past 1 — the same grid that was rock-solid at low resolution now explodes, with no other setting touched.
Sim speed caps stepping at a fixed compute budget (120 substeps per animation frame, standing in for a fixed CPU/GPU time slice). When the stability-mandated Δt is so small that keeping up with real time would need more substeps than the budget allows, the simulation deliberately falls behind — the same "not enough headroom" bottleneck the article's Resource Allocation and Profiling sections describe, made visible as a number instead of a stutter.

Real solvers (CFD, seismic, EM/FDTD) live or die by this same inequality — it is the single most common reason a "just make Δt smaller" fix silently makes a simulation ten times slower, and the single most common reason an "unstable" bug is really just an oversized timestep.

Positive displacement Negative displacement

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