Courant–Friedrichs–Lewy condition

Remarks

Proof idea: $u_j^n$ depends only on the values $u_0(x_i)$ with $x_{j - pn} \le x_i \le x_{j + qn}$. In the limit this is the interval $[x - p t/\lambda,\ x + q t/\lambda]$, the numerical domain of dependence of $(x, t)$. The exact solution $u(x, t) = u_0(x - a t)$ depends only on the point $x - at$. If $x - at$ lies outside the interval, change $u_0$ near $x - at$ only: the limit of the scheme stays the same, but $u(x, t)$ changes.

The condition is necessary but not sufficient. The forward-time centred-space scheme has $p = q = 1$ and violates no CFL bound when $\abs{a}\lambda \le 1$, yet it diverges for every $\lambda > 0$. The Lax–Friedrichs method converges exactly when $\abs{a} \lambda \le 1$. For $a \ge 0$, the upwind scheme $u_j^{n+1} = u_j^n - a\lambda(u_j^n - u_{j-1}^n)$ has $p = 1$ and $q = 0$, and it converges exactly when $a \lambda \le 1$. For systems, the same argument gives $\Lambda \Delta t \le \Delta x$ for three-point schemes, where $\Lambda$ is the largest characteristic speed.

Going deeper

For a student

The condition was found by Courant, Friedrichs and Lewy in 1928, in work on difference equations of mathematical physics. In practice it sets the largest stable time step of explicit methods, such as $\Delta t \le \Delta x / \Lambda$. Implicit schemes have a numerical domain of dependence equal to the whole grid, so the argument places no restriction on them. Sufficient conditions come from stability analysis, and the Lax equivalence theorem links stability with convergence for consistent linear schemes. In the Glimm scheme and Godunov's scheme, a CFL bound also keeps neighbouring Riemann problems from interacting.