Why Simulations Blow Up: The Stability Limit

A computer steps the heat equation forward in small time steps on a grid. Take steps too big for the grid spacing, past r = kΔt/Δx² = ½, and the sharpest wiggle grows instead of fading, until the simulation explodes into a sawtooth.

How it works

Finite differences for the heat equation and their stability condition

In one sentence

The explicit scheme $u_j^{m+1} = u_j^m + r(u_{j+1}^m - 2u_j^m + u_{j-1}^m)$ for $u_t = k\,u_{xx}$ is consistent with error $O(\Delta t + \Delta x^2)$ . By von Neumann analysis its amplification factors $G_n = 1 - 4r\sin^2(n\pi\Delta x/2)$ satisfy $|G_n| \le 1$ for all $n$ exactly when $r = k\Delta t/\Delta x^2 \le \frac12$ , and by the Lax equivalence theorem consistency plus stability gives convergence.

General formula

u_j^{m+1} = {\htmlData{sym=0}{u_j^m}} + {\htmlData{sym=1}{r}}\left(u_{j+1}^m - 2{\htmlData{sym=0}{u_j^m}} + u_{j-1}^m\right), \qquad {\htmlData{sym=1}{r}} = \frac{{\htmlData{sym=2}{k}}\,{\htmlData{sym=3}{\Delta t}}}{{\htmlData{sym=4}{\Delta x}}^2} \le \frac{1}{2}

Each step mixes a point with its neighbors. Mix too strongly and the scheme overshoots, flipping the sharpest wiggle back and forth with growing amplitude.

What the symbols mean

$u_j^m$
Grid value
The temperature at grid point j after m time steps.
$r$
Mesh ratio
Diffusivity times the time step, divided by the grid spacing squared.
$k$
Thermal diffusivity
How fast heat spreads.
$\Delta t$
Time step
How far the simulation advances each step.
$\Delta x$
Grid spacing
The distance between neighboring grid points.

Replace $u_t$ by a forward difference in time and $u_{xx}$ by a centered difference in space. Each new value is the old one plus $r$ times the local curvature, with $r = k\,\Delta t/\Delta x^2$ .
On the grid the solution is a sum of discrete sine modes, and each step multiplies mode $n$ by its amplification factor $G_n = 1 - 4r\sin^2\frac{n\pi\Delta x}{2}$ .
The true heat equation shrinks every mode. The scheme does too only if $|G_n| \le 1$ for every $n$ , which requires $r \le \tfrac12$ .
Past that, the sharpest wiggle (a sawtooth) has $G < -1$ : it flips sign and grows every step. Halving $\Delta x$ forces a time step four times smaller, which is why implicit methods exist.

Learn the math

Finite Difference MethodsPDE · Lesson 18, Beyond the Basics