Numerical Methods · Design background

Why 2D Needs a Different Trick Than 1D

Adding a second spatial dimension doesn't just add work — it tightens the rules. ADI is the standard way out, and it's built entirely from pieces you already have.

▶ Play the lab 📖 Design background

1From one dimension to two

The 1D fin-stability lab derived an explicit stability limit from a single diffusion direction. A 2D plate diffuses heat in both $x$ and $y$ at once:

$$ \frac{\partial \theta}{\partial t} = \alpha\left(\frac{\partial^2\theta}{\partial x^2}+\frac{\partial^2\theta}{\partial y^2}\right) $$

Explicit time-stepping has to respect both directions' tendency to amplify error simultaneously, and the two effects add:

$$ \Delta t \le \frac{1}{2\alpha\left(\frac{1}{\Delta x^2}+\frac{1}{\Delta y^2}\right)} $$

With equal spacing in both directions, that's exactly half the 1D limit. Every dimension you add to a problem squeezes explicit time-stepping further — in 3D, it's worse again.

2The ADI idea: be implicit one direction at a time

A fully implicit 2D step would be unconditionally stable, but solving for every point's new value in terms of every other point's new value means a large, genuinely 2D linear system — expensive. Alternating Direction Implicit finds a middle path: split one time step into two half-steps, and in each half-step, be implicit in only one direction.

The ADI recipe (Peaceman-Rachford)

Half-step 1: for each row (fixed $y$), solve implicitly in $x$ — a 1D tridiagonal system — while treating the $y$-direction term explicitly using the old values.
Half-step 2: for each column (fixed $x$), solve implicitly in $y$ instead — using the intermediate result from half-step 1 — while now treating $x$ explicitly.

Each half-step is just the 1D implicit scheme from the fin-stability lab, applied row-by-row or column-by-column with the exact same Thomas algorithm. Swapping which direction is implicit each half-step is what makes the combined method stable in both directions, even though neither half-step alone is.

▶ In the lab

Set ADI to 5× the explicit-only limit and it stays smooth; set 2D Explicit to just 1.5× that same limit and it diverges within a few dozen steps. Open the lab →

3Stability is not the same thing as accuracy

It's tempting to read "unconditionally stable" as "any time step is fine" — it isn't. ADI's truncation error still grows with $\Delta t$, the same as any time-marching scheme; a huge step just won't diverge, it'll quietly give you a less accurate answer for the early transient (though it still relaxes to the correct steady state, since that's a fixed point of the scheme regardless of step size). Stability answers "will this number mean anything at all"; accuracy answers "how close is it to the truth." They're different questions, and a method can score well on one and only adequately on the other.

Key takeaways

The explicit stability limit gets tighter with every spatial dimension added — 2D is twice as constrained as 1D at equal grid spacing.
ADI splits one 2D step into two 1D implicit sweeps, alternating which direction is implicit — each sweep reuses the ordinary 1D tridiagonal solve.
The result is unconditionally stable in 2D, at roughly the cost of two 1D solves per step instead of one expensive 2D solve.
Unconditional stability doesn't mean unconditional accuracy — a very large step is still safe, just less precise during the transient.

▶ Play

Open the ADI lab

Explicit vs. Implicit (1D)

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