Stencil Consistency Lab

Pick the weights of a finite difference recipe and watch what happens when the grid spacing h shrinks. Red terms blow up, grey terms fade away, and the one green term that survives is the differential equation.

The recipe

a (weight on u_i+1, = α in Q9A)

b (weight on u_i)

c (weight on u_i−1)

β (power of h underneath)

h (gap between beads) Slide right to make h bigger, left to shrink it toward 0.

Try:

Walk through Q9A with me

Term ledger

Each row is one piece of the Taylor series. After dividing by h^β, each piece carries some power of h. A negative power means h is stuck underneath, so the piece grows into a giant as h shrinks.

Piece	Taylor coefficient	Power of h after ÷h^β	Size at this h	As h → 0

Giant explodes, must be cancelled Survivor becomes the ODE Crumb fades to 0 (the error) Cancelled coefficient is 0

Verdict

Beads on a string

The computer only sees beads spaced h apart. The recipe looks at three of them: left, middle, right. Shrink h and watch them squeeze together.

What happens as h shrinks

Moving right means smaller h. Lines going up are giants, lines going down are crumbs, a flat line is the survivor.