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 Incoming links: Leapfrog finite-difference scheme for the diffusion equation

Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 39A / iii / Solution Created 2026-09-24 Updated 2026-10-14

The leapfrog finite-difference scheme for the diffusion equation uses centered time and space differences:

\frac{u _{j}^{n + 1} - u _{j}^{n - 1}}{2Δ t} = \frac{u _{j - 1}^{n} - 2 u _{j}^{n} + u _{j + 1}^{n}}{( Δ x ) ^{2}} .

A Fourier mode with amplification factor

z

gives

z^{2} + 8 μ sin^{2} (θ /2) z - 1 = 0.

b = 4 μ sin^{2} (θ /2)

, the roots are

z_{\pm} = - b \pm 1 + b^{2}

. For any

μ > 0

and any nonzero spatial frequency,

b > 0

and

∣ z_{-} ∣ = b + 1 + b^{2} > 1.

Thus

leapfrog is unstable for the heat equation for every μ > 0.

Although the other root decays, the growing parasitic root is excited by generic starting data or perturbations, so it cannot be discarded in a stability test.

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Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 41E / c / Solution Created 2026-09-24 Updated 2026-10-03

For the same Fourier mode, the leapfrog finite-difference scheme for the diffusion equation gives

r^{n + 1} = r^{n - 1} - 2 q r^{n}, p (r) = r^{2} + 2 q r - 1.

The two amplification roots are

r_{\pm} = - q \pm 1 + q^{2} .

For every nonzero mode

q > 0

, one has

∣ r_{-} ∣ = q + 1 + q^{2} > 1.

Thus every

μ > 0

admits a growing Fourier mode, and

the leapfrog scheme is unstable for every μ > 0.

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