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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-66/3/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 66 / 3 / b / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-06

The Lagrange interpolation polynomials for these nodes are

ℓ_{1} (τ) = 2 τ^{2} - 3 τ + 1, ℓ_{2} (τ) = 4 τ - 4 τ^{2}, ℓ_{3} (τ) = 2 τ^{2} - τ .

Their integrals give the Lobatto IIIA method tableau

0 1/2 1 0 5/24 1/6 1/6 0 1/3 2/3 2/3 0 - 1/24 1/6 1/6 .

To verify its nonlinear order, not just its scalar linear order, set

e = (1, 1, 1)^{T}

,

c = A e

and

C = diag (c)

. Direct multiplication verifies the fourth-order conditions for a Runge-Kutta method:

b^{T} e = 1, b^{T} c = \frac{1}{2}, b^{T} c^{2} = \frac{1}{3}, b^{T} A c = \frac{1}{6}, b^{T} c^{3} = \frac{1}{4}, b^{T} C A c = \frac{1}{8}, b^{T} A c^{2} = \frac{1}{12}, b^{T} A^{2} c = \frac{1}{24} .

c

are componentwise. These eight Butcher order conditions establish order at least four for general smooth ODEs. On the Dahlquist test equation, elimination of the stages gives

R (z) = \frac{1 + z /2 + z ^{2} /12}{1 - z /2 + z ^{2} /12}, R (z) - e^{z} = - \frac{z ^{5}}{720} + O (z^{6}) .

A nonzero fifth-order step defect rules out order five. Thus the method has exactly order four.

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