Heun's method is the two-stage explicit Runge-Kutta method with

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Ordinary differential equations: order, stability and stiffness

Problem

c_2 = 1

, and its stability function is

R_{\rm H}(z) = 1 + z + \tfrac12z^2

with

z = h\lambda

, for the test equation

y' = \lambda y

. Kutta's three-stage method has

c = (0, \tfrac12, 1)

a_{21} = \tfrac12

a_{31} = -1

a_{32} = 2

and

b = (\tfrac16, \tfrac23, \tfrac16)

. ((a)) Apply the three-stage method to the test equation to find its stability function

R_3(z)

. Find the point where the negative real axis leaves its stability region, and the largest stable step for a mode with

\lambda = -8000\ \mathrm{s^{-1}}

. ((b)) Show that

|R_{\rm H}(iy)| > 1

for every real

y \ne 0

. Use this to find the growth factor per step when Heun's method integrates an undamped oscillator of angular frequency

\omega = 50\ \mathrm{rad\,s^{-1}}

with step

h = 1\ \mathrm{ms}

. ((c)) Show that

|R_3(iy)|^2 = 1 - \tfrac{1}{12}y^4 + \tfrac{1}{36}y^6

, and find the factor per step by which the three-stage method multiplies the amplitude of the same oscillator at the same step.

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