A two-stage explicit Runge-Kutta method has

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

k_1 = f(t_n, y_n)

k_2 = f(t_n + c_2h,\, y_n + ha_{21}k_1)

and

y_{n+1} = y_n + h(b_1k_1 + b_2k_2)

, and satisfies the conditions for order 2,

b_1 + b_2 = 1

b_2c_2 = \tfrac12

and

a_{21} = c_2

. First take an equation

y' = g(t)

whose right-hand side does not depend on

y

, so that the method is a quadrature rule for

\int g\,\mathrm dt

. ((a)) Start one step from the exact value

y(t_n)

. Show that the exact value minus the method's value is

h^3g''(t_n)\bigl(\tfrac16 - \tfrac{c_2}4\bigr) + O(h^4)

. ((b)) Find

c_2

a_{21}

b_1

and

b_2

so that this error is

O(h^4)

for every smooth

g

. ((c)) With the method of (b), take one step of

h = 0.3

y' = t^2

from

y(0) = 0

(all dimensionless) and compare with the exact value. Give also the errors, exact minus computed, of Heun's method (

c_2 = 1

) and of the midpoint method (

c_2 = \tfrac12

) for the same step. ((d)) Now take

y' = y

with

y(0) = 1

and

h = 0.3

. Find the error in one step, exact minus computed, of the method of (b). Using the conditions for order 3 quoted in the chapter, explain why no choice of

c_2

removes it.

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