Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 252

With k 1 hf i , and using expression (3.283) for k 2 , (3.289) for k 3 , and (3.294) for

k 4 , we find the sum

=

hf

3! D 2 f

f y Df

h 2
2! Df

h 3 1

6 ( k 1

+ 2 k 2 + 2 k 3 + k 4 ) = + + +

3 Df Df y r = r i +

4! D 3 f

h 4

O h 5 f y D 2 f f 2 + + +

y Df

+

= y i + 1 − y i , (3.295)

in agreement with the Taylor expansion (3.272), and the Runge-Kutta scheme
(3.281).
The governing equations for the spheroidal and torsional deformations, (3.102)
to (3.107), (3.110) to (3.113), (3.114) to (3.118), (3.119) and (3.120), and (3.121)
to (3.124), summarised in Section 3.4, are all systems of linear, first-order ordinary
di

ff

erential equations. The simple, single, first-order equation (3.261) is replaced
by the vector di

ff

erential equation
d y ( r )
dr =

f ( r , y ) = A ( r ) · y ( r ), (3.296)

where y ( r ) is the solution vector. For the sixth-order spheroidal system repre-
sented by equations (3.102) through (3.107), for example, the vector y ( r )is
[y 1 ( r ),y 2 ( r ),y 3 ( r ),y 4 ( r ),y 5 ( r ),y 6 ( r )] T ,and A ( r )isa6 × 6coe cient matrix ex-

pressing the linear nature of the di y ( r ).

The Runge-Kutta scheme (3.281), for continuing the solution vector from radius
r i to radius r i

ff erential system through the product A ( r ) ·

+ h , becomes the vector relation

1
6 ( k 1

y i + 1 − y i = + 2 k 2 + 2 k 3 + k 4 ), (3.297)

with

k 1 = h A ( r i ) · y ( r i ),

h A r i

y r i

h
2

h
2

k 1
2

k 2 = + · + +

,

h A r i

y r i

k 2
2

h
2

h
2

k 3 = + · + + ,

k 4 = h A ( r i + h ) · [ y ( r i + h ) + k 3 ]. (3.298)

er-
ential system. For a system of order n ,thereare n linearly independent solutions,
and every solution of the system is expressible as a linear combination of these
n fundamental solutions (Cole (1968), Theorem 3-2.2, p. 43). This leads to the

More generally, we will be dealing with the fundamental solutions of the di ff