Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 247

and
⎩

r = r i =

f ⎭

∂ 3 ∂

3 f
∂ r 3 +

3 f
∂ r 2

3 f
∂ r ∂y

3 f
∂y ∂ r + f ∂

3 f ∂ 3 f 2 ∂

2 + f 3 ∂

∂y + ∂y

3

∂ ∂ f

∂y

2 ∂ f

2 f
∂ r 2 +

2 f 2 f

∂y

+ ∂ f
∂y

2 f ∂

∂ r ∂y + f 2 ∂ ∂ r + f ∂ f

+

2

∂y

r = r i

3 ∂ f

∂

2 f
∂ r ∂y +

2 f
∂y

f ∂ f
∂y

f ∂

+

∂ r +

2

r = r i . (3.271)
With the abbreviation f y = ∂ f /∂y, substituting (3.269) and (3.271) in (3.266), it is
found that

⎩ ⎭

∂ f
∂y

2

∂ f
∂y

3 Df D ∂ f
∂y

D 3 f D 2 f = + + Df +

hf +

3! D 2 f + f y Df

h 2
2! Df +

h 3

y i + 1 − y i =

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

(3.272)

To lowest order, extension of the solution from radius r i to r i + 1 is

O h 2 . y i + 1 − y i = hf ( r i ,y i ) + (3.273)

This is simply a trapezoidal extension of y( r ) assuming constant slope, d y/ dr
= f ( r i ,y i ).

A better approximation would allow for the change of slope from that at the
beginning of the interval to an estimate of the slope elsewhere on the interval.
Taking a weighted average of the two, we would write

y i + 1 − y i = w 1 hf i + w 2 hf ( r i + α 2 h ,y i + β 21 hf i )
= w 1 k 1 + w 2 k 2 , (3.274)

where w 1 , w 2 , α 2 , β 21 are parameters to be determined. Taylor expansion in two
variables gives

r = r i +··· . f

+ α 2 h ∂ f

∂ r + β 21 hf i ∂ f

f ( r i + α 2 h ,y i + β 21 hf i ) = (3.275)

∂y

Then,

r = r i + h (w 1 + w 2 ) f

+ w 2 h 2

O h 3

α 2 ∂ f

∂ r + β 21 f i ∂ f

y i + 1 − y i =

∂y

+ w 2 h 2 D 2 f r = r i +

h (w 1

O h 3 , = + w 2 ) f (3.276)