Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 260

1, the power series expansions require evaluation of the poly-
nomials p 1 ( n ), p 2 ( n ), defined by (3.153) and (3.154), and the polynomials q 1 ( n ),
q 2 ( n ) defined by (3.150) and (3.151). These are evaluated by the double precision
function subprogrammes P1, P2, Q1 and Q2, listed below.

As well, for n ≥

DOUBLE PRECISION FUNCTION P1(AN,LAMBDA,MU)
C Finds value of function P1 for series solution at geocentre.
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
DOUBLE PRECISION LAMBDA,MU
P1=2.D0*AN*(AN*(AN+2.D0)*LAMBDA+(AN*(AN+2.D0)-1.D0)*MU)
RETURN
END

DOUBLE PRECISION FUNCTION P2(AN,LAMBDA,MU)
C Finds value of function P2 for series solution at geocentre.
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
DOUBLE PRECISION LAMBDA,MU
P2=AN*(AN+5.D0+(AN+3.D0)*(LAMBDA/MU))
RETURN
END

DOUBLE PRECISION FUNCTION Q1(AN,LAMBDA,MU)
C Finds value of function Q1 for series solution at geocentre.
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
DOUBLE PRECISION LAMBDA,MU
Q1=2.D0*AN*((AN+2.D0)*LAMBDA+(AN+1.D0)*MU)
RETURN
END

DOUBLE PRECISION FUNCTION Q2(AN,LAMBDA,MU)
C Finds value of function Q2 for series solution at geocentre.
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
DOUBLE PRECISION LAMBDA,MU
Q2=2.D0*(AN+1.D0)+(AN+3.D0)*(LAMBDA/MU)
RETURN
END

The power series expansions of the fundamental solutions, regular at the geo-
centre, take the density and elastic constants to be independent of radius, out to a
minimum radius, RMIN. They are then used there to calculate the initial solutions,
and their derivatives, required for Runge-Kutta integration. RMIN is found so that
the maximum ratio of the second term to first term in the power series expansions,
ERRMAX, is bounded above by the specified relative error, EPS, at radius RMIN.
Numerical experimentation shows that a conservative value of RMIN = 1.2 km eas-

10 − 5 . In addition, an estimate of the initial step-
size for the Runge-Kutta integration is required. Since the Runge-Kutta scheme
used is fourth order, the error term is proportional to the stepsize to the fifth power.
Thus, the initial stepsize is taken as the fifth root of the ratio EPS

ily meets this criterion for EPS =

ERRMAX. The
evaluation of ERRMAX requires determination of the maximum relative ratio of
the elements of the arrays of second term expansion coe

/

cients to the correspond-
ing elements in the solution arrays. This is accomplished by the subroutine REL.