D203 Second-order Differential Equations (Runge--Kutta--Nyström)

Subroutine subprograms RRKNYS and DRKNYS advance the solution of the system of $n\; \ge 1$ simultaneous second-order differential equations

$\{d2y$_idx²} = f_i(x,y₁,...,y_n,y'₁,...,y'_n), (i = 1,2,...,n)

where $y\text{'}$_i= dy_i/dx , by a single step of length h in the independent variable x.

On computers other than CDC or Cray, only the double-precision version DRKNYS is available. On CDC and Cray computers, only the single-precision version RRKNYS is available.

Structure:

SUBROUTINE subprograms
User Entry Names : RRKNYS, DRKNYS
Obsolete User Entry Names: RKNYS $\equiv$ RRKNYS
External References: User-supplied SUBROUTINE subprogram

Usage:

For $t=R$ (type REAL), $t=D$ (type DOUBLE PRECISION),

    CALL tRKNYS(N,H,X,Y,YP,SUB,W)

N

( INTEGER) Number n of equations.

H

(type according to t) The step-length h.

X

On entry, X must be equal to the initial value of the independent variable x. On exit, X is equal to x+h.

Y

(type according to t) One-dimensional array of length $\ge N$ . On entry, $Y(i),(i=1,...,N)$ , must contain $y$_i(x) . On exit, $Y(i),(i=1,...,N)$ , contains approximate values $y$_i(x+h) .

YP

(type according to t) One-dimensional array of length $\ge N$ . On entry, $YP(i),(i=1,...,N)$ , must contain $y\text{'}$_i(x) . On exit, $YP(i),(i=1,...,N)$ , contains approximate values $y\text{'}$_i(x+h) .

SUB

Name of a user-supplied SUBROUTINE subprogram, declared EXTERNAL in the calling program.

W

(type according to t) Array containing at least 6*N elements required as working-space.

    SUBROUTINE SUB(X,Y,YP,F)

$F(I)=f$_I(X,Y(1),...,Y(N),YP(1),...,YP(N))(I = 1,2,...,N).

Method:

Using boldface quantities to denote vectors of length n, the computational sequence is as follows:

$k$1	$=$	$\{12\}h2f(x,y(x),y\text{'}(x))$
$k$2	$=$	$\{12\}h2f(x+\{12\}h,y(x)+\{12\}hy\text{'}(x)+\{14\}k$1,y'(x)+{1h}k1)
$k$3	$=$	$\{12\}h2f(x+\{12\}h,y(x)+\{12\}hy\text{'}(x)+\{14\}k$1,y'(x)+{1h}k2)
$k$4	$=$	$\{12\}h2f(x+h,y(x)+hy\text{'}(x)+k$3,y'(x)+{2h}k3)
$y(x+h)$	$=$	$y(x)+hy\text{'}(x)+\{13\}(k$1+ k2+ k3)
$y\text{'}(x+h)$	$=$	$y\text{'}(x)+\{13h\}(k$1+ 2k2+ 2k3+k4)

The error per step is proportional to $h5$ .

Error handling:

For $N\; \le 0$ or $H=0$ , control is returned to the calling program without any change in Y or YP.

References:

1. L. Collatz, The numerical treatment of differential equations, (Springer-Verlag Berlin 1960) 537

D300