D200 First-order Differential Equations (Runge--Kutta)

Subroutine subprograms RRKSTP and DRKSTP advance the solution of the system of $n\; \ge 1$ simultaneous first-order differential equations

$\{dy$_idx} = f_i(x,y₁,...,y_n), (i = 1,2,...,n)

by a single step of length h in the independent variable x.

On CDC and Cray computers, the double-precision version DRKSTP is not available.

Structure:

SUBROUTINE subprograms
User Entry Names : RKSTP, DRKSTP
Obsolete User Entry Names : RKSTP $\equiv$ RRKSTP
Files Referenced : Unit 6
External References: user-supplied SUBROUTINE subprogram

Usage:

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

    CALL tRKSTP(N,H,X,Y,SUB,W)

N

( INTEGER) Number n of equations.

H

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

X

(type according to t) 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) .

SUB

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

W

(type according to t) Array containing at least 3*N elements required as working-space.

    SUBROUTINE SUB(X,Y,F)

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

Method:

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

$k$1	$=$	$hf(x,y(x)),$
$k$2	$=$	$hf(x+\{12\}h,y(x)+\{12\}k$1),
$k$3	$=$	$hf(x+\{12\}h,y(x)+\{12\}k$2),
$k$4	$=$	$hf(x+h,y(x)+k$3); y(x+h) = y(x)+{16}(k1+ 2k2+ 2k3+ k4)

The error per step is proportional to $h5$ .

Error handling:

acts as do nothing.

References:

1. F.B. Hildebrand, Introduction to numerical analysis, (McGraw-Hill, New--York 1956) Sect. 6.16.

D201

K.S. Kölbig