Runge-Kutta Method for a System of Equation

This program computes the initial value for the equation:

dYi/dx = fi (Yj(x),x) , x>a  where (i -1,2,. .., N)
Yi (a) = Yai

using a 4th order Runge-Kutta method. You will need to enter an end point (b) so that x is bounded by (a,b). You will also need to define the functions fi ( Yj (x),x ) starting at line 2000. You must also specify a local error tolerance (E) and step size (H) for the independent variable (x). Two options are available:

1. Fixed step size. The program checks the local error tolerance at each integration step, prompting an error message if the tolerance is exceeded.

2. Controlled step size. The program automatically controls the step size, increasing it or decreasing it to meet the prescribed tolerance.

Program Notes

As written, this program will solve for systems of the 4th order at most. To increase this, the DIM statement in line 1 must be modified as follows:

Y(N), YO(N), W(N), Z(N), S(N), Ql(N) where N is the new dimension.
 

Downloads:

basic program - qbrkmfoe.bas (Qbasic)

basic program (liberty basic) lbrkmfoe.bas

Test Case:

for the following equations input at end of program:

[TWZEONZE]
F = Y(2)
RETURN
[TWZETHZE]
F = -.2*Y(2)
RETURN
[TWZEFIZE]
F = Y(4)
RETURN
[TWZESEZE]
F = -9.81-.2*Y(4)
RETURN
[TWZENIZE]
F =  0
 

RUNGE-KUTTA METH0D FOR A
FIRST ORDER EQUATION

NUMBER OF EQUATIONS
M=4

ENTER LIMITS OF INTEGRATION
A =0
B =1

ENTER INITIAL DATA
Y1 (A) = 0
Y2 (A) = 3
Y3 (A) = 0
Y4 (A) = 5

INITIAL STEPSIZE
H =.1

LOCAL ERROR TOLERANCE X AS IN 1E-X
E = 6

STEPSIZE CONTROL?(Y/N)?y

SOLUTION
********
X = 0
Y1 =0
Y2 =3
Y3 =0
Y4 =5

X = 0.1
Y1 =0.2970199
Y2 =2.94059602
Y3 =0.44630854
Y4 =3.92973829

X = 0.3
Y1 =0.873532
Y2 =2.8252936
Y3 =1.0231348
Y4 =1.85237304

X = 0.5
Y1 =1.42743873
Y2 =2.71451225
Y3 =1.19268778
Y4 =-0.14353756

X = 0.7
Y1 =1.95962647
Y2 =2.60807471
Y3 =0.97093688
Y4 =-2.06118738

X = 0.9
Y1 =2.47094683
Y2 =2.50581063
Y3 =0.37322537
Y4 =-3.90364507

X = 1.1
Y1 =2.96221803
Y2 =2.40755639
Y3 =-0.58570515
Y4 =-5.67385897

END OF EXECUTION