Runge-Kutta fourth order with negative stepsize

Solution 1:

If you have a procedure

y_next = rk4step(f,t,y,h) 

that works correctly for positive h, then this same procedure also works correctly for negative h. Remember that the time step remains

t_next = t+h

The only problem that may arise is the control of the loop. If the sampling times are given as array, then the loop

 for k in range(1,len(t)):
      y[k] = rk4step(f,t[k-1],y[k-1], t[k]-t[k-1])

will work independent of the direction of the time sample points.

If the loop control is based on the end time, then while t < tf works for positive h, for negative h one has to switch the sign or include h as in while 0<(tf-t)*h.