runge_kutta_4 A Correctness Example and Test

runge_kutta_4_correct.py

@(@\newcommand{\B}[1]{{\bf #1}} \newcommand{\R}[1]{{\rm #1}}@)@ runge_kutta_4 A Correctness Example and Test
Discussion
Source Code

Discussion
Define @(@ y : \B{R} \rightarrow \B{R}^n @)@ by @[@ y_j (t) = t^{j+1} @]@ It follows that the derivative of @(@ y(t) @)@ satisfies the runge_kutta_4 ODE equation where @(@ y(0) = 0 @)@ and @(@ f(t, y) @)@ is given by @[@ f(t , y)_j = y_j '(t) = \left\{ \begin{array}{ll} 1 & {\; \rm if \;} j = 0 \\ (j+1) y_{j-1} (t) & {\; \rm otherwise } \end{array} \right. @]@

Source Code




 
from pycppad import *
def pycppad_test_runge_kutta_4_correct() :
	def fun(t , y) :
		n        = y.size
		f        = numpy.zeros(n)
		f[0]     = 1.
		index    = numpy.array( range(n-1) ) + 1
		f[index] = (index + 1) * y[index-1] 
		return f
	n  = 5              # size of y(t) (order of method plus 1)
	ti = 0.             # initial time
	dt = 2.             # a very large time step size to test correctness
	yi = numpy.zeros(n) # initial value for y(t); i.e., y(0)

	# take one 4-th order Runge-Kutta integration step of size dt 
	yf = runge_kutta_4(fun, ti, yi, dt)

	# check the results
	t_jp = 1.                                # t^0 at t = dt
	for j in range(n-1) :
		t_jp = t_jp * dt                    # t^(j+1) at t = dt
		assert abs( yf[j] - t_jp ) < 1e-10  # check yf[j] = t^(j+1)

Input File: example/runge_kutta_4_correct.py