lines 5-40 of file: xrst/rk4_step.xrst

{xrst_begin rk4_step}
{xrst_spell
   kutta
   runge
}


A Fourth Runge-Kutta ODE Step
#############################

Problem
*******
Given
:math:`f : \B{R}^n \rightarrow \B{R}^n` ,
:math:`y(0) = y_0 \in \B{R}^n` , and
:math:`y^{(1)} (t) = f( y(t) )` for all :math:`t` .
Approximate :math:`y(h)` for some :math:`h \neq 0` .

Method
******
The following algorithm computes
the approximation :math:`y^1 = y(h) + O( h^5 )` where
:math:`O( h^5 )` means that the error in the approximation is order
:math:`h` to the fifth power.

.. math::

   k_1 & = f \left( y_0 \right) \\
   k_2 & = f \left( y_0 + \frac{h}{2} k_1  \right) \\
   k_3 & = f \left( y_0 + \frac{h}{2} k_2  \right) \\
   k_4 & = f \left( y_0 + h k_3 \right) \\
   y_1 & = y_0 + \frac{h}{6} \left( k_1 + 2 k_2 + 2 k_3 + k_4 \right)



{xrst_end rk4_step}