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I want a int of how to set the resolution (using Newton Raphson method) in order to solve this equation :

$$ k_1 = f(t_i+ \frac{1}{2} h, y_1 + \frac{1}{2}k1) $$

I'm using python but I want understand the step

This formula is used in order to compute the 2nd order IMPLICIT Runge kutta methods, suitable for solving stiff system of differential equations. after compute $k1$ the solution of equations is $$ y_{i+1} = y_i + k1 h$$

EDIT It is an implicit Runge Kutta method!!

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  • $\begingroup$ At second sight this looks like a fixed-point equation in $k_1$. $\endgroup$ – mvw Aug 12 '18 at 16:11
  • $\begingroup$ Something must be wrong in your problem setting. ODE resolution with Runge-Kutta does not require this kind of twist. $\endgroup$ – Yves Daoust Aug 12 '18 at 16:14

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