# Integrating 2nd order ODE using midpoint rule

by Niles
Tags: integrating, midpoint, order, rule
 P: 1,863 Hi I am trying to integrate Newtons equations for my system $$a = \frac{F}{m} = \frac{d^2x}{dt^2}$$ This is only for the first coordinate of the particle. I wish to do it for y and z as well, but let us just work with x for now to make it simple. The force in the x-direction depends on the velocity in the x-direction, vx, and the y- and z-coordinate. In other words $$F=F(v_x, y, z)$$ Now, I wish to solve this equation, and I have currently implemented an Euler method. This is how I iterate $$v_{n+1} = v_n + dt\cdot a(v_{x,n},y_n,z_n) \\ x_{n+1} = x_{n} + dt\cdot v_{n}$$ I now want to improve the error, and use a 2nd order Runge-Kutta method, i.e. the midpoint rule as briefly summarized here: http://www.efunda.com/math/num_ode/num_ode.cfm I am not quite sure how to do this. In the link they say that now I should generally write $$y_{n+1} = y_{n} + dt\cdot f(x_n + dt/2, y_n + k_1/2)$$ where $$k_1 = dt\cdot f(x_n, y_n).$$ This is where my confusion arises: What does $f(x_n + dt/2, y_n + k_1/2)$ correspond to for me? I would really appreciate a hint or two with this. Best, Niles.