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I am trying to integrate Newtons equations for my system

[tex]

a = \frac{F}{m} = \frac{d^2x}{dt^2}

[/tex]

This is only for the first coordinate of the particle. I wish to do it foryandzas 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, v_{x}, and they- andz-coordinate. In other words

[tex]

F=F(v_x, y, z)

[/tex]

Now, I wish to solve this equation, and I have currently implemented an Euler method. This is how I iterate

[tex]

v_{n+1} = v_n + dt\cdot a(v_{x,n},y_n,z_n) \\

x_{n+1} = x_{n} + dt\cdot v_{n}

[/tex]

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

[tex]

y_{n+1} = y_{n} + dt\cdot f(x_n + dt/2, y_n + k_1/2)

[/tex]

where

[tex]

k_1 = dt\cdot f(x_n, y_n).

[/tex]

This is where my confusion arises: What does [itex]f(x_n + dt/2, y_n + k_1/2)[/itex] correspond to for me?

I would really appreciate a hint or two with this.

Best,

Niles.

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# Integrating 2nd order ODE using midpoint rule

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