Integrating 2nd order ODE using midpoint rule

  1. Hi

    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 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
    [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.
     
  2. jcsd
  3. Here you have it explained:
    Computational physics

    page 292, "13.4 More on finite difference methods, Runge-Kutta methods"
     
  4. Thanks!
     
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