Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Integrating 2nd order ODE using midpoint rule

  1. Apr 28, 2012 #1

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

  2. jcsd
  3. Apr 29, 2012 #2
    Here you have it explained:
    Computational physics

    page 292, "13.4 More on finite difference methods, Runge-Kutta methods"
  4. May 4, 2012 #3
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Integrating 2nd order ODE using midpoint rule