- #1
Niles
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Hi
I am trying to integrate Newtons equations for my system
<br /> a = \frac{F}{m} = \frac{d^2x}{dt^2}<br />
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
<br /> F=F(v_x, y, z)<br />
Now, I wish to solve this equation, and I have currently implemented an Euler method. This is how I iterate
<br /> v_{n+1} = v_n + dt\cdot a(v_{x,n},y_n,z_n) \\<br /> x_{n+1} = x_{n} + dt\cdot v_{n}<br />
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
<br /> y_{n+1} = y_{n} + dt\cdot f(x_n + dt/2, y_n + k_1/2)<br />
where
<br /> k_1 = dt\cdot f(x_n, y_n).<br />
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.
Niles.
I am trying to integrate Newtons equations for my system
<br /> a = \frac{F}{m} = \frac{d^2x}{dt^2}<br />
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
<br /> F=F(v_x, y, z)<br />
Now, I wish to solve this equation, and I have currently implemented an Euler method. This is how I iterate
<br /> v_{n+1} = v_n + dt\cdot a(v_{x,n},y_n,z_n) \\<br /> x_{n+1} = x_{n} + dt\cdot v_{n}<br />
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
<br /> y_{n+1} = y_{n} + dt\cdot f(x_n + dt/2, y_n + k_1/2)<br />
where
<br /> k_1 = dt\cdot f(x_n, y_n).<br />
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.
Niles.
