2nd Order Differential Equation with Improved Euler Method (Heun's)

[Fluidman117]

Homework Statement

I would like to solve a 2nd Order Differential Equation using the Improved Euler Method. The 2nd ODE is a Mass-Spring-Damper equation. I tried coming up with an solution for the Improved Euler Method, but not entirely sure. Can you help me and have a look if this is correct?
This solution assumes that inital conditions for x and u are known.

Homework Equations

Forced Mass-Spring-Damper equation:

m\ddot{x} + b\dot{x}+kx = F

The Attempt at a Solution

Converting the equation to a pair of first order differential equations:
u=\dot{x}
\dot{u}=\ddot{x}

And thus we have:
\dot{u}=\frac{1}{m}\left[F-bu-kx\right]

The forward Euler solution would result in:
x_{n+1}=x_{n}+dt*u_{n}
u_{n+1}=u_{n}+dt*\dot{u_{n}}=u_{n}+dt*\frac{1}{m}\left[F_{n}-bu_{n}-kx_{n}\right]

And the Improved Euler solution would be:
x_{n+1}=x_{n}+dt*u_{n}
u_{n+1}=u_{n}+dt*\dot{u_{n}}=u_{n}+\frac{dt}{2} \left[ \frac{1}{m} \left[F_{n}-bu_{n}-kx_{n}\right]+\frac{1}{m}\left[F_{n+1}-bu_{n+1}-kx_{n+1}\right] \right]

In the last equation, does the last F needs to be F_{n+1} like I have it?

Thanks a bunch, if you have time to have a look at it!

 

[STEMucator]

Your improved Euler solution looks fine to me. You indeed want the 'n+1'.