0
$\begingroup$

I am learning on how to use various numerical methods to approximate solutions to differential equations. We are using R, to actually iterate these functions, but I am having difficulties wrapping my head around the "trapezoidal" method.

I have been given the following:

$$ \frac{dy}{dt} = -y, \hspace{1mm} y(0)=1 $$

And the trapezoidal method is shown as

$$ y_{k+1} = y_k + \frac{1}{2}h(f(t_k,y_k)+f(t_{k+1},y_{k+1})). $$

This is implicit as the solution depends on $f(t_{k+1},y_{k+1})$, so I am unsure of how to go about this, and turn it into an iterative process, without know the value at $y_{k+1}$.

Thanks for any help.

$\endgroup$
0
$\begingroup$

You have to solve an equation for $y_{k+1}$, which in general itself requires some iterative procedure. This is a basic limitation of implicit methods.

But in this particular case you can solve the equation explicitly: the original method reads

$$y_{k+1}=y_k - h \frac{y_k + y_{k+1}}{2}$$

and you can solve it:

$$y_{k+1}=\frac{1-h/2}{1+h/2} y_k.$$

$\endgroup$
  • $\begingroup$ This makes sense, and is far simpler than I was imagining. Do I then plug this form of $y_{k+1}$ into the right side of the first equation, or would I just iterate upon this explicit form? $\endgroup$ – sokomov Feb 22 at 21:06
  • $\begingroup$ @sokomov You can just iterate with the explicit form. $\endgroup$ – Ian Feb 22 at 23:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.