I already showed that the Lotka Volterra Equations preserve the weighted area $ dx \wedge dy/xy$ see here. Now I need that a modification of the Forward Euler Method preserves the same weighted area. Here is the modification:

$$ \frac{x_{n+1}-x_n}{\Delta t} = x_n -x_ny_n $$ $$\frac{y_{n+1}-y_n}{\Delta t} = -y_n -x_{n+1}y_n $$

side-question: why is this Euler scheme still explicite?

Proof that this modification preserves the weighted area $dx \wedge dy/xy$:

to simplify notitation :

$$ X = \Delta tx +x - \Delta txy$$ and $$ Y = -\Delta ty+ y + \Delta tXy $$ where we set $X:=x_{n+1}, Y:=y_{n+1}$, $x:= x_n$ and $ y := y_n$ solved for the unknown $X$ and $Y$.

now I showed $dX \wedge dX = 0$, $dY \wedge dY = 0$ and compute $dX \wedge dY$ and compute $XY$.

I found out that:

$$ \frac{1}{XY}{dX \wedge dY} = \frac{1}{xy}{dx \wedge dy} $$

Question: why is this enough to say that the modification preserves the weighted area above?


Your Answer

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

Browse other questions tagged or ask your own question.