How can the stability of this numerical method be proven?

[juaninf]

Please anyone help me with stability proof this next numerical method \dfrac{u^{n+1} - u^{n}}{\vartriangle t} = (u^{n+1}u^n)I am trying make :

<br /> \begin{equation*}<br /> \begin{split}<br /> u_2^{n+1} - u_1^{n+1} &amp; = \displaystyle\frac{u_2^{n}}{1-\vartriangle tu_2^{n}} - \displaystyle\frac{u_1^{n}}{1-\vartriangle tu_1^{n}} \\<br /> &amp; = \displaystyle\frac{u_2^{n}-u_1^{n}}{(1-\vartriangle tu_1^{n})(1-\vartriangle tu_2^{n})}\\<br /> &amp; \geq{\displaystyle\frac{u_2^{n}-u_1^{n}}{e^{-\vartriangle t(u_2^{n}+u_1^{n})}}}\\<br /> \end{split}<br /> \end{equation*}<br />

but I not have more idea :( please help me

 

[mathman]

Your notation is confusing. I don't know what the right side of your first equation is supposed to mean.

 

[juaninf]

product <br /> \dfrac{u^{n+1} - u^{n}}{\vartriangle t} = (u^{n+1}u^n)<br /> this numerical method is from u&#039;(t)=u^2

 

[mathman]

I don't know what (uⁿ⁺¹uⁿ) is supposed to mean!

 

[juaninf]

is product of iterations (u^{n+1})*(u^n)

 

[mathman]

On the face of it the first equation doesn't make any sense. Dimensionally, it looks completely wrong.