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 :

$$\begin{equation*} \begin{split} u_2^{n+1} - u_1^{n+1} & = \displaystyle\frac{u_2^{n}}{1-\vartriangle tu_2^{n}} - \displaystyle\frac{u_1^{n}}{1-\vartriangle tu_1^{n}} \\ & = \displaystyle\frac{u_2^{n}-u_1^{n}}{(1-\vartriangle tu_1^{n})(1-\vartriangle tu_2^{n})}\\ & \geq{\displaystyle\frac{u_2^{n}-u_1^{n}}{e^{-\vartriangle t(u_2^{n}+u_1^{n})}}}\\ \end{split} \end{equation*}$$

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 $$\dfrac{u^{n+1} - u^{n}}{\vartriangle t} = (u^{n+1}u^n)$$ this numerical method is from $$u'(t)=u^2$$

 

[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.