So I have the pde $u_t + (vu)_x =0$ , $x \epsilon $R, t>0
and $u(x,0)=u_0(x)$ , $x \epsilon $R. as well as $u_j^0 =u_0(x_j)$ and $v_j =v(x_j)$
and I have to discuss the stability of this:
$\frac{u_j^{n+1} -u_j^n}{\Delta t} +v_j \frac{u_j^n -u_{j-1}^n}{h} +u_j^n \frac{v_j -v_{j-1}}{h} =0$
To do that I think that I have to prove that
$sup_{x\epsilon R} |u(x,t)| \leqslant sup_{x\epsilon R}|u_0(x)|$ is true
which I'm pretty sure i've managed through induction and the triangle inequalities..
I know that it is the advection equation ish, and I think its the backwards version of it.
I have tried rearranging the formula for = $u_j^{n+1}$ and I don't think that helped.
Should I show that that formula equals the supremum formula? but even then I don't know how that would explain whether or not it is stable.
Thanks for any help in advance