I am solving the IBVP $u_t+uu_x=Du_{xx}$ using Crank Nicolson implicit scheme, with periodic boundary conditions. Discretization leads to a system of type $Au^{n+1}=Bu^n+F(u^{n+1})$, where $A$ and $B$ are variable matrices and $F$ is a nonlinear vector.
I read different articles with saying that in order to start,let $u^{n+1}=u^n$ on the right hand side only. but we need $F$ at each iteration and I am pretty confused here.
Let $n=0$ then $u^1=u^0$ and we get $u^1$, but what will be the next step because we have $u^1$ only and for $u^2$ we need $u^1$ and $u^2$ both. Is there any one who can help me? Your cooperation will be appreciated.
Kind regards,
Bibigul