# implicit method (crank-Nicolson) I not understand the procedure [closed]

I'm trying to understand the passage through this equation can be written for easily solved with the fortran alghorithm in particular i don't understood the meaning of L_x and L_xx ... what (-1,0,1) stands for ? could somebody explain me ? thanks a lot !

the equation is reported here (explicit scheme is clear .. obviously) enter link description here

## 1 Answer

$L_x$ and $L_{xx}$ are shorthands (operators) to denote the more extended notation: $$L_x u_i=(u_{i+1}-u_{i-1})$$ and $$L_{xx}u_i = (u_{i-1}-2u_i+u_{i+1})$$. Therefore $L_x$ can be written in the node $i$ as the vector $L_x=(-1,0,1)$ (see the coefficients of $u_{i-1}$ which is $-1$, $u_i$ zero and $u_{i+1}$ one). The same results for $L_{xx}$.

For a better comprehension one could formally write, for a given stencil: $\tilde{u}_i=(u_{i-1}, u_i,u_{i+1})^{T}$ the scalar product: $$L_x \tilde{u}_i=(-1,0,1)(u_{i-1}, u_i,u_{i+1})^{T}=u_{i+1}-u_{i-1}$$

Hope this helps. For a description of the procedure of the CN scheme you can see: https://scicomp.stackexchange.com/a/27190/22590

• thank you very much !! now it's more clear ! .. just the last thing ! could you explain me what is (in Fortran code) the last terms of d(j) .. dt* (lambda * U1(j) + alpha*U1(j)**3 ) where does lambda and alpha comes from ? what is their physicals meaning ? (a sort of under-relaxation ?? or what else ?) Thanks – Marco Ghiani Sep 24 '17 at 11:12