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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
$L_x$ 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
