Implicit Scheme
It means that we can rewrite the problem as
\begin{equation*}
V_{j} = \mathbf{A}^{-1}V_{j-1}=\mathbf{A}^{-1}\left(\mathbf{A}^{-1}V_{j-2}\right)=\dots = \mathbf{A}^{-j}V_0.
\end{equation*}
This is an implicit scheme since it relies on determining the vector
u_{i,j-1} instead of
u_{i,j+1} .
If
\alpha does not depend on time
t , we need
to invert a matrix only once. Alternatively we can solve this system of equations using our methods
from linear algebra.
These are however very cumbersome ways of solving since they involve
\sim O(N^3) operations
for a
N\times N matrix.
It is much faster to solve these linear equations using methods for tridiagonal matrices,
since these involve only
\sim O(N) operations.