Implicit Scheme
It means that we can rewrite the problem as
Vj=A−1Vj−1=A−1(A−1Vj−2)=⋯=A−jV0.
This is an implicit scheme since it relies on determining the vector
ui,j−1 instead of
ui,j+1.
If
α 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
∼O(N3) operations
for a
N×N matrix.
It is much faster to solve these linear equations using methods for tridiagonal matrices,
since these involve only
∼O(N) operations.