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Implicit Scheme

It means that we can rewrite the problem as Vj=A1Vj1=A1(A1Vj2)==AjV0. This is an implicit scheme since it relies on determining the vector ui,j1 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.