Parts of Code for the Crank-Nicolson Scheme

We can code in an efficient way the Crank-Nicolson algortihm by first multplying the matrix $$\begin{equation*} \tilde{V}_{j-1}=\left(2\hat{I}-\alpha\hat{B}\right)V_{j-1}, \end{equation*}$$ with our previous vector $V_{j-1}$ using the matrix-vector multiplication algorithm for a tridiagonal matrix, as done in the forward-Euler scheme. Thereafter we can solve the equation $$\begin{equation*} \left(2\hat{I}+\alpha\hat{B}\right) V_{j}= \tilde{V}_{j-1}, \end{equation*}$$ using our method for systems of linear equations with a tridiagonal matrix, as done for the backward Euler scheme.

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