Implicit Scheme

We obtain now $$\begin{equation*} u_{i,j-1}= -\alpha u_{i-1,j}+(1-2\alpha)u_{i,j}-\alpha u_{i+1,j}. \end{equation*}$$ Here $u_{i,j-1}$ is the only unknown quantity. Defining the matrix $\mathbf{A}$ $$\begin{equation*} \mathbf{A}=\begin{bmatrix}1+2\alpha&-\alpha&0& 0 &\dots\\ -\alpha&1+2\alpha&-\alpha & 0 & \dots \\ \dots & \dots & \dots & \dots &\dots \\ \dots & \dots & \dots & \dots & -\alpha \\ 0 & 0 &\dots &-\alpha& 1+2\alpha\end{bmatrix}, \end{equation*}$$ we can reformulate again the problem as a matrix-vector multiplication $$\begin{equation*} \mathbf{A}V_{j} = V_{j-1} \end{equation*}$$

• «
• 1
• ...
• 26
• 27
• 28
• 29
• 30
• 31
• 32
• 33
• 34
• 35
• 36
• 37
• 38
• 39
• 40
• 41
• 42
• 43
• ...
• 98
• »