I have the following system where $k = 0,1,..., N-1$, $j = 0,1,...,n$ and $v$ is a constant
$u_k^{j+1} = u_k^j - v (u_k^j)^2 + v u_k^j u_{k-1}^j$
and I would like to write this in matrix-vector form if possible.
I have tried to do this myself but I have the term $u_k^j$ appearing in the matrix.
Thank you.
You can get a bilinear form, i.e.
$$ \mathbf{U}^{j+1} = \mathbf{U}^{j} - (\mathbf{U}^{j})^T B \, \mathbf{U}^{j} $$ with $$ B = v \cdot \begin{pmatrix} 1 & -1 & 0 & \cdots & 0 \\ 0 & 1 & -1 & \cdots & 0 \\ 0 & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & \ddots & 1 & -1 \\ 0 & \cdots & \cdots & 0 & 1 \end{pmatrix}. $$
If that helps you with your particular problem depends on the question you want to answer.