 Solution

Choose a fixed time increment $h=\Delta t>0$, let $t_n=nh$, and approximate $\mathbf{c}(t_n)$ by ${\mathbf{c}}_n$. The Backward Euler method evaluates the transport term at the new time:Rearrange to the linear systemAlthough $A$ is a singular matrix, $B$ is an invertible matrix for every $h>0$. In each row, its diagonal entry is $1+2h$ and the sum of the absolute values of its off-diagonal entries is $2h$; hence it is a strictly diagonally dominant matrix. Its Gershgorin discs have centre $1+2h$ and radius $2h$, excluding zero. In particular, the zero eigenvalue of $A$ becomes an eigenvalue $1$ of $B$.

The algorithm uses Reusing LU factors for multiple right-hand sides:

Both triangular solves use the factors of $I-hA$. The LU factorization of the singular transport matrix $A$ from part (a) is not a solver for this new linear system. No matrix inverse needs to be formed. Since $A$ and $h$ are fixed, the same LU factorization serves every step; only the right-hand side changes. A varying $\mathbf{b}(t)$ would give $r_n={\mathbf{c}}_n-h\mathbf{b}(t_{n+1})$ without changing these factors.

As a check, multiplying the update by $1^T$ givesexactly reproducing the inventory balance at the sampled times, apart from roundoff error. The Backward Euler method has temporal global error $O(h)$ for smooth solutions, so its A-stability does not remove the need to select $h$ for accuracy.