1. **Problem Statement:** Solve the linear system $Ap = q$ where $A$ is a graph Laplacian-like matrix of size $n=500$ nodes representing arterial networks, using the Jacobi iterative method starting from initial guess $p^{(0)}$ equal to the mean pressure. 2. **Jacobi Method Formula:** The Jacobi iteration updates the solution as: $$p^{(k+1)} = D^{-1}(q - (L+U)p^{(k)})$$ where $A = D + L + U$, with $D$ the diagonal, $L$ the strictly lower, and $U$ the strictly upper parts of $A$. 3. **Handling Ill-conditioning:** Since $\text{cond}(A) \approx 10^6$, the system is ill-conditioned due to vessel compliance. Ill-conditioning slows convergence and can cause numerical instability. 4. **Step (a) - 50 Iterations:** Perform 50 Jacobi iterations starting from $p^{(0)} = \text{mean pressure}$: - Compute $p^{(k+1)}$ using the formula above. - Monitor convergence but expect slow progress due to ill-conditioning. 5. **Step (b) - Preconditioning with Diagonal Scaling:** Precondition by scaling the system with $D^{-1}$: $$B = I - D^{-1}A$$ Jacobi iteration matrix is $B$. The error at iteration $k$ is: $$e^{(k)} = p^{(k)} - p^* = B^k e^{(0)}$$ where $e^{(0)} = p^{(0)} - p^*$ is the initial error. 6. **Step (c) - Validation Against CFD Simulation:** Compare the Jacobi solution after iterations with CFD simulation results. The error in pulse wave propagation should be less than 1%, confirming accuracy. **Summary:** - Use Jacobi iteration formula. - Start from mean pressure. - Perform 50 iterations. - Precondition with diagonal scaling to improve convergence. - Error evolves as $e^{(k)} = B^k e^{(0)}$. - Validate solution with CFD, ensuring error <1%.