Factorizations and fast diagonalization for the heat equation

This work investigates diagonalization-based methods for efficiently solving linear evolution problems, with a particular focus on the heat equation. The plain diagonalization of the differential operator, though effective for elliptic problems where fast diagonalization can be used, exhibits instability when applied to the heat equation. To address this difficulty, we examine three alternative approaches, based on LU factorization, a suitable arrowhead factorization, and a low-rank modification. These methods introduce more robust factorizations of the time derivative, ensuring both computational efficiency and stability.