ETD

• convection-diffusion equation
• hss method
• Krylov method
• numerical method
• parabolic pde
• splitting

In this thesis we investigate the use of the HSS method, first proposed by Bai, Golub and Ng for solving positive definite linear systems, for the time-stepping of parabolic PDEs that are semi-discretized with either finite differences or finite elements. This use of the HSS method is in fact analogous to the Peaceman-Rachford method, and is second-order accurate. We choose as our test problem the Convection-Diffusion equation, which models the diffusion of a quantity like heat or the concentration of a contaminant through a moving fluid. In order to efficiently solve the many linear systems that arise in the HSS method, we make use of an important result by Faber and Manteuffel which characterizes the matrices which admit a short-term recurrence Krylov method. Finally, after some considerations on preconditioner choices and extrapolation techniques, we experimentally compare a few variants of the HSS method across various parameter ranges, in order to assess their cost and performance, and provide guidelines on when their use is appropriate.