This is the AM-SURE Summer Research at NYU Courant. The research will be conducted under the supervision of Florian Wechsung.

The project will focus on developing a faster converging algorithm for solving multiple linear system simultaneously. In addition, we will study this in the context of fluid flow simulations, with the goal of improving an existing solver for Navier-Stokes and Stokes equations.

This report will give a general explanation of four analogous algorithms-- conjugate gradient, block conjugate gradient, preconditioned conjugate gradient, and preconditioned block conjugate gradient. We compare equations that describe the time and memory costs for preconditioned conjugate gradient and preconditioned block conjugate gradient. We plot graphs for convergence and time spent of these algorithms. We also find out another method to avoid the singular matrix issue that we encounter in the block version algorithms.