  We propose quantum methods for solving differential equations that are based
on a gradual improvement of the solution via an iterative process, and are
targeted at applications in fluid dynamics. First, we implement the Jacobi
iteration on a quantum register that utilizes a linear combination of unitaries
(LCU) approach to store the trajectory information. Second, we extend quantum
methods to Gauss-Seidel iterative methods. Additionally, we propose a
quantum-suitable resolvent decomposition based on the Woodbury identity. From a
technical perspective, we develop and utilize tools for the block encoding of
specific matrices as well as their multiplication. We benchmark the approach on
paradigmatic fluid dynamics problems. Our results stress that instead of
inverting large matrices, one can program quantum computers to perform
multigrid-type computations and leverage corresponding advances in scientific
computing.
