Directional diffusion splitting method for advection-diffusion-reaction model

In the context of investigating the potential application of quantum computers to the finite element method, we propose a specific approach for solving the two-dimensional parabolic advection-diffusion-reaction boundary value problem by transforming it into a set of corresponding one-dimensional problems. This method employs specialized operator splitting, resulting in a recurrent fractional-step scheme for time integration. While this scheme offers significant parallelization potential, the main goal is to map the original large elliptic 2D problem arising at each time step into a set of smaller 1D problems of the same kind. This transformation, combined with recently proposed quantum-assisted stabilization schemes, allows us to pass smaller linear systems to a quantum linear solver, making them suitable for near-term quantum computers. We combine the proposed algorithm with the finite element method to solve the resulting one-dimensional problems, although these problems can also be addressed using other discretization techniques, such as the finite volume or finite difference methods.