Multigrid waveform relaxation methods (see [4]) applied to parabolic evolution equations enable the simultaneous solution of multiple time steps. Thus, they can improve the scaling behavior compared to traditional time-stepping methods. However, using spatial point smoothers in this context requires a forward sweep in time for each degree of freedom in space, which is equivalent to solving a scalar ordinary differential equation (ODE). This can be parallelized using cyclic reduction. Instead, in this talk we explore the idea of employing associative scan algorithms to solve scalar ODEs in parallel (cf. [2]) and applying this approach to the smoother. Furthermore, we discuss how this generalizes to patch-based smoothing strategies. This strategy can be used in a parallel-in-time Navier-Stokes solver for low Reynolds numbers that uses a global-in-time Newton linearization and a Schur complement splitting in each linear step. To achieve robust convergence behavior with respect to the number of time steps an LSC preconditioner combined with the augmented Lagrangian methodology is necessary [3]. This, in turn, requires patch smoothers to solve the velocity block. The use of time-parallel waveform relaxation smoothers in this setting enables a space-time parallel Navier-Stokes solver with lower computational overhead than other time-concurrent multigrid methods, such as space-time multigrid [1]. REFERENCES [1] Gander M.J., Neumüller M. (2016). Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems, SIAM J. Sci. Comput., 38(4), pp. A2173 - A2208, 2016. [2] Iacob C., Razavi H., Särkkä S. A parallel-in-time Newton’s method-based ODE solver, arXiv, https://doi.org/10.48550/arXiv.2511.01465, 2025. [3] Lohmann C., Turek S., Augmented Lagrangian Acceleration of Global–in–Time Pressure Schur Complement Solvers for Incompressible Oseen Equations, J. Fluid Mech., 26(27), 2024. [4] Lubich C., Ostermann A., Multi-grid Dynamic Iteration for Parabolic Equations, BIT, 27(2), pp. 216–234, 1987