Granular flows have attracted increasing attention in recent decades due to their relevance across a wide range of natural and industrial settings. They are key to understanding natural hazards (landslides, avalanches, debris flows) and to modeling industrial operations in mining (crushing, grinding, sorting), pharmaceuticals (powder mixing, tablet formation), and food processing (e.g., rice, barley, and malt handling) [1]. The μ(I)-rheology is widely used to model granular flows, but its strong nonlinearity can lead to numerical challenges, including non-physical instabilities in certain inertial-number regimes and difficulties associated with pressure-dependent viscosity. Several numerical approaches exist for simulating these flows, ranging from particle-based methods (DEM, SPH, MPS) to continuum-based formulations (FEM, FVM)—with MPM and PFEM as hybrid alternatives—each with distinct advantages and limitations. In this work, we adopt FEM for spatial discretization due to its suitability for complex geometries. We present a novel implicit–explicit (IMEX) time-stepping scheme that addresses the pressure nonlinearity by treating convection, the pressure gradient, and the μ(I)-dependent viscosity explicitly, while keeping the remaining terms implicit. The resulting method is fully decoupled and linearized: at each time step, it reduces the incompressible non-Newtonian Navier–Stokes system to a linear advection–diffusion–reaction problem for the velocity and a Poisson problem for the pressure. The scheme avoids Newton/Picard iterations, is computationally efficient, and is proven unconditionally stable (i.e., no CFL restriction). Numerical results are demonstrated on debris-flow benchmarks.