## 1 Introduction
These promising results may be related to the use of a coarse mesh, whereby simulations avoid the ill-posedness by damping the fast-growing high wavenumbers [33]. In any case, the quasi-static regime near the static/flowing interface is known to be very complex, involving strong and weak force chains and local rearrangement of particles, e.g., [36; 37] and has been widely studied, in particular using discrete element methods, e.g., [1; 38; 39]. This regime is not accurately described by the proposed viscoplastic laws involving a simple yield stress. Questions however remain as to whether such'simple' viscoplastic laws are able to describe quantitatively the change with time of the static/flowing interface position and the velocity profiles observed experimentally for flows over an initially static bed and how the viscosity affects these processes. Based on such a viscoplastic model with yield stress, an analytic expansion in the non-averaged thin-layer regime is provided in [40], giving a theoretical basis for equations describing the static/flowing interface dynamics in a dry granular material. A key point in this approach that fundamentally differs from previous thin-layer models (see [21] for a review) is that even if the flow is assumed to be thin, the normal coordinate \(Z\) is still present (there is no depth-averaging). The static/flowing interface is free in this approach, in the sense that it has no explicit equation defining its evolution. Its time variation is instead implicitly determined by an extra boundary condition on the velocity at this interface. The model from [40] being still too complicated, however a formal decoupling of the coordinates \(X\) in the down-slope direction and \(Z\) normal to the topography is proposed in [41], leading to a model with only the coordinate \(Z\), but including a source term \(S\) that represents the opposite of the net force acting on the flow, including gravity, pressure gradient, and internal friction. We propose here to evaluate the dynamics produced by the free interface model of [41], for the most  
numerical methods that give similar grid-independent results, thereby showing that our model is well-posed for the case considered here involving only \(Z\) dependency. The paper is organized as follows. The model of [40, 41] for the case of a constant source (corresponding to simple shear flows of the viscoplastic model) is recalled in Section 2. Then, in Section 3, we derive an analytical solution for the inviscid model with a constant source term, which partly reproduces the experimental observations and shows explicitly how the static/flowing interface position and the velocity profiles are related to the flow characteristics and to the initial and boundary conditions. In Section 4, we introduce two new numerical methods for the simulation of the viscous model, with a constant viscosity or the variable viscosity associated with the \(\mu(I)\) rheology. The numerical results show that, as opposed to the inviscid model, viscosity makes it possible to reproduce the initial penetration of the static/flowing interface within the static bed and the exponential shape of the velocity profiles near this interface. Finally, in Section 5, these results are discussed based on comparison with former experimental and numerical studies on granular flows, showing the key features and the limits of the non-averaged thin-layer model for providing a better understanding and modeling of laboratory and natural flows.