Development of a Semi-Lagrangian advection scheme in the Finite Element Model Elmer (v9.0): Application to Ice dynamics

Abstract. Transport processes are of great importance in geophysical applications, including atmospheric, oceanic, or ice flow dynamics. An Eulerian view is commonly adopted in models dealing with the mathematical representation of fluid dynamics. In such a framework, transport processes are accounted for by prescribing advection terms within the partial differential equations (PDEs) of the model. Yet, advection terms are prone to cause instabilities in the numerical solution of these equations, notably when using the Finite Element Method with a standard Galerkin approach. Various methods have been developed to overcome these instabilities, but often at the price of spurious artificial diffusion. To avoid such unwanted numerical smoothing, a commonly used technique is the Discontinuous Galerkin method, which allows for discontinuous solutions; hence, a better tracking of fine features with steep gradients without relying on artificial diffusion. In this study, we explore an alternative approach which lies in Semi-Lagrangian methods, combining elements of both Eulerian and Lagrangian approaches by updating particle positions based on the Eulerian velocity field from the previous time step. The method does not rely on any artificial diffusion and can accurately capture advection. Here, we present a computationally efficient Semi-Lagrangian algorithm to track the motion of particles in complex 3D geometries that is suitable for highly parallel computing. We illustrate the accuracy and power of the Semi-Lagrangian (SL) algorithm by comparing it to a Discontinuous Galerkin (DG) method developed within the open-source multi-physics code Elmer. We show that both the DG and SL methods provide accurate transport solutions with a different sensitivity to resolution. We conclude that, for practical use, the choice between the SL and the DG methods will depend on specific simulation requirements and the trade-off between acceptable diffusion and computational efficiency.