### Introduction to the SLIM model

SLIM is a hydrodynamical model based on finite element method (FEM). The main advantage of FEM formulation is that it allows the use of unstructured grids. The computational grid can therefore  be refined arbitrarily in the areas of interest thus focusing the computational power where it is needed, without the need of nested grids. The latter gives rise to multi-scale modelling as the spatial resolution may vary greatly within the same grid and a single model is able to resolve both the large-scale features, such as in the open sea, but also small-scale phenomena in shallow areas, coasts, estuaries and rivers. Moreover, coast lines can be represented as piecewise linear curves in contrast to staircase-like boundaries of the structured grids. In global scale applications the poles have traditionally posed a problem  as they represent a singularity in the coordinate system. Such difficulties are absent with unstructured grids.

#### 1D, 2D and 3D models

 1D mesh 2D mesh 3D mesh

SLIM consists of a 1D river model, a 2D depth averaged model and 3D barotropic/baroclinic model. It therefore can be applied on a wide range of problems. Currently the 1D and 2D models can be coupled to simulate an entire sea - estuary - river network continuum in one model. SLIM utilises a generic 3D coordinate system where the curvature of Earth can easily be taken into account, which renders it suitable for large geophysical applications.

The 1D river model consists of linear river segments where variable river width and cross-section are taken into account. River segments can be joined to model a river network with accurate computation of bifurcation by the means of a Riemann solver.

In the 2D model the domain is divided into triangular elements allowing accurate representation of complex topography. The meshes are generated with GMSH software.

The 3D model uses triangular prismatic elements that are formed by extruding the 2D mesh in the vertical direction. The governing equations are solved using the mode-splitting technique, i.e. the 2D depth-averaged system is solved first and the vertical structure is updated afterwards.

#### Discrete Galerkin formulation

The spatial derivative operators are discretised with the Discontinuous Galerkin (DG) finite element method for both the free surface elevation and the velocity field. The numerical solution is thus a piecewise polynomial function that is discontinuous at the element interfaces. The inter-element fluxes are solved with an approximate Riemann solver. The DG-FEM approach can be seen as a mixture of finite volume and finite element methods and it has several advantages: Because characteristic variables are upwinded across the element interfaces, DG-FEM well-suited for advection dominated problems and does not suffer from oscillations or excessive numerical dissipation. Moreover, due to the completely discontinuous elements, DG-FEM is highly parallelisable and local mass conservation is ensured. DG-FEM is also very flexible in terms of mesh topologies and element types, which makes it an attractive approach for hp-adaptivity (adaptation of the mesh resolution and/or the polynomial degree of the solution).

#### Time stepping

Currently available time-marching schemes are explicit and semi-explicit Runge-Kutta schemes as well as diagonally implicit Runge-Kutta using Newton-Raphson iteration. The latter is very advantageous for large multi-scale simulations as it allows taking long time steps independently of the spatial resolution.

#### Further information

The subsequent pages offer more in-depth information on the following topics: