Abstract Salinization of coastal aquifers is a current problem in many regions worldwide. Simulation of this process provides an important tool for the forecast of drinking water resources. The consideration of the unsaturated phreatic zone has an essential influence on the accuracy of the prediction. However, for the large temporal and spatial scales of the aquifers, the presence of the unsaturated subdomains creates challenging difficulties for the numerical methods. In this work, we investigate two approaches for simulating haline density-driven flow in partially saturated aquifers. The first approach, based on the Richards equation, uses a representation of the saturation field and employs an adaptive linearly implicit time discretization scheme. The second approach explicitly represents the water table using a level-set method and relies on weak coupling combined with a standard implicit Euler scheme. Both approaches employ a finite-volume discretization for spatial representation of fluid flow and salt transport. The implementation is based on the UG4 toolkit together with the parallel groundwater flow simulator d3f++. The resulting linear systems are solved using a geometric multigrid method. We compare the aforementioned approaches with respect to theoretical and numerical aspects. The performance of both methods is evaluated in three numerical experiments. In particular we demonstrate robustness of the linearly implicit scheme and introduce a stabilization for the level-set method. The high-performance computing (HPC) potential of the approaches is assessed and demonstrated as well.

Surface-subsurface flow models for hydrological applications solve a coupled multiphysics problem. This usually consists of some form of the Richards and shallow water equations. A typical setup couples these two nonlinear partial differential equations in a partitioned approach via boundary conditions. Full interaction between the subsolvers is ensured by an iterative coupling procedure. This can be accelerated using relaxation. In this paper, we apply continuous and fully discrete linear analysis techniques to study an idealized, linear, 1D-0D version of a surface-subsurface model. These result in explicit expressions for the convergence factor and an optimal relaxation parameter, depending on material and discretization parameters. We test our analysis results numerically for fully nonlinear 2D-1D experiments based on existing benchmark problems. The linear analysis can explain fast convergence of iterations observed in practice for different materials and test cases, even though we are not able to capture various nonlinear effects.

In recent years, cone-adapted shearlets have been established as a powerful tool for the detection of directional information in images. It has also turned out that an interpretation of cone-adapted shearlets in the realm of coorbit space theory exists. Then, cone-adapted shearlets are defined by requiring a certain sparsity pattern of the frame expansion coefficients. In this note, we discuss some structural properties of these spaces.