Klaus Johannsen

The saltpool benchmark problem – numerical simulation of saltwater upconing in a porous medium

Recently, a series of laboratory experiments was carried out in which a typical variable-density flow problem in a porous medium was investigated. A stable layering of saltwater below freshwater is affected by the recharge and discharge of freshwater on the top. The experiments were conducted with 1% and 10% initial salt-mass fraction contrasts. In this paper, we define a mathematical model problem, which is able to reproduce the experimental results within a reasonable accuracy. To this end, the sensitivity of the model with respect to model parameters is investigated. An inverse modelling leads to an appropriate choice of the model parameters and the definition of the mathematical benchmark problem. It will become clear that, in the 10% case, the transversal dispersion coefficient plays an important role. Detailed numerical investigations are carried out and reference solutions are obtained. For the high concentration case a very high spatial grid resolution using up to 16 million grid points is necessary. Error bounds are derived for the solutions without any a priori assumption on regularity and convergence.

A Parallel Software-Platform for Solving Problems of Partial Differential Equations using Unstructured Grids and Adaptive Multigrid Methods

The goal of this work is the development of a parallel software-platform for solving partial differential equation problems. State-of-the-art numerical methods have been developed and implemented for the efficient and comfortable solution of real-world application problems. Emphasis is laid on the following topics: distributed unstructured grids, adaptive grid refinement, derefinement/coarsening, robust parallel multigrid methods, various FE and FV discretizations, dynamic load balancing, mapping and grid partitioning. Some important application examples will be presented including structural mechanics, two-phase flow in porous media, Navier-Stokes problems (CFD) and density-driven groundwater flow.

Advances in High-Performance Computing: Multigrid Methods for Partial Differential Equations and its Applications

The program package UG provides a software platform for discretizing and solving partial differential equations. It supports high level numerical methods for unstructured grids on massively parallel computers. Various applications of complex up to real-world problems have been realized, like Navier-Stokes problems with turbulence modeling, combustion problems, two-phase flow, density driven flow and multi-component transport in porous media. Here we report on new developments for a parallel algebraic multigrid solver and applications to an eigenvalue solver, to flow in porous media and to a simulation of Navier-Stokes equations with turbulence modeling.

An Aligned 3D-Finite-Volume Method for Convection-Diffusion Problems

A new construction for the finite volume method is presented. It is applicable to 2D and 3D unstructured meshes and uses finite volumes aligned to a given velocity. In the case of a stationary scalar convection-diffusion equation it provides a better approximation than standard finite volumes. In the special case of grid-alignment the discretization of the convective part results in a two-point-star if full upwinding is used. Numerical results are presented demonstrating the improved approximation properties.

High-accuracy Simulation of Density Driven Flow in Porous Media

A three-dimensional test case for the density driven flow equations in porous media recently proposed by Oswald, Scheidegger and Kinzelbach is investigated numerically. It was shown in [14] that the results from the physical experiment can be reproduced if parameters are adjusted correctly and that a mathematical benchmark can be defined as an idealization of the physical experiment. Intensive numerical investigations were carried out, with calculations on up to 16.7 million grid points that required the efficient solution of linear systems with more than 35 million unknowns.

The importance of dispersive mixing for modelling of density-dependent and reactive transport

It is well known that dispersion is a major difficulty in the modelling of transport in groundwater, that is the physical dispersion itself as well as numerical dispersion. This study addresses the topic of how important the process of dispersive mixing could be especially for coupled problems like density-dependent flow and reactive transport. This is based on evaluating the physical process of dispersive mixing in laboratory visualisation experiments, and using this data to assess the behaviour of specific codes and the underlying conceptual model.

A robust smoother for convection-diffusion problems with closed characteristics

Abstract In this paper we analyze a model problem for the convection-diffusion equation where the reduced problem has closed characteristics. A full upwinding finite difference scheme is used to discretize the problem. Additionally to the strength of the convection, an arbitrary amount of crosswind-diffusion can be added on the discrete level. We present a smoother which is robust w.r.t. the strength of convection and the amount of crosswind-diffusion. It is of Gauss–Seidel type using a downwind ordering. To handle the cyclic dependencies a frequency-filtering algorithm is used. The algorithm is of nearly optimal complexity ?(n log n). It is proved that it fulfills a robust smoothing property.