A local discontinuous Galerkin scheme for Darcy flow with internal jumps - Rupp, Andreas; Knabner, Peter; Dawson, Clint (Computational Geosciences, 2018)
Keywords: Local discontinuous Galerkin method; Instationary Darcy problem; Jump condition; Henry’s Law; Different orders of approximation spaces

Analysis of a mixed discontinuous Galerkin method for instationary Darcy flow - Aizinger, Vadym; Rupp, Andreas; Schütz, Jochen; Knabner, Peter (Computational Geosciences, 2018)
Keywords: Mixed discontinuous Galerkin method; Stability and error analysis; Local discontinuous Galerkin method; Instationary Darcy problem

Application of a cellular automaton method to model the structure formation in soils under saturated conditions: A mechanistic approach - Rupp, Andreas; Guhra, Tom; Meier, Andreas; Prechtel, Alexander; Ritschel, Thomas; Ray, Nadja; Totsche, Kai Uwe (Frontiers in Environmental Science, 2019)
Soil functions are closely related to the structure of soil microaggregates. Yet, the mechanisms controlling the establishment of soil structure are diverse and partly unknown. Hence, the understanding of soil processes and functions requires the connection of the concepts on the formation and consolidation of soil structural elements across scales that are hard to observe experimentally. At the bottom level, the dynamics of microaggregate development and restructuring build the basis for transport phenomena at the continuum scale. By modeling the interactions of specific minerals and/or organic matter, we aim to identify the mechanisms that control the evolution of structure and establishment of stationary aggregate properties. We present a mechanistic framework based on a cellular automaton model to simulate the interplay between the prototypic building units of soil microaggregates quartz, goethite, and illite subject to attractive and repulsive electrostatic interaction forces. The resulting structures are quantified by morphological measures. We investigated shielding effects due to charge neutralisation and the aggregate growth rate in response to the net system charge. We found that the fraction as well as the size of the interacting oppositely charged constituents control the size, shape, and amount of occurring aggregates. Furthermore, the concentration in terms of the liquid solid ratio has been shown to increase the aggregation rate. We further adopt the model for an assessment of the temporal evolution of aggregate formation due to successive formation of particle dimers at early stages in comparison to higher order aggregates at later stages. With that we show the effect of composition, charge, size ratio, time and concentration on microaggregate formation by the application of a mechanistic model which also provides predictions for soil aggregation behaviour in case an observation is inhibited by experimental limitation
Keywords: Individual based modeling; microaggregate formation; mineral surface charge; morphological characteristics; soil structure simulation; self-organisation; cellular automaton

Beyond Kozeny-Carman: Predicting the Permeability in Porous Media - Schulz, Raphael; Ray, Nadja; Zech, Simon; Rupp, Andreas; Knabner, Peter (Transport in Porous Media, 2019)

Bound-preserving flux limiting schemes for DG discretizations of conservation laws with applications to the Cahn–Hilliard equation - Frank, Florian; Rupp, Andreas; Kuzmin, Dmitri (Computer Methods in Applied Mechanics and Engineering, 2020)
Many mathematical models of computational fluid dynamics involve transport of conserved quantities which must lie in a certain range to be physically meaningful. The analytical or numerical solution u of a scalar conservation law is said to satisfy a maximum principle (MP) if global bounds umin and umax exist such that umin ≤ u ≤ umax holds in the domain of definition. To enforce such inequality constraints at least for element averages in the context of discontinuous Galerkin (DG) methods, the numerical fluxes must be defined and constrained in an appropriate manner. In this work, we introduce a general framework for calculating fluxes that produce non-oscillatory DG approximations and preserve relevant global bounds for element averages even if the exact solution of the PDE violates them due to modeling errors. The proposed methodology is based on a combination of flux and slope limiting. The (optional) slope limiter adjusts the gradients to impose local bounds on pointwise values of the high-order DG solution which is used to calculate the fluxes. The flux limiter constrains changes of element averages so as to prevent violations of global bounds. Since manipulations of the target flux may introduce a consistency error, it is essential to guarantee that physically admissible fluxes remain unchanged. We propose two kinds of flux limiters which meet this requirement. The first one is of monolithic type and its time-implicit version requires the iterative solution of a nonlinear problem. Only a fully converged solution is provably bound-preserving. The explicit version of this limiter is subject to a time step restriction which we derive in this article. The second limiter is an iterative version of the multidimensional flux-corrected transport (FCT) algorithm. Our fractional step limiting (FSL) approach guarantees the MP property for each iterate but avoidable consistency errors may occur if the iterative process is terminated too early. Practical applicability of the proposed iterative limiters is demonstrated by numerical studies for the advection equation (hyperbolic, linear) and the Cahn-Hilliard equation (parabolic, nonlinear).<br

Convergence order estimates of the local discontinuous Galerkin method for instationary Darcy flow - Rupp, Andreas; Knabner, Peter (Numerical Methods For Partial Differential Equations, 2017)
Keywords: instationary Darcy problem; error analysis; stability analysis; local discontinuous Galerkin method

Discontinuous Galerkin method for coupling hydrostatic free surface flows to saturated subsurface systems - Reuter, Balthasar; Rupp, Andreas; Aizinger, Vadym; Knabner, Peter (Computers & Mathematics with Applications, 2019)
Keywords: Darcy flow; Coupled model; Hydrostatic equations; Discrete energy stability analysis; Three-dimensional shallow water equations with free surface; Local discontinuous Galerkin method

Discrete-Continuum Multiphase Model for Structure Formation in Soils Including Electrostatic Effects - Rupp, Andreas; Totsche, Kai Uwe; Prechtel, Alexander; Ray, Nadja (Frontiers in Environmental Science, 2018)

Discrete-continuum multiscale model for transport, biomass development and solid restructuring in porous media - Ray, Nadja; Rupp, Andreas; Prechtel, Alexander (Advances in Water Resources, 2017)
Upscaling transport in porous media including both biomass development and simultaneous structural changes in the solid matrix is extremely challenging. This is because both affect the medium's porosity as well as mass transport parameters and flow paths. We address this challenge by means of a multiscale model. At the pore scale, the local discontinuous Galerkin (LDG) method is used to solve differential equations describing particularly the bacteria's and the nutrient's development. Likewise, a sticky agent tightening together solid or bio cells is considered. This is combined with a cellular automaton method (CAM) capturing structural changes of the underlying computational domain stemming from biomass development and solid restructuring. Findings from standard homogenization theory are applied to determine the medium's characteristic time- and space-dependent properties. Investigating these results enhances our understanding of the strong interplay between a medium's functional properties and its geometric structure. Finally, integrating such properties as model parameters into models defined on a larger scale enables reflecting the impact of pore scale processes on the larger scale. (C) 2017 Elsevier Ltd. All rights reserved.
Keywords: Soil structure; Biomass development; Transport properties; Mathematical modeling; Cellular automaton; Multiscale methods

FESTUNG: A MATLAB / GNU Octave Toolbox for the Discontinuous Galerkin Method. Part IV: Generic Problem Framework and Model-Coupling Interface - Reuter, Balthasar; Rupp, Andreas; Aizinger, Vadym; Frank, Florian; Knabner, Peter (Communications in Computational Physics, 2020)
This is the fourth installment in our series on implementing the discontinuous Galerkin (DG) method as an open source MATLAB / GNU Octave toolbox. Similarly to its predecessors, this part presents new features for application developers employing DG methods and follows our strategy of relying on fully vectorized constructs and supplying a comprehensive documentation. The specific focus of the current work is the newly added generic problem implementation framework and the highly customizable model-coupling interface for multi-domain and multi-physics simulation tools based on this framework. The functionality of the coupling interface in the FES-TUNG toolbox is illustrated using a two-way coupled free-surface / groundwater flow system as an example application.
CRIS-Team WoS Importer:2020-06-26

Numerical benchmark study for flow in highly heterogeneous aquifers - Alecsa, Cristian D.; Boros, Imre; Frank, Florian; Knabner, Peter; Nechita, Mihai; Prechtel, Alexander; Rupp, Andreas; Suciu, Nicolae (Advances in Water Resources, 2020)
This article presents numerical investigations on accuracy and convergence properties of several numerical approaches for simulating steady state flows in heterogeneous aquifers. Finite difference, finite element, discontinuous Galerkin, spectral, and random walk methods are tested on two-dimensional benchmark flow problems. Realizations of log-normal hydraulic conductivity fields are generated by Kraichnan algorithms in closed form as finite sums of random periodic modes, which allow direct code verification by comparisons with manufactured reference solutions. The quality of the methods is assessed for increasing number of random modes and for increasing variance of the log-hydraulic conductivity fields with Gaussian and exponential correlation. Experimental orders of convergence are calculated from successive refinements of the grid. The numerical methods are further validated by comparisons between statistical inferences obtained from Monte Carlo ensembles of numerical solutions and theoretical first-order perturbation results. It is found that while for Gaussian correlation of the log-conductivity field all the methods perform well, in the exponential case their accuracy deteriorates and, for large variance and number of modes, the benchmark problems are practically not solvable with reasonably large computing resources, for all the methods considered in this study.<br

Old and New Approaches Predicting the Diffusion in Porous Media - Ray, Nadja; Rupp, Andreas; Schulz, Raphael; Knabner, Peter (Transport in Porous Media, 2018)
Keywords: Upscaling; Tortuosity; Diffusion in porous media

Simulating Structure Formation in Soils across Scales using Discontinuous Galerkin Methods - Rupp, Andreas (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2019)
This thesis deals with mathematical modeling, analysis, and numerical realization of microaggregates in soils. These microaggregates have the size of a few hundred micrometers and can be understood as the fundamental building units of soil. Thus, understanding their dynamically evolving, three-dimensional structure is crucial for modeling and interpreting many soil parameters such as diffusivities and flow paths that come into play in CO2-sequestration or oil recovery scenarios. Moreover, effects and knowledge deduced from the model are transfered to scales which are more relevant for applications. The quality of these averaged models is of general interest, since simulations for the field-scale that resolve the pore-scale are not applicable for economical reasons. Thus, this book compares parameterizations of diffusivities with mathematically rigorous results and gives suggestions to improve the formulas that can be found in the literature. Last but not least, it is imperative to apply a proper numerical method to implement the model in silico. The local discontinuous Galerkin (LDG) method seems to be suitable for this task, since it is locally mass-conservative and is stable for discontinuous data - that might, for example, originate from the discrete movement of the geometry or from the sharp boundaries between the different phases. Additionally, this method has no problems with complicated transfer conditions. These aspects are demonstrated in a mathematically rigorous way, and the method is improved upon by reducing the linear system of equations resulting from the discretization. This is a real enhancement, since it does not diminish the order of convergence but decreases the computational costs.
Keywords: microaggregate development; discontinuous Galerkin; upscaling; diffusion; homogenization; Darcy’s equation