Building Blocks for Arbitrary High Order Discontinuous Galerkin Schemes

In this article we propose the use of the ADER methodology of solving generalized Riemann problems to obtain a numerical flux, which is high order accurate in time, for being used in the Discontinuous Galerkin framework for hyperbolic conservation laws. This allows direct integration of the semi-discrete scheme in time and can be done for arbitrary order of accuracy in space and time. The resulting fully discrete scheme in time does not need more memory than an explicit first order Euler time-stepping scheme. This becomes possible because of an extensive use of the governing equations inside the numerical scheme itself via the so-called Cauchy–Kovalewski procedure. We give an efficient algorithm for this procedure for the special case of the nonlinear two-dimensional Euler equations. Numerical convergence results for the nonlinear Euler equations results up to 8th order of accuracy in space and time are shown     

Erschließung eines Marmorkarstvorkommens als mitteltiefer Erdwärmesondenspeicher im Tuxertal,Tirol

Borehole heat exchangers can be economically beneficial for meeting heating and cooling demands of houses or buildings. In karst aquifers development of thermal storage and exchange systems may be problematic in terms of groundwater protection and storage design, due to possibly high groundwater velocities. The new development of the Hochstegen marble unit in the Tux Valley (Zillertal, Austria) was designed in two stages for the requested cooling and heating demands. An enhanced geothermal response test was completed using optical frequency domain reflectometry in an exploration drillhole. Additional studies focussing on local geology and hydrology were also conducted. Geothermal parameters obtained at precise depths allowed differentiating between conductive and convective heat flow and were correlated with the lithostratigraphically-conditioned karst characteristics. The borehole heat exchanger field was developed with nine 400 m deep dual U?shaped tube probes in 2013 for 1 GWh/a extraction and 400 MWh/a induction. Along with borehole geophysics and geothermal response tests, the study has provided relevant geothermal data for improving storage design and exploration.     

The numerical modeling of acoustic wave propagation using the multiple pressure variables approach

Based on an asymptotic analysis of the compressible Navier–Stokes equations, the basic concept of the Multiple Pressure Variables (MPV) approach is to split up the pressure into three different terms representing global background effects, acoustic waves and the incompressible pressure, respectively. Special attention is payed to aeroacoustic phenomena and the numerical treatment of the generation and spreading of sound waves. To this aim, further perturbation analysis is done about the incompressible limit solution. This gives hints how to extract acoustic waves responsible for the noise generation from the flow field.This work has been supported by Deutsche Forschungsgemeinschaft (DFG).     

Efficient Parallelization of a Shock Capturing for Discontinuous Galerkin Methods using Finite Volume Sub-cells

We present a shock capturing procedure for high order Discontinuous Galerkin methods, by which shock regions are refined in sub-cells and treated by finite volume techniques. Hence, our approach combines the good properties of the Discontinuous Galerkin method in smooth parts of the flow with the perfect properties of a total variation diminishing finite volume method for resolving shocks without spurious oscillations. Due to the sub-cell approach the interior resolution on the Discontinuous Galerkin grid cell is nearly preserved and the number of degrees of freedom remains the same. This structure allows the interpretation of the data either as DG solution or as finite volume solution on the subgrid. In this paper we explain the efficient implementation of this coupled method on massively parallel computers and show some numerical results.     

Arbitrary High-Order Discontinuous Galerkin Schemes for the Magnetohydrodynamic Equations

In this paper, we propose a discontinuous Galerkin scheme with arbitrary order of accuracy in space and time for the magnetohydrodynamic equations. It is based on the Arbitrary order using DERivatives (ADER) methodology: the high order time approximation is obtained by a Taylor expansion in time. In this expansion all the time derivatives are replaced by space derivatives via the Cauchy-Kovalevskaya procedure. We propose an efficient algorithm of the Cauchy-Kovalevskaya procedure in the case of the three-dimensional magneto-hydrodynamic (MHD) equations. Parallel to the time derivatives of the conservative variables the time derivatives of the fluxes are calculated. This enables the analytic time integration of the volume integral as well as that of the surface integral of the fluxes through the grid cell interfaces which occur in the discrete equations. At the cell interfaces the fluxes and all their derivatives may jump. Following the finite volume ADER approach the break up of all these jumps into the different waves are taken into account to get proper values of the fluxes at the grid cell interfaces. The approach under considerations is directly based on the expansion of the flux in time in which the leading order term may be any numerical flux calculation for the MHD-equation. Numerical convergence results for these equations up to 7th order of accuracy in space and time are shown.     

Wasserwirtschaftliche Schlussfolgerungen aus der Stichtagsbeprobung eines 225 km2 großen Einzugsgebietes im NW Tauernfenster, Österreich

Grundwasser - Wie in anderen Alpenregionen auch, erfolgt die Trinkwasserversorgung in den Zillertaler Alpen vorwiegend privat oder in Genossenschaften aus nahe gelegenen Quellen, typischerweise mit...     

Erratum zu: Wasserwirtschaftliche Schlussfolgerungen aus der Stichtagsbeprobung eines 225 km2 großen Einzugsgebietes im NW Tauernfenster, Österreich

Grundwasser - Erratum zu: Grundwasser – Zeitschrift der Fachsektion Hydrogeologie 2019 https://doi.org/10.1007/s00767-019-00436-9 Im Beitrag, S. 2 im Abschnitt Untersuchungsgebiet,...     

Erkundung und Beweissicherung für eine geothermale Erschließung eines Alpinen Karstaquifers im Tuxertal, Österreich

Karst aquifers may on one hand improve the efficiency of geothermal systems due to increased permeabilities, but on the other hand, high groundwater velocities can reduce the efficiency of the underground heat storage capacity. The marble karst aquifer of the Hochstegen formation was explored and developed for the first time as an intermediate-depth geothermal energy storage system at Finkenberg, Tux valley (Tyrol, Austria).Geological field studies and a spring monitoring program for the project revealed characteristic hydro-chemical signatures related to the catchments in specific tectonic units depending on their lithology. Observations showed that the catchment area of the Hochstegen formation karst aquifer extends up to 2650 m a.s.l. southwest of Finkenberg. In the boreholes, karstification was detected to 400 m below surface (Sass et al., 2016). A monitoring program involving seven springs downgradient of the boreholes has shown that the geothermal project has had no long-term impact on groundwater quality.     

Microjet-Initiated Nano-Gaseous Layer Pinch-Off from the Surface of a Bubble and Subsequent Breakup

This paper describes the first observation and study of the pinch-off and breakup of nano-gaseous layers from the surface of an accelerated bubble by a microjet. The microjet is generated by the interaction between two bubbles in a microfluidic channel. Femtoliter-sized bubbles are developed from the breakup of the nano-gaseous layers due to hydrodynamic instability. The dynamics of this whole process were demonstrated experimentally and studied theoretically. The ability to control the pinch-off and breakup of a thin layer, especially in micro/nanofluidic systems, offers an opportunity to conduct precise manipulation on nanoscale materials for a wide range of cell biological applications.     

A low Mach number scheme based on multi-scale asymptotics

The low Mach number regime is characterized by a large discrepancy between the flow velocity and the speed of sound, leading to physical effects on different length scales and of different orders of magnitude. A single time scale, multiple space scale asymptotic analysis provides detailed insight into the limit behavior of solutions of the compressible Euler equations as the Mach number tends to zero. This analysis shows that “the pressure” splits up into three parts with different physical meanings. This knowledge is then used to develop a numerical scheme including multiple pressure variables to account for the different effects. The numerical method is a semi-implicit predictor-corrector algorithm. In the predictor step, the asymptotic equations are used to guess the global and large scale effects. Then the corrector step can be viewed as an incompressible solver with compressibility effects acting as source terms. Received: 28 July 1999 / Accepted: 21 September 1999