Vadym Aizinger

A discontinuous/continuous low order finite element shallow water model on the sphere

We study the applicability of a new finite element in atmosphere and ocean modeling. The finite element under investigation combines a second order continuous representation for the scalar field with a first order discontinuous representation for the velocity field and is therefore different from continuous and discontinuous Galerkin finite element approaches. The specific choice of low order approximation spaces is attractive because it satisfies the Ladyzhenskaya-Babuska-Brezzi condition and is, at the same time, able to represent the crucially important geostrophic balance. The finite element is used to solve the viscous and inviscid shallow water equations on a rotating sphere. We introduce the spherical geometry via a stereographic projection. The projection leads to a manageable number of additional terms, the associated scaling factors can be exactly represented by second order polynomials. We perform numerical experiments considering steady and unsteady zonal flow, flow over topography, and an unstable zonal jet stream. For ocean applications, the wind driven Stommel gyre is simulated. The experiments are performed on icosahedral geodesic grids and analyzed with respect to convergence rates, conservation properties, and energy and enstrophy spectra. The results match quite well with results published in the literature and encourage further investigation of this type of element for three-dimensional atmosphere/ocean modeling.

Scale separation in fast hierarchical solvers for discontinuous Galerkin methods

We present a method for solution of linear systems resulting from discontinuous Galerkin (DG) approximations. The two-level algorithm is based on a hierarchical scale separation scheme (HSS) such that the linear system is solved globally only for the cell mean values which represent the coarse scales of the DG solution. The system matrix of this coarse-scale problem is exactly the same as in the cell-centered finite volume method. The higher order components of the solution (fine scales) are computed as corrections by solving small local problems. This technique is particularly efficient for DG schemes that employ hierarchical bases and leads to an unconditionally stable method for stationary and time-dependent hyperbolic and parabolic problems. Unlike p-multigrid schemes, only two levels are used for DG approximations of any order. The proposed method is conceptually simple and easy to implement. It compares favorably to p-multigrid in our numerical experiments. Numerical tests confirm the accuracy and robustness of the proposed algorithm.

A three-dimensional discontinuous Galerkin model applied to the baroclinic simulation of Corpus Christi Bay

In this work, we present results of a numerical study of Corpus Christi Bay, Texas and surrounding regions and compare simulated model results to recorded data. The validation data for the year 2000 include the water elevation, velocity, and salinity at selected locations. The baroclinic computations were performed using the University of Texas Bays and Estuaries 3D (UTBEST3D) simulator based on a discontinuous Galerkin finite element method for unstructured prismatic meshes. We also detail some recent advances in the modeling capabilities of UTBEST3D, such as a novel turbulence scheme and the support for local vertical discretization on parts of the computational domain. All runs were conducted on parallel clusters; an evaluation of parallel performance of UTBEST3D is included.

Robust shallow water models

Shallow water models have been used extensively for decades with various applications in hydroscience and engineering such as flood prediction and management, river restoration and engineering, environmental hydraulics and morphodynamics. In recent years, there has been a special focus on developing robust solvers which are capable of simulating a wide range of complex hydrodynamic problems including dam breaks (shock propagation, wetting and drying, small water depths), as well as flow transitions (subcritical–supercritical flow and vice versa, hydraulic jumps). In addition to classical finite-volume and finite-element schemes, newer methods such as smoothed particle hydrodynamics (SPH) are now being applied and extended. Furthermore, we have seen rapid development in surveying techniques (e.g. airborne and terrestrial laser scanning) which describe the geometry and physical properties of our systems, providing us with increasingly high-resolution information. Substantial improvements have also been made in high-performance computing, with notable advances including Graphic Processing Unit (GPU)-based parallel computing as well as scaling methods, relying, for example, on frictionor porosity-based approaches. Linkage of robust numerical methods, highresolution data, high-performance computing, scaling methods and new information and communication technologies will lead to the next generation of shallow water models opening up new application fields such as rainfall– runoff simulation in urban and rural catchments, real-time flood and tsunami prediction and management, as well as the spreading of impulse water waves generated by landslides. In view of these developments, we have encouraged submissions on both the development of numerical models and interesting applications, for the XX International Conference on Computational Methods in Water Resources held on June 10–13, 2014 in Stuttgart, Germany, especially for the sessions on ‘Robust shallow water models’, ‘Numerical methods for waves, circulation and transport in the coastal Ocean’ and ‘High-performance computing, visualization and scientific workflow’. This special issue contains papers which have been selected by the guest editors among about 20 contributions. The corresponding authors have been offered the opportunity to contribute to this special issue through a paper about their conference contribution (abstract, oral, or poster presentation). The papers have all undergone a full peer-review process as regular journal submissions and, finally, we are happy to present this special issue consisting of ten papers. This special issue widens the already existing journal focus on shallow water models and their applications, see, for example, Delfs et al. (2013), Shi et al. (2014), Carbajal et al. (2014) or Huang et al. (2014). This special issue starts with two contributions dealing with floods. Beisiegel and Behrens (2015) have developed a robust discontinuous Galerkin (DG) model using special basis functions and a slope limiter. Flooding and drying is simulated through applying a stable scheme, and the model’s advantages are demonstrated in a numerical study. & Reinhard Hinkelmann reinhard.hinkelmann@wahyd.tu-berlin.de

Analysis of a mixed discontinuous Galerkin method for instationary Darcy flow

We present an a priori stability and convergence analysis of a new mixed discontinuous Galerkin scheme applied to the instationary Darcy problem. The analysis accounts for a spatially and temporally varying permeability tensor in all estimates. The proposed method is stabilized using penalty terms in the primary and the flux unknowns.

A hierarchical scale separation approach for the hybridized discontinuous Galerkin method

In this work, the hierarchical scale separation (HSS) method developed for linear systems resulting from discontinuous Galerkin (DG) discretizations has been extended to hybridized discontinuous Galerkin (HDG) schemes. The HSS method is related to p-multigrid techniques for DG systems but is conceptually much simpler. Our extension of the HSS scheme to the HDG method tested using a convectiondiffusion equation for a range of benchmark problems demonstrated excellent performance on a par with that of the HSS method for a non-hybridized DG approximation. In the limiting case of a pure convection equation, the measured convergence rate of the HSS scheme was significantly better than the rates demonstrated in the non-hybridized case.

On numerical modelling of impulse water waves generated by submarine landslides

An algorithm is presented for numerical modelling of impulse water waves generated by submarine landslides moving along irregular bottom profiles. A spatially nonuniform submarine landslide moving on a spatially nonuniform slope is modelled by a “quasi-deformed” rigid body. In addition, a simplified model is studied for the particular case of landslide and bottom profiles dependent on one spatial coordinate only. Both models are used for the comparative analysis of numerical and experimental data for submarine rigid landslides moving along a plane slope. The simplified model is applied to analyse the dependences of wave characteristics on various parameters for submarine landslides moving along a sea bottom slope with monotonically increasing depth and in a bounded reservoir. The full model is used to study the landslide trajectories and the wave patterns for the model submarine landslide of a spatially irregular shape moving in the model reservoir with a spatially irregular bottom. All numerical computations are performed using the nonlinear shallow water equations with moving bottom and the time-stepping algorithm provided by a predictor–corrector scheme on adaptive grids.

A Discontinuous Galerkin Method for the Subjective Surfaces Problem

The work formulates and evaluates the local discontinuous Galerkin method for the subjective surfaces problem based on the curvature driven level set equation. A new mixed formulation simplifying the treatment of nonlinearities is proposed. The numerical algorithm is evaluated using several artificial and realistic test cases.

Large-scale turbulence modelling via alpha-regularisation for atmospheric simulations

We study the possibility of obtaining a computational turbulence model by means of non-dissipative regularisation of the compressible atmospheric equations for climate-type applications. We use an α-regularisation (Lagrangian averaging) of the atmospheric equations. For the hydrostatic and compressible atmospheric equations discretised using a finite volume method on unstructured grids, deterministic and non-deterministic numerical experiments are conducted to compare the individual solutions and the statistics of the regularised equations to those of the original model. The impact of the regularisation parameter is investigated. Our results confirm the principal compatibility of α-regularisation with atmospheric dynamics and encourage further investigations within atmospheric model including complex physical parametrisations.

Locally Filtered Transport for computational efficiency in multi-component advection-reaction models

Abstract This work introduces the Locally Filtered Transport (LFT) method for numerical transport models. Locally turning off the transport computation in areas of nearly uniform concentration is proposed as a new approach for reducing computational cost in ecosystem models that require transport of tens to hundreds of constituent concentrations. The proposed method is locally mass conservative just as the discontinuous Galerkin finite element scheme it is based on. The performance of the method is illustrated using numerical examples including an advection-reaction ecosystem simulation with a simple nitrogen, phytoplankton, and zooplankton (NPZ) model.