• Skip navigation
  • Skip to navigation
  • Skip to the bottom
Simulate organization breadcrumb open Simulate organization breadcrumb close
Department of Mathematics
  • FAUTo the central FAU website
  • de
  • UnivIS
  • StudOn
  • meincampus
  • CRIS
  • emergency help

Department of Mathematics

Navigation Navigation close
  • Department
    • Chairs and Professorships
    • Organisation
    • Development Association
    • System Administration
    • Contact and Directions
    • Actual
    Portal Department of Mathematics
  • Research
    • Research Projects
    • Publications
    • Preprint Series Applied Mathematics
    Portal Research
  • Study
    • Advice and Services
    • Prospective students
    • Current students
    • International
    Portal Study
  • Events
  1. Home
  2. Applied Mathematics 1
  3. Research
  4. UTBEST3D

UTBEST3D

In page navigation: Applied Mathematics 1
  • Staff Members A-Z
    • Dr. Rufat Badal
    • Apratim Bhattacharya
    • Astrid Bigott
    • Prof. Dr. Martin Burger
    • Sebastian Czop
    • Prof. Dr. Günther Grün
    • Lea Föcke
    • Prof. Dr. Manuel Friedrich
    • Samira Kabri
    • Lorenz Klein
    • Jonas Knoch
    • Prof. Dr. Serge Kräutle
      • CV
      • Research
    • Prof. Dr. Wilhelm Merz
      • Research
    • Dr. Stefan Metzger
    • PD Dr. Maria Neuss-Radu
      • Research
      • Anne Petzold
    • Dr. Alexander Prechtel
      • Research
      • Teaching
    • Dr. Nadja Ray
    • Doris Schneider
      • Conference Preview
      • Research
    • Dr. habil. Raphael Schulz
      • Research
    • Joscha Seutter
    • Dr. Daniel Tenbrinck
    • Cornelia Weber
    • Lukas Weigand
    • Simon Zech
  • Teaching
    • Lectures, Seminars and Tutorium
    • Scripts
  • Workshop on Recent Developments in Modelling, Analysis, and Simulation of Processes in Porous Media
  • Research
    • Overview of Habilitation and Dissertation Theses
    • Research Group Porous Media
      • Multicomponent reactive transport
      • Multiscale problems in life sciences
      • Geophysical flows
      • Multiphase flow in porous media
      • Emergence in porous media
      • Stochastic modelling of porous media
    • Software
      • Software
        • SiMRX
        • flexBox
    • Richy 1D
    • Richy 2D/3D
    • FESTUNG
    • Projects
    • UTBEST3D
    • EconDrop3D
    • Research group Prof. Dr. Grün
      • Prof. Dr. Günther Grün
      • Research Interests
      • Projects
      • Prof. Dr. Günther Grün
        • Projects and Publications
        • Research interests
  • Former Members
    • Dr. Vadym Aizinger
      • Research
        • Software
    • Dr. Leon Bungert
    • Dr. Tobias Elbinger
    • Dr. Antonio Esposito
    • Dr. Hubertus Grillmeier
      • Research
    • Dr. Alicja Kerschbaum
    • Prof. Dr. Peter Knabner
      • Curriculum Vitae
      • Research
      • Teaching
        • Books
          • Mathematical Modeling
          • Numerical Methods for Elliptic and Parabolic Partial Differential Equations
        • Earlier Lectures
    • Dr. habil. Nicolae Suciu
    • Dr. Markus Knodel
      • Research
    • Alice Lieu, PhD
    • Dr. Balthasar Reuter
      • LaTeX templates
      • Research
      • Teaching
    • Dr. Andreas Rupp
      • Research
    • Dr. Oliver Sieber
    • Dr. Philipp Wacker
    • Dr. Patrick Weiß
    • Dr. Philipp Werner
  • Upcoming events
    • 50 Years Applied Mathematics
    • Math meets Reality
    • Mathematical Modeling of Biomedical Problems
    • PDEs meet uncertainty
    • Short Course: Stochastic Compactness and SPDEs
    • Workshop on Modelling, Analysis and Simulation of Processes in Porous Media
      • Program – Workshop on Modelling, Analysis and Simulation of Processes in Porous Media
      • Program – Workshop Porous Media

UTBEST3D

UTBEST3D (University of Texas Bays and Estuaries Simulator — 3D)
Main Developers Vadym Aizinger (University of Erlangen-Nuernberg), Clint Dawson(University of Texas at Austin)
Numerical Method Discontinuous Galerkin Method
Programming Language/Libraries C++, MPI, OpenMP, METIS
Application Baroclinic/barotropic hydrostatic equations of coastal/regional ocean in primitive variables

Details>>

UTBEST3D (University of Texas Bays and Estuaries Simulator — 3D)

Fig. 1 Prismatic vertical mesh.

The numerical solution algorithm in UTBEST3D considers the system of hydrostatic primitive equations (1) with a free surface. A prismatic mesh (Fig. 1) is obtained by projecting a given triangular mesh in the vertical direction to provide a continuous piecewise linear representations of the topography and of the free surface. The vertical columns are then subdivided into layers. If a bottommost prism is degenerate, it is merged with the one above it. Due to the discontinuous nature of the approximation spaces, no constraints need to be enforced on the element connectivity. Hanging nodes and mismatching elements are allowed and have no adverse effects on stability or conservation properties of the scheme. This flexibility with regard to mesh geometry is exploited in several key parts of the algorithm: vertical mesh construction in areas with varying topography (Fig. 4), local mesh adaptivity (Fig. 2), wetting/drying (Fig. 3).

\partial_t {\bf u}_H +\nabla \cdot \left( {\bf u}_H {\bf u} – D_{\bf u} \nabla{\bf u} \right) + \nabla_H P -f_C {\bf k} \times {\bf u}_H =0
\partial_t T + \nabla \cdot \left({\bf u} T – D_T \nabla T \right) = 0
\partial_t S + \nabla \cdot \left({\bf u} S – D_S \nabla S \right) = 0
\partial_t \xi + \nabla_H \cdot \int_{z_b}^{\xi} {\bf u} dz =0\hspace{45mm}(1)
\nabla \cdot {\bf u} =0
\partial_z P= – \frac{\rho}{\rho_0}g
\rho = \rho(T,S,z)

\nabla_H = (\partial_x,\partial_y), {\bf k} = (0,0,1)^T, D_{X, X \in \{{\bf u},S,T\}} – eddy viscosity/diffusivity, \xi – free surface elevation, z_b – bathymetry, {\bf u}= (u,v,w) – velocity, {\bf u}_H= (u,v), T – temperature, S – salinity, P – pressure, \rho – density, \rho_0 – reference density, g – gravity acceleration, f_C – Coriolis coefficient.

Our DG discretization is based on the local discontinuous Galerkin method (Cockburn, Shu, 1998) that represents a direct generalization of the cell-centered finite volume method, the latter being just the piecewise constant DG discretization. One of the features of this method is a much smaller numerical diffusion exhibited by the linear and higher order DG approximations compared to the finite difference or finite volume discretizations. The method guarantees the element-wise conservation of all primary unknowns including tracers, supports an individual choice of the approximation space for each prognostic and diagnostic variable, demonstrates excellent stability properties, and possesses proven local adaptivity skills (Fig. 2).

 

Supercritical flow (Froude number 2.5) in a constricted channel with a dynamically adapted 2D grid. Two levels of adaptive refinement.

UTBEST3D is written in C++ to provide clean interfaces between geometrical, numerical, computational, and communication parts of the code. The object-oriented coding paradigm is designed to enable a labor efficient development lifecycle of the model. The programming techniques were carefully chosen and tested with the view of guaranteeing a smooth portability to different hardware architectures, operating systems, compilers, and software environments; the package relies on no external libraries in the serial configuration and only makes use of METIS (Karypis, Kumar, 1999) grid partitioning library in the parallel configuration.

Wetting/drying simulation of a storm surge during hurricane Ike. Visible fractionally subdivided elements at the wet/dry interface.

Geometry and mesh of the steep seamount test case. Bathymetry and horizontal mesh, vertical mesh.

Bibliography

  • Vadym Aizinger,  A discontinuous Galerkin method for two- and three-dimensional shallow-water equations, PhD Dissertation, THE UNIVERSITY OF TEXAS AT AUSTIN, 2004.
  • V. Aizinger, C. Dawson, The local discontinuous Galerkin method for three-dimensional shallow water flow, Computer methods in applied mechanics and engineering, 196, 734-746, 2007.
  • B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35, 2440-2463, 1998.
  • C. Dawson, V. Aizinger, A Discontinuous Galerkin Method for Three-Dimensional Shallow Water Equations, Journal of Scientific Computing, 22, 245-267, 2005.
  • G. Karypis and V. Kumar, A fast and high quality multilevel scheme for partitioning irregular graphs,  SIAM Journal on Scientific Computing, 20(1), 359-392, 1999.
Friedrich-Alexander-Universität
Erlangen-Nürnberg

Schlossplatz 4
91054 Erlangen
  • Contact and Directions
  • Internal Pages
  • Staff Members A-Z
  • Imprint
  • Privacy
  • EN/DE
Up