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Research

In page navigation: Applied Mathematics 1
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Research

Research interests

The main focus of my work is the development of high order numerical schemes, particularly the discontinuous Galerkin (DG) method, and application of these methods to the simulation of circulation and transport in coastal and global ocean.

A number of very interesting though difficult issues have to be addressed in order to fully exploit the potential of DG methods and still achieve competitive computational performance for this type of application.

The problems of interest in deep and coastal ocean studies tend to be very large; the time scales being simulated can lie anywhere between minutes (as in fast moving flooding and storm fronts) and centuries — the latter typically found in climate applications. Highly complex bottom topography and coast lines call for algorithms that can efficiently utilize unstructured grids. Computational domains change with time in the vertical direction (free surface) and may also have moving lateral boundaries (flooding), thus dynamic meshes must be supported by the numerical algorithm.

The DG methods possess many desirable properties that make them an attractive choice for a new generation of ocean models. These include

  • robust treatment of shocks and discontinuities;
  • local conservation of all variables;
  • choice of high or low order approximation spaces;
  • support for non-conforming meshes and hanging nodes;
  • approximation spaces easily accomodating h- and p-refinement;
  • locality of approximation stencil.

A much higher computational cost of DG algorithms even compared to other unstructured methods such as the classical finite elements makes the task of formulating and implementing DG methods for ocean applications quite challenging. Several strategies that build on the strengths of the method appear to hold promise with regard to increasing method’s efficiency for realistic problems. My research concerns all of these strategies:

  • using adaptivity to efficiently utilize computational resources in order to obtain an accurate solution in the areas of interest;
  • exploiting locality of the DG approximation stencil to achieve good scalability on parallel clusters and hybrid computational architectures;
  • improving accuracy and resolution of the numerical solution by means of dynamic mesh and approximation space optimization — all this without violating the local conservation properties of the scheme.

Publications

2022

  • Faghih-Naini S., Aizinger V.:
    p-adaptive discontinuous Galerkin method for the shallow water equations with a parameter-free error indicator.
    In: GEM - International Journal on Geomathematics 13 (2022)
    ISSN: 1869-2672
    DOI: 10.1007/s13137-022-00208-3
    BibTeX: Download
  • Zint D., Grosso R., Aizinger V., Faghih-Naini S., Kuckuk S., Köstler H.:
    Automatic Generation of Load-Balancing-Aware Block-Structured Grids for Complex Ocean Domains.
    2022 SIAM International Meshing Roundtable
    In: Robinson, Trevor ; Moxey, David ; Tomov, Vladimir Z. (ed.) (ed.): Proceedings of the 2022 SIAM International Meshing Roundtable 2022
    DOI: 10.5281/zenodo.6562440
    BibTeX: Download

2021

  • Kenter T., Shambhu A., Faghih-Naini S., Aizinger V.:
    Algorithm-hardware co-design of a discontinuous Galerkin shallow-water model for a dataflow architecture on FPGA
    2021 Platform for Advanced Scientific Computing Conference, PASC 2021 (Online, 05-07-2021 - 09-07-2021)
    In: Proceedings of the Platform for Advanced Scientific Computing Conference, PASC 2021 2021
    DOI: 10.1145/3468267.3470617
    BibTeX: Download

2020

  • Reuter B., Rupp A., Aizinger V., Frank F., Knabner P.:
    FESTUNG: A MATLAB / GNU Octave Toolbox for the Discontinuous Galerkin Method. Part IV: Generic Problem Framework and Model-Coupling Interface
    In: Communications in Computational Physics 28 (2020), p. 827-876
    ISSN: 1815-2406
    DOI: 10.4208/cicp.OA-2019-0132
    BibTeX: Download

2019

  • Hajduk H., Kuzmin D., Aizinger V.:
    New directional vector limiters for discontinuous Galerkin methods
    In: Journal of Computational Physics 384 (2019), p. 308-325
    ISSN: 0021-9991
    DOI: 10.1016/j.jcp.2019.01.032
    BibTeX: Download
  • Koldunov NV., Aizinger V., Rakowsky N., Scholz P., Sidorenko D., Danilov S., Jung T.:
    Scalability and some optimization of the Finite-volumE Sea ice-Ocean Model, Version 2.0 (FESOM2)
    In: Geoscientific Model Development 12 (2019), p. 3991-4012
    ISSN: 1991-959X
    DOI: 10.5194/gmd-12-3991-2019
    BibTeX: Download
  • Reuter B., Rupp A., Aizinger V., Knabner P.:
    Discontinuous Galerkin method for coupling hydrostatic free surface flows to saturated subsurface systems
    In: Computers & Mathematics with Applications 77 (2019), p. 2291-2309
    ISSN: 0898-1221
    DOI: 10.1016/j.camwa.2018.12.020
    URL: https://www1.am.uni-erlangen.de/research/publications/Jahr_2018/2018_ReuterRuppAizingerKn_DiscontGalerkinMethForCouplHydrostFreeSurfaceFlowsToSaturSubsurfSyst
    BibTeX: Download
  • Zint D., Grosso R., Aizinger V., Köstler H.:
    Generation of Block Structured Grids on Complex Domains for High Performance Simulation
    9th International Conference of Numerical Geometry, Grid Generation and Scientific Computing (Dorodnicyn Computing Center, Moscow, 03-12-2018 - 05-12-2018)
    In: Springer International Publishing, Cham (ed.): Numerical Geometry, Grid Generation and Scientific Computing, Cham: 2019
    DOI: 10.1007/978-3-030-23436-2_6
    BibTeX: Download

2018

  • Aizinger V., Rupp A., Schütz J., Knabner P.:
    Analysis of a mixed discontinuous Galerkin method for instationary Darcy flow
    In: Computational Geosciences 22 (2018), p. 179-194
    ISSN: 1420-0597
    DOI: 10.1007/s10596-017-9682-8
    URL: https://www1.am.uni-erlangen.de/research/publications/Jahr_2018/2018_AizingerRuppSchuetzKn_AnalysisOfAMixedDiscontinuousGalerkinForDarcyFlow
    BibTeX: Download
  • Fried M., Aizinger V., Bungert L.:
    Comparison of two local discontinuous Galerkin formulations for the subjective surfaces problem
    In: Computing and Visualization in Science (2018)
    ISSN: 1432-9360
    DOI: 10.1007/s00791-018-0291-4
    URL: https://link.springer.com/article/10.1007%2Fs00791-018-0291-4
    BibTeX: Download
  • Hajduk H., Hodges BR., Aizinger V., Reuter B.:
    Locally Filtered Transport for computational efficiency in multi-component advection-reaction models
    In: ENVIRON MODELL SOFTW 102 (2018), p. 185-198
    ISSN: 1364-8152
    DOI: 10.1016/j.envsoft.2018.01.003
    URL: https://www1.am.uni-erlangen.de/research/publications/Jahr_2018/2018_HajdukHodgesAizingerReuter_LocallyFilteredTransport.pdf
    BibTeX: Download
  • Jaust A., Reuter B., Aizinger V., Schütz J., Knabner P.:
    FESTUNG: A MATLAB/GNU Octave toolbox for the discontinuous Galerkin method. Part III: Hybridized discontinuous Galerkin (HDG) formulation
    In: Computers & Mathematics with Applications 75 (2018), p. 4505-4533
    ISSN: 0898-1221
    DOI: 10.1016/j.camwa.2018.03.045
    URL: https://www1.am.uni-erlangen.de/research/publications/Jahr_2018/2018_JaustReuterAizingerSchuetzKn_FestungAMatlabGnuPartIIIHDG
    BibTeX: Download
  • Marx A., Conrad M., Aizinger V., Prechtel A., van Geldern R., Barth J.:
    Groundwater data improve modelling of headwater stream CO2 outgassing with a stable DIC isotope approach
    In: Biogeosciences 15 (2018), p. 3093-3106
    ISSN: 1726-4170
    DOI: 10.5194/bg-15-3093-2018
    URL: https://www.biogeosciences.net/15/3093/2018/
    BibTeX: Download

2017

  • Aizinger V., Kosik A., Kuzmin D., Reuter B.:
    Anisotropic slope limiting for discontinuous Galerkin methods
    In: International Journal For Numerical Methods in Fluids (2017), p. in press
    ISSN: 0271-2091
    DOI: 10.1002/fld.4360
    URL: https://www1.am.uni-erlangen.de/research/publications/Jahr_2017/2017_AizingerKosikKuzminReuter_AnisoSlopeLimForDG.pdf
    BibTeX: Download
  • Ditter A., Schönwetter D., Kuzmin A., Fey D., Aizinger V.:
    Memory Analysis and Performance Modeling for HPC Applications on Embedded Hardware via Instruction Accurate Simulation
    Federated Conference on Software Development and Object Technologies, SDOT 2015 (Žilina, 19-11-2015 - 20-11-2015)
    In: Proceedings of the 2015 Federated Conference on Software Development and Object Technologies, Cham: 2017
    DOI: 10.1007/978-3-319-46535-7_2
    BibTeX: Download

2016

  • Bungert L., Aizinger V., Fried M.:
    A Discontinuous Galerkin Method for the Subjective Surfaces Problem
    In: Journal of Mathematical Imaging and Vision (2016), p. in press
    ISSN: 1573-7683
    DOI: 10.1007/s10851-016-0695-z
    BibTeX: Download
  • Geveler M., Reuter B., Aizinger V., Göddeke D., Turek S.:
    Energy efficiency of the simulation of three-dimensional coastal ocean circulation on modern commodity and mobile processors
    In: Computer Science - Research and Development 31 (2016), p. 225-234
    ISSN: 1865-2034
    DOI: 10.1007/s00450-016-0324-5
    URL: https://www1.am.uni-erlangen.de/research/publications/Jahr_2016/2016_GevelerReuterAizingerGoeddekeTurek_EnergyEfficienceSimulation3DCoastalOceanCommodityMobileProc.pdf
    BibTeX: Download
  • Reuter B., Aizinger V., Wieland M., Frank F., Knabner P.:
    FESTUNG: A MATLAB /GNU Octave toolbox for the discontinuous Galerkin method. Part II: Advection operator and slope limiting
    In: Computers & Mathematics with Applications 72 (2016), p. 1896-1925
    ISSN: 0898-1221
    DOI: 10.1016/j.camwa.2016.08.006
    URL: https://www1.am.uni-erlangen.de/research/publications/Jahr_2016/2016_ReuterAizingerWielandFrankKn_FestungAMatLabGnuPartII
    BibTeX: Download
  • Schönwetter D., Ditter A., Aizinger V., Reuter B., Fey D.:
    Cache Aware Instruction Accurate Simulation of a 3-D Coastal Ocean Model on Low Power Hardware
    6th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (Lisbon)
    DOI: 10.5220/0006006501290137
    BibTeX: Download
    (Conference report)
  • Schönwetter D., Ditter A., Kleinert B., Hendricks A., Aizinger V., Fey D.:
    Virtualization Guided Tsunami and Storm Surge Simulations for Low Power Architectures
    DOI: 10.1007/978-3-319-31295-8_7
    BibTeX: Download

2015

  • Frank F., Reuter B., Aizinger V., Knabner P.:
    FESTUNG: A MATLAB/GNU Octave toolbox for the discontinuous Galerkin method. Part I: Diffusion operator
    In: Computers & Mathematics with Applications 70 (2015), p. 11 - 46
    ISSN: 0898-1221
    DOI: 10.1016/j.camwa.2015.04.013
    URL: https://www1.am.uni-erlangen.de/research/publications/Jahr_2015/2015_FrankReuterAizingerKn_FestungAMatlabGnuPartIDiffusionOperator
    BibTeX: Download
  • Reuter B., Aizinger V., Köstler H.:
    A multi-platform scaling study for an OpenMP parallelization of a discontinuous Galerkin ocean model
    In: Computers & Fluids 117 (2015), p. 325 - 335
    ISSN: 0045-7930
    DOI: 10.1016/j.compfluid.2015.05.020
    URL: https://www1.am.uni-erlangen.de/research/publications/Jahr_2015/2015_ReuterAizingerKoestler_MultiPlatformScalingOpenMPdGoceanModel.pdf
    BibTeX: Download
  • Schönwetter D., Ditter A., Kleinert B., Hendricks A., Aizinger V., Köstler H., Fey D.:
    Tsunami and Storm Surge Simulation Using Low Power Architectures - Concept and Evaluation
    5th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (Colmar, 21-07-2015 - 23-07-2015)
    In: Proceedings of the 5th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, Colmar, Alsace, France: 2015
    DOI: 10.5220/0005566603770382
    BibTeX: Download

2000

  • Dawson C., Aizinger V., Cockburn B.:
    The Local Discontinuous Galerkin method for contaminant transport problems
    In: Cockburn B, Karniadakis GE, Shu C (ed.): Discontinuous Galerkin Methods, Springer, 2000, p. 309-314 (Lecture Notes in Computational Science and Engineering)
    DOI: 10.1007/978-3-642-59721-3_26
    BibTeX: Download

Projects

  • Rechenleistungsoptimierte Software-Strategien für auf unstrukturierten Gittern basierende Anwendungen in der Ozeanmodellierung

    (Third Party Funds Single)

    Term: 01-01-2017 - 30-09-2020
    Funding source: DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)
    Abstract

    Um akkurate Ozean, Atmosphären oder Klima Simulationen durchzuführen werden sehr effiziente numerische Verfahren und große Rechenkapazitäten benötigt, die in vielen Teilen der Welt und bei vielen Forschungsgruppen in diesen Anwendungsfeldern nicht verfügbar sind. Solche Beschränkungen führen auch dazu, dass Modelle und Softwarepakete basierend auf strukturierten Gittern derzeit in der Ozeanwissenschaft immer noch vorherrschend sind.In diesem Projekt soll zum einen die Rechenzeit für Modelle, die auf unstrukturierten Gittern und einer diskontinuierlichen Galerkin finite Elemente Methode basieren, deutlich reduziert werden, und zum anderen die Produktivität bei der Softwareentwicklung gesteigert werden. Das erste Ziel soll durch einen neuen Ansatz zur parallelen Gebietszerlegung und durch adaptive numerische Verfahren erreicht werden.Für das zweite Ziel kommen moderne Software Design Strategien zum Einsatz, vor allem Codegenerierung und automatische Optimierung von rechenintensiven Programmteilen. Die Fortschritte bei der Rechenzeit und dem Software Design, die aus dem Projekt resultieren, können einen wichtigen Beitrag leisten, um unstrukturierte Gitter für alle Forscher aus den Ozeanwissenschaften nutzbar zu machen, auch wenn sie nur Zugang zu moderat parallelen Systemen und nicht zu Höchstleistungsrechnern haben.

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