The Combination Technique for the Initial Value Problem in Linear Gyrokinetics

  • Christoph KowitzEmail author
  • Dirk Pflüger
  • Frank Jenko
  • Markus Hegland
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 88)


The simulation of hot fusion plasmas via the five-dimensional gyrokinetic equations is computationally intensive with one reason being the curse of dimensionality. Using the sparse grid combination technique could reduce the computational effort. For the computation of the full grid solutions, the plasma turbulence code GENE is used. It is shown that the combination technique is applicable to linear gyrokinetics by retrieving combination coefficients with a least squares approach. The retrieved sparse grid solution is actually close to the full grid one. Also, combination schemes were found which provided promising results with respect to the computational effort and accuracy.


Grid Point Magnetic Field Line Vlasov Equation Sparse Grid Linear Computation 
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  1. 1.
    Beer, M., Cowley, S., Hammett, G.: Field-aligned coordinates for nonlinear simulations of tokamak turbulence. Physics of Plasmas 2(7), 2687 (1995)CrossRefGoogle Scholar
  2. 2.
    Brizard, A., Hahm, T.: Foundations of nonlinear gyrokinetic theory. Reviews of modern physics 79(2), 421 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bungartz, H.J., Griebel, M.: Sparse grids. Acta Numerica 13, 147–269 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cohen, B., Williams, T., Dimits, A., Byers, J.: Gyrokinetic simulations of E ×B velocity-shear effects on ion-temperature-gradient modes. Physics of Fluids B: Plasma Physics 5, 2967 (1993)CrossRefGoogle Scholar
  5. 5.
    Dannert, T.: Gyrokinetische Simulation von Plasmaturbulenz mit gefangenen Teilchen und elektromagnetischen Effektien. Ph.D. thesis, Technische Universität München (2005)Google Scholar
  6. 6.
    Dannert, T., Jenko, F.: Gyrokinetic simulation of collisionless trapped-electron mode turbulence. Physics of Plasmas 12, 072,309 (2005)CrossRefGoogle Scholar
  7. 7.
    Garcke, J.: A dimension adaptive sparse grid combination technique for machine learning. In: W. Read, J.W. Larson, A.J. Roberts (eds.) Proceedings of the 13th Biennial Computational Techniques and Applications Conference, CTAC-2006, ANZIAM J., vol. 48, pp. C725–C740 (2007)Google Scholar
  8. 8.
    Garcke, J.: An optimised sparse grid combination technique for eigenproblems. In: Proceedings of ICIAM 2007, PAMM, vol. 7, pp. 1022,301–1022,302 (2008)Google Scholar
  9. 9.
    Garcke, J., Griebel, M.: On the computation of the eigenproblems of hydrogen and helium in strong magnetic and electric fields with the sparse grid combination technique. Journal of Computational Physics 165(2), 694–716 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Garcke, J., Griebel, M.: Classification with sparse grids using simplicial basis functions. Intelligent Data Analysis 6(6), 483–502 (2002)zbMATHGoogle Scholar
  11. 11.
    Görler, T.: Multiscale effects in plasma microturbulence. Ph.D. thesis, Universität Ulm (2009)Google Scholar
  12. 12.
    Görler, T., Lapillonne, X., Brunner, S., Dannert, T., Jenko, F., Merz, F., Told, D.: The global version of the gyrokinetic turbulence code GENE. Journal of Computational Physics 230(18), 7053–7071 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Griebel, M., Huber, W.: Turbulence simulation on sparse grids using the combination method. In: N. Satofuka, J. Periaux, A. Ecer (eds.) Parallel Computational Fluid Dynamics, New Algorithms and Applications, pp. 75–84. North-Holland, Elsevier (1995)Google Scholar
  14. 14.
    Griebel, M., Schneider, M., Zenger, C.: A combination technique for the solution of sparse grid problems. In: P. de Groen, R. Beauwens (eds.) Iterative Methods in Linear Algebra, pp. 263–281. IMACS, Elsevier, North Holland (1992). Also as SFB Bericht, 342/19/90 A, Institut für Informatik, TU München, 1990Google Scholar
  15. 15.
    Hegland, M.: Adaptive sparse grids. In: K. Burrage, R.B. Sidje (eds.) Proc. of 10th Computational Techniques and Applications Conference CTAC-2001, vol. 44, pp. C335–C353 (2003)Google Scholar
  16. 16.
    Hegland, M., Garcke, J., Challis, V.: The combination technique and some generalisations. Linear Algebra and its Applications 420(2–3), 249–275 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Kammerer, M., Merz, F., Jenko, F.: Exceptional points in linear gyrokinetics. Physics of Plasmas 15, 052,102 (2008)CrossRefGoogle Scholar
  18. 18.
    Lapillonne, X., Brunner, S., Sauter, O., Villard, L., Fable, E., Görler, T., Jenko, F., Merz, F.: Non-linear gyrokinetic simulations of microturbulence in tcv electron internal transport barriers. Plasma Physics and Controlled Fusion 53, 054,011 (2011)CrossRefGoogle Scholar
  19. 19.
    Lederer, H., Tisma, R., Hatzky, R., Bottino, A., Jenko, F.: Application enabling in DEISA: Petascaling of plasma turbulence codes. Parallel Computing: Architectures, Algorithms and Applications 38, 713–720 (2008)Google Scholar
  20. 20.
    Liu, F., Zhou, A.: Two-scale finite element discretizations for partial differential equations. J. Comput. Math 24(3), 373–392 (2006)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Merz, F.: Gyrokinetic simulation of multimode plasma turbulence. Ph.D. thesis, Universität Münster (2009)Google Scholar
  22. 22.
    Merz, F., Kowitz, C., Romero, E., Roman, J., Jenko, F.: Multi-dimensional gyrokinetic parameter studies based on eigenvalue computations. Computer Physics Communications 1, 1–9 (2011)Google Scholar
  23. 23.
    Roman, J.E., Kammerer, M., Merz, F., Jenko, F.: Fast eigenvalue calculations in a massively parallel plasma turbulence code. Parallel Computing 36(5–6), 339–358 (2010). Parallel Matrix Algorithms and ApplicationsGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Christoph Kowitz
    • 1
    Email author
  • Dirk Pflüger
    • 2
  • Frank Jenko
    • 3
  • Markus Hegland
    • 4
  1. 1.Institute for Advanced StudyTechnische Universität MünchenMunichGermany
  2. 2.Institute for Parallel and Distributed SystemsUniversity of StuttgartStuttgartGermany
  3. 3.Max-Planck-Institut für PlasmaphysikGarchingGermany
  4. 4.Australian National UniversityCanberraAustralia

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