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Computational methods for plasma fluid models

Published online by Cambridge University Press:  19 September 2016

G. Fuhr*
Affiliation:
Aix Marseille Univ, CNRS, PIIM, Faculte de Saint Jerome, C631, 13397 Marseille Cedex 20, France
P. Beyer
Affiliation:
Aix Marseille Univ, CNRS, PIIM, Faculte de Saint Jerome, C631, 13397 Marseille Cedex 20, France
S. Benkadda
Affiliation:
Aix Marseille Univ, CNRS, PIIM, Faculte de Saint Jerome, C631, 13397 Marseille Cedex 20, France
*
Email address for correspondence: [email protected]

Abstract

Challenges in plasma physics are wide. Investigation and advances are made in experiments but at the same time, to understand and to reach the experimental limits, accurate numerical simulations are required from systems of nonlinear equations. The numerical challenges of solving the associated fluid equations are discussed in this paper. Using the framework of the finite difference discretization, the most widely used methods for the problems linked to the diffusion or advection operators are presented.

Type
Research Article
Copyright
© Cambridge University Press 2016 

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References

Amdahl, G. M. 1967 Validity of the single processor approach to achieving large scale computing capabilities. In Proceedings of the April 18–20, 1967, Spring Joint Computer Conference, AFIPS ’67 (Spring), pp. 483485. ACM.Google Scholar
Arakawa, A. 1966 Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part I. J. Comput. Phys. 1 (1), 119143.Google Scholar
Balay, S., Abhyankar, S., Adams, M. F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W. D., Kaushik, D. et al. 2016 PETSc webpage. http://www.mcs.anl.gov/petsc.Google Scholar
Beyer, P., Benkadda, S. & Garbet, X. 2000 Proper orthogonal decomposition and galerkin projection for a three-dimensional plasma dynamical system. Phys. Rev. E 61 (1), 813823.Google Scholar
Biskamp, D. 1993 Nonlinear Magnetohydrodynamics. Cambridge University Press.Google Scholar
Boris, J. P. & Book, D. L. 1997 Flux-corrected transport. J. Comput. Phys. 135 (2), 172186.Google Scholar
Braginskii, S. I. 1965 Transport processes in a plasma. Rev. Plasma Phys. 1, 205311.Google Scholar
Brock, D. & Horton, W. 1982 Toroidal drift-wave fluctuations driven by ion pressure gradients. Plasma Phys. 24 (3), 271.Google Scholar
Butcher, J. C. 2008 Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley.Google Scholar
Canuto, C., Hussaini, M., Quarteroni, A. & Zang, T. 1988 Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics. Springer.Google Scholar
Chapman, B., Jost, G. & Van der Pas, R. 2007 Using OpenMP: Portable Shared Memory Parallel Programming. MIT Press.Google Scholar
Coiffier, J. 2012 Fundamentals of Numerical Weather Prediction. Cambridge University Press.Google Scholar
Courant, R., Friedrichs, K. & Lewy, H. 1928 über die partiellen differenzengleichungen der mathematischen physik. Math. Ann. 100 (1), 3274.Google Scholar
Courant, R., Isaacson, E. & Rees, M. 1952 On the solution of nonlinear hyperbolic differential equations by finite differences. Commun. Pure Appl. Maths 5 (3), 243255.Google Scholar
Crouseilles, N., Kuhn, M. & Latu, G. 2015 Comparison of numerical solvers for anisotropic diffusion equations arising in plasma physics. J. Sci. Comput. 65 (3), 10911128.Google Scholar
D’haeseleer, W. D. 1991 Flux Coordinates and Magnetic Field Structure: A Guide to a Fundamental Tool of Plasma Structure, Springer Series in Computational Physics. Springer.Google Scholar
Durran, D. R. 1999 Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer.Google Scholar
Durran, D. R. 2010 Numerical Methods for Fluid Dynamics, 2nd edn. Springer.Google Scholar
Ferszinger, H. 2002 Computational Methods for Fluid Dynamics. Springer.Google Scholar
FFTW 2015 FFTW webpage. http://www.fftw.org/.Google Scholar
Freidberg, J. P. 2007 Plasma Physics and Fusion Energy. Cambridge University Press.Google Scholar
Godunov, S. K. 1959 Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47, 271300.Google Scholar
Griffiths, D. F. 2010 Numerical Methods for Ordinary Differential Equations. Springer.Google Scholar
Gustafson, J. L. 1988 Reevaluating Amdahl’s law. Commun. ACM 31 (5), 532533.Google Scholar
Hairer, E. 2006 Geometric Numerical Integration. Springer.Google Scholar
Hirsch, C. 2007 Numerical Computation of Internal and External Flows, Volume 1, Fundamentals of Computational Fluid Dynamics, 2nd edn. Butterworth-Heinemann.Google Scholar
Horton, W. & Estes, R. D. 1980 Fluid simulation of ion pressure gradient driven drift modes. Plasma Phys. 22 (7), 663.Google Scholar
Jardin, S. 2010 Computational Methods in Plasma Physics. CRC Press.Google Scholar
Jardin, S. 2011 Review of implicit methods for the magnetohydrodynamic description of magnetically confined plasmas. J. Comput. Phys. 231, 822838.Google Scholar
Karniadakis, G. E., Israeli, M. & Orszag, S. A. 1991 High-order splitting methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 97 (2), 414443.Google Scholar
Kuhn, M., Latu, G., Genaud, S. & Crouseilles, N. 2013 Optimization and parallelization of emerged on shared memory architecture. In Proceedings of the 2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, SYNASC ’13, pp. 503510. IEEE Computer Society.Google Scholar
Lax, P. & Wendroff, B. 1960 Systems of conservation laws. Commun. Pure Appl. Maths 13 (2), 217237.Google Scholar
Leveque, R. J. 1992 Numerical Methods for Conservation Laws, Lectures in Mathematics. Birkhäuser.Google Scholar
Leveque, R. J. 2002 Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.Google Scholar
Lomax, H., Pulliam, T. H. & Zingg, D. W. 2001 Fundamentals of Computational Fluid Dynamics. Springer.Google Scholar
Message Passing Interface Forum2015 MPI: A Message-Passing Interface Standard Version 3.1. High Performance Computing Center Stuttgart (HLRS).Google Scholar
MUMPS2015 MUMPS webpage. http://mumps-solver.org/.Google Scholar
Naulin, V. & Nielsen, A. H. 2003 Accuracy of spectral and finite difference schemes in 2d advection problems. SIAM J. Sci. Comput. 25 (1), 104126.Google Scholar
OpenMP2015 OpenMP tutorial from LLNL. https://computing.llnl.gov/tutorials/openMP/.Google Scholar
Patterson, G. S. & Orszag, S. A. 1971 Spectral calculations of isotropic turbulence: efficient removal of aliasing interactions. Phys. Fluids 14 (11), 25382541.Google Scholar
Platzman, G. W. 1961 An approximation to the product of discrete functions. J. Meteorol. 18 (1), 3137.Google Scholar
Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. 2007 Numerical Recipes: The Art of Scientific Computing, 3rd edn. Cambridge University Press.Google Scholar
Rosen, J. S.1967 The Runge–Kutta equations by quadrature methods. NASA Tech. Rep. TR-R-275.Google Scholar
Schneider, K., Kolomenskiy, D. & Deriaz, E. 2013 Is the CFL Condition Sufficient? Some Remarks. pp. 139146. Birkhäuser Boston.Google Scholar
Scott, B. D. 1997 Three-dimensional computation of drift Alfvén turbulence. Plasma Phys. Control. Fusion 39 (10), 1635.Google Scholar
Shu, C.-W. 1998 Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws. pp. 325432. Springer.Google Scholar
Shu, C.-W. & Osher, S. 1988 Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77 (2), 439471.Google Scholar
Smith, G. D. 1978 Numerical Solution of Partial Differential Equations, 2nd edn. Clarendon.Google Scholar
Suresh, A. & Huynh, H. T. 1997 Accurate monotonicity-preserving schemes with Runge–Kutta time stepping. J. Comput. Phys. 136 (1), 8399.Google Scholar
Wesson, J. 1997 Tokamaks, 2nd edn. Clarendon.Google Scholar
Zhengfu, X. & Shu, C.-W. 2005 Anti-diffusive flux corrections for high order finite difference WENO schemes. J. Comput. Phys. 205 (2), 458485.Google Scholar