American Institute of Mathematical Sciences

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April  2012, 5(2): 271-282. doi: 10.3934/dcdss.2012.5.271

Guiding-center simulations on curvilinear meshes

Jean-Philippe Braeunig 1, , Nicolas Crouseilles 1, , Michel Mehrenberger 1, and  Eric Sonnendrücker 1,

1. 

INRIA Nancy-Grand Est, 615 rue du Jardin Botanique, 54600 Villers-lès-Nancy, IRMA, Université de Strasbourg, 7 rue René-Descartes, 67084 Strasbourg Cedex, France, France, France, France

Received  July 2009 Revised  December 2009 Published  September 2011

The purpose of this work is to design simulation tools for magnetised plasmas in the ITER project framework. The specific issue we consider is the simulation of turbulent transport in the core of a Tokamak plasma, for which a 5D gyrokinetic model is generally used, where the fast gyromotion of the particles in the strong magnetic field is averaged in order to remove the associated fast time-scale and to reduce the dimension of 6D phase space involved in the full Vlasov model. Very accurate schemes and efficient parallel algorithms are required to cope with these still very costly simulations. The presence of a strong magnetic field constrains the time scales of the particle motion along and accross the magnetic field line, the latter being at least an order of magnitude slower. This also has an impact on the spatial variations of the observables. Therefore, the efficiency of the algorithm can be improved considerably by aligning the mesh with the magnetic field lines. For this reason, we study the behavior of semi-Lagrangian solvers in curvilinear coordinates. Before tackling the full gyrokinetic model in a future work, we consider here the reduced 2D Guiding-Center model. We introduce our numerical algorithm and provide some numerical results showing its good properties.
Keywords: plasma, curvilinear coordinates, guiding-center., Semi-Lagrangian scheme, simulation.
Mathematics Subject Classification: Primary: 65M08, 65M25; Secondary: 78A3.
Citation: Jean-Philippe Braeunig, Nicolas Crouseilles, Michel Mehrenberger, Eric Sonnendrücker. Guiding-center simulations on curvilinear meshes. Discrete & Continuous Dynamical Systems - S, 2012, 5 (2) : 271-282. doi: 10.3934/dcdss.2012.5.271
References:
[1]

A. J. Brizard and T. S. Hahm, Foundations of nonlinear gyrokinetic theory,, Rev. Mod. Phys., 79 (2007), 421.  doi: 10.1103/RevModPhys.79.421.  Google Scholar

[2]

N. Crouseilles, M. Mehrenberger and E. Sonnendrücker, Conservative semi-Lagrangian schemes for Vlasov equations,, J. Comput. Phys., 229 (2010), 1927.  doi: 10.1016/j.jcp.2009.11.007.  Google Scholar

[3]

C. Z. Cheng and G. Knorr, The integration of the Vlasov equation in conÞguration space,, J. Comput. Phys., 22 (1976), 330.  doi: 10.1016/0021-9991(76)90053-X.  Google Scholar

[4]

A. M. Dimits, et al., Comparisons and physics basis of tokamak transport models and turbulence simulations,, Phys. Plasmas, 7 (2000).  doi: 10.1063/1.873896.  Google Scholar

[5]

N. Crouseilles, G. Latu and E. Sonnendrücker, A Vlasov solver based on local cubic spline interpolation on patches,, J. Comput. Phys., 228 (2009), 1429.  doi: 10.1016/j.jcp.2008.10.041.  Google Scholar

[6]

, X. Garbet,, private communication, (2009).   Google Scholar

[7]

V. Grandgirard, et al., Global full-f gyrokinetic simulations of plasma turbulence,, Plasma Phys. Control. Fus., 49B (2007), 173.  doi: 10.1088/0741-3335/49/12B/S16.  Google Scholar

[8]

V. Grandgirard, M. Brunetti, P. Bertrand, N. Besse, X. Garbet, P. Ghendrih, G. Manfredi, Y. Sarazin, O. Sauter, E. Sonnendrücker, J. Vaclavik and L. Villard, A drift-kinetic semi-Lagrangian 4D code for ion turbulence simulation,, J. Comput. Physics, 217 (2006), 395.  doi: 10.1016/j.jcp.2006.01.023.  Google Scholar

[9]

M. Shoucri, A two-level implicit scheme for the numerical solution of the linearized vorticity equation,, Int. J. Numer. Meth. Eng., 17 (1981).  doi: 10.1002/nme.1620171007.  Google Scholar

[10]

E. Sonnendrücker, J. Roche, P. Bertrand and A. Ghizzo, The semi-Lagrangian method for the numerical resolution of the Vlasov equation,, J. Comput. Physics, 149 (1999), 201.  doi: 10.1006/jcph.1998.6148.  Google Scholar

[11]

M. Zerroukat, N. Wood and A. Staniforth, The parabolic spline method (PSM) for conservative transport problems,, Int. J. Numer. Meth. Fluids, 51 (2006), 1297.  doi: 10.1002/fld.1154.  Google Scholar

[12]

M. Zerroukat, N. Wood and A. Staniforth, Application of the parabolic spline method (PSM) to a multi-dimensional conservative semi-Lagrangian transport scheme (SLICE),, J. Comput. Phys, 225 (2007), 935.  doi: 10.1016/j.jcp.2007.01.006.  Google Scholar

show all references

References:
[1]

A. J. Brizard and T. S. Hahm, Foundations of nonlinear gyrokinetic theory,, Rev. Mod. Phys., 79 (2007), 421.  doi: 10.1103/RevModPhys.79.421.  Google Scholar

[2]

N. Crouseilles, M. Mehrenberger and E. Sonnendrücker, Conservative semi-Lagrangian schemes for Vlasov equations,, J. Comput. Phys., 229 (2010), 1927.  doi: 10.1016/j.jcp.2009.11.007.  Google Scholar

[3]

C. Z. Cheng and G. Knorr, The integration of the Vlasov equation in conÞguration space,, J. Comput. Phys., 22 (1976), 330.  doi: 10.1016/0021-9991(76)90053-X.  Google Scholar

[4]

A. M. Dimits, et al., Comparisons and physics basis of tokamak transport models and turbulence simulations,, Phys. Plasmas, 7 (2000).  doi: 10.1063/1.873896.  Google Scholar

[5]

N. Crouseilles, G. Latu and E. Sonnendrücker, A Vlasov solver based on local cubic spline interpolation on patches,, J. Comput. Phys., 228 (2009), 1429.  doi: 10.1016/j.jcp.2008.10.041.  Google Scholar

[6]

, X. Garbet,, private communication, (2009).   Google Scholar

[7]

V. Grandgirard, et al., Global full-f gyrokinetic simulations of plasma turbulence,, Plasma Phys. Control. Fus., 49B (2007), 173.  doi: 10.1088/0741-3335/49/12B/S16.  Google Scholar

[8]

V. Grandgirard, M. Brunetti, P. Bertrand, N. Besse, X. Garbet, P. Ghendrih, G. Manfredi, Y. Sarazin, O. Sauter, E. Sonnendrücker, J. Vaclavik and L. Villard, A drift-kinetic semi-Lagrangian 4D code for ion turbulence simulation,, J. Comput. Physics, 217 (2006), 395.  doi: 10.1016/j.jcp.2006.01.023.  Google Scholar

[9]

M. Shoucri, A two-level implicit scheme for the numerical solution of the linearized vorticity equation,, Int. J. Numer. Meth. Eng., 17 (1981).  doi: 10.1002/nme.1620171007.  Google Scholar

[10]

E. Sonnendrücker, J. Roche, P. Bertrand and A. Ghizzo, The semi-Lagrangian method for the numerical resolution of the Vlasov equation,, J. Comput. Physics, 149 (1999), 201.  doi: 10.1006/jcph.1998.6148.  Google Scholar

[11]

M. Zerroukat, N. Wood and A. Staniforth, The parabolic spline method (PSM) for conservative transport problems,, Int. J. Numer. Meth. Fluids, 51 (2006), 1297.  doi: 10.1002/fld.1154.  Google Scholar

[12]

M. Zerroukat, N. Wood and A. Staniforth, Application of the parabolic spline method (PSM) to a multi-dimensional conservative semi-Lagrangian transport scheme (SLICE),, J. Comput. Phys, 225 (2007), 935.  doi: 10.1016/j.jcp.2007.01.006.  Google Scholar

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