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Application of High Dimensional B-Spline Interpolation in Solving the Gyro-Kinetic Vlasov Equation Based on Semi-Lagrangian Method

Part of: Applications to specific types of physical systems Numerical approximation and computational geometry

Published online by Cambridge University Press:  06 July 2017

Xiaotao Xiao ,
Lei Ye ,
Yingfeng Xu  and
Shaojie Wang
Xiaotao Xiao*
Affiliation:
Institute of Plasma Physics, Chinese Academy of Science, Hefei, Anhui 230031, China
Lei Ye*
Affiliation:
Institute of Plasma Physics, Chinese Academy of Science, Hefei, Anhui 230031, China
Yingfeng Xu*
Affiliation:
Institute of Plasma Physics, Chinese Academy of Science, Hefei, Anhui 230031, China
Shaojie Wang*
Affiliation:
Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
*
*Corresponding author. Email addresses:xtxiao@ipp.ac.cn (X. Xiao), lye@ipp.ac.cn (L. Ye), xuyingfeng@ipp.ac.cn (Y. Xu), wangsj@ustc.edu.cn (S. Wang)
*Corresponding author. Email addresses:xtxiao@ipp.ac.cn (X. Xiao), lye@ipp.ac.cn (L. Ye), xuyingfeng@ipp.ac.cn (Y. Xu), wangsj@ustc.edu.cn (S. Wang)
*Corresponding author. Email addresses:xtxiao@ipp.ac.cn (X. Xiao), lye@ipp.ac.cn (L. Ye), xuyingfeng@ipp.ac.cn (Y. Xu), wangsj@ustc.edu.cn (S. Wang)
*Corresponding author. Email addresses:xtxiao@ipp.ac.cn (X. Xiao), lye@ipp.ac.cn (L. Ye), xuyingfeng@ipp.ac.cn (Y. Xu), wangsj@ustc.edu.cn (S. Wang)
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Abstract

The computation efficiency of high dimensional (3D and 4D) B-spline interpolation, constructed by classical tensor product method, is improved greatly by precomputing the B-spline function. This is due to the character of NLT code, i.e. only the linearised characteristics are needed so that the unperturbed orbit as well as values of the B-spline function at interpolation points can be precomputed at the beginning of the simulation. By integrating this fixed point interpolation algorithm into NLT code, the high dimensional gyro-kinetic Vlasov equation can be solved directly without operator splitting method which is applied in conventional semi-Lagrangian codes. In the Rosenbluth-Hinton test, NLT runs a few times faster for Vlasov solver part and converges at about one order larger time step than conventional splitting code.

Keywords

High dimensional interpolationsemi-Lagrangiangyro-kinetic simulationnumerical Lie transform

MSC classification

Secondary: 65D05: Interpolation 65Z05: Applications to physics 68U20: Simulation 82D10: Plasmas
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 3 , September 2017 , pp. 789 - 802
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Lin, Z., Turbulent Transport Reduction by Zonal Flows: Massively Parallel Simulations, Science 281 (5384) (1998) 1835–1837. doi:10.1126/science.281.5384.1835.Google Scholar
[2] Idomura, Y., Tokuda, S., Kishimoto, Y., Global gyrokinetic simulation of ion temperature gradient driven turbulence in plasmas using a canonical Maxwellian distribution, Nuclear Fusion 43 (4) (2003) 234–243. doi:10.1088/0029-5515/43/4/303.Google Scholar
[3] Jolliet, S., Bottino, A., Angelino, P., Hatzky, R., Tran, T., Mcmillan, B., Sauter, O., Appert, K., Idomura, Y., Villard, L., A global collisionless PIC code in magnetic coordinates, Computer Physics Communications 177 (5) (2007) 409–425. doi:10.1016/j.cpc.2007.04.006.CrossRefGoogle Scholar
[4] Jenko, F., Dorland, W., Kotschenreuther, M., Rogers, B. N., Electron temperature gradient turbulence., Physical review letters 85 (26 Pt 1) (2000) 5579–82.Google Scholar
[5] Electron temperature gradient driven turbulence, Physics of Plasmas 7 (5) (2000) 1904. doi:10.1063/1.874014.Google Scholar
[6] Candy, J., Waltz, R., An Eulerian gyrokinetic-Maxwell solver, Journal of Computational Physics 186 (2) (2003) 545–581. doi:10.1016/S0021-9991(03)00079-2.Google Scholar
[7] Sonnendrücker, E., Roche, J., Bertrand, P., Ghizzo, A., The Semi-Lagrangian Method for the Numerical Resolution of the Vlasov Equation, Journal of Computational Physics 149 (2) (1999) 201–220. doi:10.1006/jcph.1998.6148.Google Scholar
[8] Grandgirard, V., Brunetti, M., Bertrand, P., Besse, N., Garbet, X., Ghendrih, P., Manfredi, G., Sarazin, Y., Sauter, O., Sonnendrücker, E., Vaclavik, J., Villard, L., A drift-kinetic Semi-Lagrangian 4D code for ion turbulence simulation, Journal of Computational Physics 217 (2) (2006) 395–423. doi:10.1016/j.jcp.2006.01.023.Google Scholar
[9] Latu, G., Grandgirard, V., Abiteboul, J., Crouseilles, N., Dif-Pradalier, G., Garbet, X., Ghendrih, P., Mehrenberger, M., Sarazin, Y., Sonnendrücker, E., Improving conservation properties of a 5D gyrokinetic semi-Lagrangian code, The European Physical Journal D 68 (11) (2014) 345. doi:10.1140/epjd/e2014-50209-1.Google Scholar
[10] Cheng, C., Knorr, G., The integration of the vlasov equation in configuration space, Journal of Computational Physics 22 (3) (1976) 330–351. doi:10.1016/0021-9991(76)90053-X.Google Scholar
[11] Qiu, J.-M., Christlieb, A., A conservative high order semi-Lagrangian WENO method for the Vlasov equation, Journal of Computational Physics 229 (4) (2010) 1130–1149. doi:10.1016/j.jcp.2009.10.016.Google Scholar
[12] Crouseilles, N., Respaud, T., Sonnendrcker, E., A forward semi-Lagrangianmethod for the numerical solution of the Vlasov equation, Computer Physics Communications 180 (10) (2009) 1730–1745. doi:10.1016/j.cpc.2009.04.024.Google Scholar
[13] de Boor, C., a practical guide to splines, Vol. 27, Springer, 2001.Google Scholar
[14] Wang, S., Transport formulation of the gyrokinetic turbulence, Physics of Plasmas 19 (6) (2012) 62504. doi:10.1063/1.4729660.Google Scholar
[15] Wang, S., Kinetic theory of weak turbulence in plasmas, Physical Review E 87 (6) (2013) 063103. doi:10.1103/PhysRevE.87.063103.Google Scholar
[16] Wang, S., Nonlinear scattering term in the gyrokinetic Vlasov equation, Physics of Plasmas 20 (8) (2013) 82312. doi:10.1063/1.4818593.Google Scholar
[17] Wang, S., Lie-transform theory of transport in plasma turbulence, Physics of Plasmas 21 (7) (2014) 072312. doi:10.1063/1.4890356.Google Scholar
[18] Xu, Y., Dai, Z., Wang, S., Nonlinear gyrokinetic theory based on a new method and computation of the guiding-center orbit in tokamaks, Physics of Plasmas 21 (4) (2014) 042505. doi:10.1063/1.4871726. URL http://scitation.aip.org/content/aip/journal/pop/21/4/10.1063/1.4871726 Google Scholar
[19] Ye, L., Xu, Y., Xiao, X., Dai, Z., Wang, S., A gyrokinetic continuum code based on the numerical Lie transform (NLT) method, Journal of Computational Physics 316 (2016) 180–192. doi:10.1016/j.jcp.2016.03.068.Google Scholar
[20] Garbet, X., Idomura, Y., Villard, L., Watanabe, T., Gyrokinetic simulations of turbulent transport, Nuclear Fusion 50 (4) (2010) 043002. URL http://stacks.iop.org/0029-5515/50/i=4/a=043002 Google Scholar
[21] Ye, L., Guo, W., Xiao, X., Wang, S., Numerical simulation of geodesic acoustic modes in a multi-ion system, Physics of Plasmas 20 (7) (2013) 072501. doi:10.1063/1.4812593.Google Scholar
[22] Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery, B. P., Numerical Recipes: The Art of Scientific Computing, 2nd Edition, Cambridge university press, London, 1986.Google Scholar
[23] Strang, G., On the construction and comparison of difference schemes, SIAM Journal on Numerical Analysis 5 (3) (1968) 506–517.Google Scholar