Hostname: page-component-7dd5485656-npwhs Total loading time: 0 Render date: 2025-10-31T06:08:02.926Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access

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

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 3 , September 2017 , pp. 789 - 802
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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