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Volume 26, Issue 3
Locally Divergence-Free Spectral-DG Methods for Ideal Magnetohydrodynamic Equations on Cylindrical Coordinates

Yong Liu, Qingyuan Liu, Yuan Liu, Chi-Wang Shu & Mengping Zhang

DOI: 10.4208/cicp.OA-2018-0187

Commun. Comput. Phys., 26 (2019), pp. 631-653.

Published online: 2019-04

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  • Abstract

In this paper, we propose a class of high order locally divergence-free spectral-discontinuous Galerkin (DG) methods for three dimensional (3D) ideal magnetohydrodynamic (MHD) equations on cylindrical geometry. Under the conventional cylindrical coordinates (r,ϕ,z), we adopt the Fourier spectral method in the ϕ-direction and discontinuous Galerkin (DG) approximation in the (r,z) plane, motivated by the structure of the particular physical flows of magnetically confined plasma. By a careful design of the locally divergence-free set for the magnetic filed, our spectral-DG methods are divergence-free inside each element for the magnetic field. Numerical examples with third order strong-stability-preserving Runge-Kutta methods are provided to demonstrate the efficiency and performance of our proposed methods.

  • Keywords

Discontinuous Galerkin method, magnetohydrodynamics (MHD), divergence-free, cylindrical coordinates.

  • AMS Subject Headings

65M60, 65M70, 76W05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-26-631, author = {Yong Liu, Qingyuan Liu, Yuan Liu, Chi-Wang Shu and Mengping Zhang}, title = {Locally Divergence-Free Spectral-DG Methods for Ideal Magnetohydrodynamic Equations on Cylindrical Coordinates}, journal = {Communications in Computational Physics}, year = {2019}, volume = {26}, number = {3}, pages = {631--653}, abstract = {

In this paper, we propose a class of high order locally divergence-free spectral-discontinuous Galerkin (DG) methods for three dimensional (3D) ideal magnetohydrodynamic (MHD) equations on cylindrical geometry. Under the conventional cylindrical coordinates (r,ϕ,z), we adopt the Fourier spectral method in the ϕ-direction and discontinuous Galerkin (DG) approximation in the (r,z) plane, motivated by the structure of the particular physical flows of magnetically confined plasma. By a careful design of the locally divergence-free set for the magnetic filed, our spectral-DG methods are divergence-free inside each element for the magnetic field. Numerical examples with third order strong-stability-preserving Runge-Kutta methods are provided to demonstrate the efficiency and performance of our proposed methods.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0187}, url = {http://global-sci.org/intro/article_detail/cicp/13140.html} }
TY - JOUR T1 - Locally Divergence-Free Spectral-DG Methods for Ideal Magnetohydrodynamic Equations on Cylindrical Coordinates AU - Yong Liu, Qingyuan Liu, Yuan Liu, Chi-Wang Shu & Mengping Zhang JO - Communications in Computational Physics VL - 3 SP - 631 EP - 653 PY - 2019 DA - 2019/04 SN - 26 DO - http://doi.org/10.4208/cicp.OA-2018-0187 UR - https://global-sci.org/intro/article_detail/cicp/13140.html KW - Discontinuous Galerkin method, magnetohydrodynamics (MHD), divergence-free, cylindrical coordinates. AB -

In this paper, we propose a class of high order locally divergence-free spectral-discontinuous Galerkin (DG) methods for three dimensional (3D) ideal magnetohydrodynamic (MHD) equations on cylindrical geometry. Under the conventional cylindrical coordinates (r,ϕ,z), we adopt the Fourier spectral method in the ϕ-direction and discontinuous Galerkin (DG) approximation in the (r,z) plane, motivated by the structure of the particular physical flows of magnetically confined plasma. By a careful design of the locally divergence-free set for the magnetic filed, our spectral-DG methods are divergence-free inside each element for the magnetic field. Numerical examples with third order strong-stability-preserving Runge-Kutta methods are provided to demonstrate the efficiency and performance of our proposed methods.

Yong Liu, Qingyuan Liu, Yuan Liu, Chi-Wang Shu and Mengping Zhang. (2019). Locally Divergence-Free Spectral-DG Methods for Ideal Magnetohydrodynamic Equations on Cylindrical Coordinates. Communications in Computational Physics. 26 (3). 631-653. doi:10.4208/cicp.OA-2018-0187
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