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The Hall Magnetohydrodynamic (MHD) equations are an extension of the standard MHD equations that include the “Hall” term from the general Ohm’s law. The Hall term decouples ion and electron motion physically on the ion inertial length scales. Implementing the Hall MHD equations in a numerical solver allows more physical simulations for plasma dynamics on length scales less than the ion inertial scale length but greater than the electron inertial length. The present effort is an important step towards producing physically correct results to important problems, such as the Geospace Environmental Modeling (GEM) Magnetic Reconnection problem. The solver that is being modified is currently capable of solving the resistive MHD equations on unstructured grids using the spectral difference scheme which is an arbitrarily high-order method that is relatively simple to parallelize. The GEM Magnetic Reconnection problem is used to evaluate whether the Hall MHD equations have been correctly implemented in the solver using the spectral difference method with divergence cleaning (SDDC) algorithm by comparing against the reconnection rates reported in the literature.
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An Arbitrarily High-order Spectral Difference Method with Divergence Cleaning (SDDC) for Compressible Magnetohydrodynamic Simulations on Unstructured Grids
Abstract This paper reports a recent development of the high-order spectral difference method with divergence cleaning (SDDC) for accurate simulations of both ideal and resistive magnetohydrodynamics (MHD) on curved unstructured grids consisting of high-order isoparametric quadrilateral elements. The divergence cleaning approach is based on the improved generalized Lagrange multiplier, which is thermodynamically consistent. The SDDC method can achieve an arbitrarily high order of accuracy in spatial discretization, as demonstrated in the test problems with smooth solutions. The high-order SDDC method combined with the artificial dissipation method can sharply capture shock interfaces with the oscillation-free property and resolve small-scale vortex structures and density fluctuations on relatively sparse grids. The robustness of the codes is demonstrated through long time simulations of ideal MHD problems with progressively interacting shock structures, resistive MHD problems with high Lundquist numbers, and viscous resistive MHD problems on complex curved domains.
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- Award ID(s):
- 2129320
- PAR ID:
- 10486230
- Publisher / Repository:
- DOI PREFIX: 10.3847
- Date Published:
- Journal Name:
- The Astrophysical Journal
- Volume:
- 932
- Issue:
- 1
- ISSN:
- 0004-637X
- Format(s):
- Medium: X Size: Article No. 16
- Size(s):
- Article No. 16
- Sponsoring Org:
- National Science Foundation
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Extending the Spectral Differencing Method with Divergence Cleaning (SDDC) to the Hall MHD EquationsThe Hall Magnetohydrodynamic (MHD) equations are an extension of the standard MHD equations that include the “Hall” term from the general Ohm’s law. The Hall term decouples ion and electron motion physically on the ion inertial length scales. Implementing the Hall MHD equations in a numerical solver allows more physical simulations for plasma dynamics on length scales less than the ion inertial scale length but greater than the electron inertial length. The present effort is an important step towards producing physically correct results to important problems, such as the Geospace Environmental Modeling (GEM) Magnetic Reconnection problem. The solver that is being modified is currently capable of solving the resistive MHD equations on unstructured grids using the spectral differencing scheme which is an arbitrarily high-order method that is relatively simple to parallelize. The GEM Magnetic Reconnection problem is used to evaluate whether the Hall MHD equations have been correctly implemented in the solver using the spectral differencing method with divergence cleaning (SDDC) algorithm by comparing against the reconnection rates reported in the literature.more » « less
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