Numerical Simulations – Cornell Laboratory of Plasma Studies

The PERSEUS code developed at Cornell University by Martin, Seyler and Gourdain represents a significant advance in XMHD simulations, in which the Hall and electron inertial terms are included in the generalized Ohm’s law (GOL). The algorithm is stable, as well as accurate, for time steps orders of magnitude larger than either the electron plasma or electron cyclotron periods, leading to computation times for XMHD that are approximately the same as for resistive MHD. The great advantage of our algorithm is that it is locally implicit and very simple, which means that no global linear algebra solvers are needed or used.

The primary differences between MHD and XMHD are the equations for the current and the electric field. In MHD the electric field $\vec{E}$ is determined by

$\vec{E} + \vec{u} \times \vec{B} = η \vec{J}$

where $\vec{u}$ is the plasma flow velocity, $\vec{B}$ is the magnetic field and $η$ is the plasma resistivity. The MHD current density is determined by Ampère’s law, $\vec{\nabla} \times \vec{B} = μ_{0} \vec{J}$ . In contrast, for XMHD the time evolution of the current density is determined from

$\partial_{t} \vec{J} + \vec{\nabla} \cdot (\vec{u} \vec{J} + \vec{J} \vec{u} - \frac{1}{e n} \vec{J} \vec{J}) + \frac{e}{m_{e}} \vec{\nabla} p_{e} = \frac{e^{2} n}{m_{e}} (\vec{E} + \vec{u} \times \vec{B} - \frac{1}{e n} \vec{J} \times \vec{B} - η \vec{J})$

and the electric field from $\partial_{t} \vec{E} = c^{2} (\vec{\nabla} \times \vec{B} - μ_{0} \vec{J})$ . Here $n, e, m_{e}$ and $p_{e}$ are the electron density, the elementary charge, the electron mass and the electron pressure respectively. The remaining equations are identical to MHD.

The complete set of 14 partial differential equations is solved by a relaxation method for which the time step can be taken to be much larger than would otherwise be allowed by an explicit time advance method. The code PERSEUS (Plasma as an Extended-mhd Relaxation System using an Efficient Upwind Scheme) has been developed at Cornell University to look at the impact of the Hall term on high energy density plasmas. Starting from the Ph.D. thesis research done by Dr. Martin, Profs. Seyler and Gourdain continue to develop the code, with help from students and colleagues. It now includes the option of a fully two-fluid model, with 2D (Cartesian or cylindrical grids) and 3D versions, using MPI to run on up to tens of thousands of processors.

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