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 $$\overrightarrow E$$ is determined by

$$\overrightarrow E+\overrightarrow u \times \overrightarrow B = \eta \overrightarrow J$$

where $$\overrightarrow u$$ is the plasma flow velocity, $$\overrightarrow B$$ is the magnetic field and $$\eta$$ is the palsma resistivity. The current density is determined by Ampère's law $$\overrightarrow \nabla \times \overrightarrow B = \mu_0 \overrightarrow J$$. In contrast, for XMHD the time evolution of the current density is determined from

$$\partial_t \overrightarrow J+\overrightarrow\nabla \cdot (\overrightarrow u \overrightarrow J+ \overrightarrow J \overrightarrow u - \frac{1}{m_e}\overrightarrow J\overrightarrow J)+\frac{1}{m_e}\overrightarrow\nabla p_e=\frac{e^2 n}{m_e}(\overrightarrow E +\overrightarrow u \times \overrightarrow B-\frac{1}{en}\overrightarrow J \times \overrightarrow B - \eta \overrightarrow J)$$

and the electric field from $$\partial_t \overrightarrow E=c^2(\overrightarrow B - \mu_0 \overrightarrow J)$$. 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 as a thesis research done by Dr. Martin, Profs. Seyler and Gourdain continue to develop the code. It is now full two fluids, with 2D (Cartesian or cylindrical grids) and 3D versions using MPI to run tens of thousands of processors.

The picture below shows the simulation in cylindrical coordinate of a thin aluminum foil on the COBRA pulsed power machine during the explosion phase. We are comparing plasma velocities, densities and temperatures with experimental results to understand if the models built into PERSEUS are accurate. The panel on the left hand side of the picture shows the plasma density on the $$log_{10}$$ scale. The panel on the right shows the ion inertial length, highlighting where the Hall term matters ($$\delta_i>1mm$$). The pin diameter is 1 mm.