Second order divergence constraint preserving entropy stable finite difference schemes for ideal two-fluid plasma flow equations

Abstract

Two-fluid plasma flow equations describe the flow of ions and electrons with different densities, velocities, and pressures. We consider the ideal plasma flow i.e. we ignore viscous, resistive, and collision effects. The resulting system of equations has flux consisting of three independent components, one for ions, one for electrons, and a linear Maxwell's equation flux for the electromagnetic fields. The coupling of these components is via source terms. In this article, we present {conservative} second-order finite difference schemes that ensure the consistent evolution of the divergence constraints on the electric and magnetic fields. The key idea is to design a numerical solver for Maxwell's equations using the multidimensional Riemann solver at the vertices, ensuring discrete divergence constraints; for the fluid parts, we use an entropy-stable discretization. The proposed schemes are co-located, second-order accurate, entropy stable, and ensure divergence-free evolution of the magnetic field. We use explicit and IMplicit-EXplicit (IMEX) schemes for time discretizations. To demonstrate the accuracy, stability, and divergence constraint-preserving ability of the proposed schemes, we present several test cases in one and two dimensions. We also compare the numerical results with those obtained from schemes with no divergence cleaning and those employing perfectly hyperbolic Maxwell (PHM) equations-based divergence cleaning methods for Maxwell's equations.

Figures (15)

• Figure 1: Part of the two-dimensional grid showing the reconstructed values from the four neighboring zones at the vertex point $\mathcal{C}=\left( i+\frac{1}{2}, j+\frac{1}{2} \right)$.
• Figure 2: \ref{['test:1d_bw']}: Plots of ion density $\rho_I$ for Larmor radius $0.1$ on $10000$ cells and Larmor radius $0.001$ on $50000$ cells.
• Figure 3: \ref{['test:1d_sol']}: Plots of ion density using the O2IMEX-MultiD and O2EXP-MultiD schemes with $1500$ cells.
• Figure 4: \ref{['test:2d_ot6']}: Plots of total density, total pressure and $|\mathbf{B}|$ with $400\times 400$ cells at time $t=3.14$.
• Figure 5: \ref{['test:2d_ot6']}: Plot of pressure cut along $y = 1.9634$ for Orszag-Tang test compared with for all schemes.

Theorems & Definitions (7)

• proposition thmcounterproposition: see kumar2012entropy
• theorem 1
• theorem 2: Divergence evolution for the explicit scheme
• proof
• remark thmcounterremark
• theorem 3: Divergence errors for the IMEX scheme
• proof