Second order divergence constraint preserving schemes for two-fluid relativistic plasma flow equations

TL;DR

This work develops second-order, co-located schemes for two-fluid relativistic plasma flows that preserve electromagnetic divergence constraints at the discrete level. By coupling entropy-stable fluid discretizations with a vertex-based, multidimensional Riemann solver for Maxwell's equations and MinMod diagonal corner reconstruction, the authors achieve robust preservation of $\nabla\cdot\mathbf{B}=0$ and $\nabla\cdot\mathbf{E}=\rho_c$ while maintaining entropy stability. The paper provides explicit and IMEX time-stepping schemes and proves discrete divergence preservation, corroborated by extensive 1D and 2D tests including Brio–Wu, Orszag–Tang, blast, and GEM reconnection problems. The results demonstrate that the proposed approach matches or exceeds PHM-based methods in divergence control while delivering second-order accuracy and reliable wave-dispersion handling, making it suitable for high-fidelity simulations of relativistic two-fluid plasmas. Overall, the framework offers a practical, accurate, and divergence-respecting tool for simulating coupled fluid-Maxwell dynamics in astrophysical and laboratory plasmas.

Abstract

Two-fluid relativistic plasma flow equations combine the equations of relativistic hydrodynamics with Maxwell's equations for electromagnetic fields, which involve divergence constraints for the magnetic and electric fields. When developing numerical schemes for the model, the divergence constraints are ignored, or Maxwell's equations are reformulated as Perfectly Hyperbolic Maxwell's (PHM) equations by introducing additional equations for correction potentials. In the latter case, the divergence constraints are preserved only as the limiting case. In this article, we present second-order numerical schemes that preserve the divergence constraint for electric and magnetic fields at the discrete level. The schemes are based on using a multidimensional Riemann solver at the vertices of the cells to define the numerical fluxes on the edges. The second-order accuracy is obtained by reconstructing the electromagnetic fields at the corners using a MinMod limiter. The discretization of Maxwell's equations can be combined with any consistent and stable discretization of the fluid parts. In particular, we consider entropy-stable schemes for the fluid part. The resulting schemes are second-order accurate, entropy stable, and preserve the divergence constraints of the electromagnetic fields. We use explicit and IMEX-based time discretizations. We then test these schemes using several one- and two-dimensional test cases. We also compare the divergence constraint errors of the proposed schemes with schemes having no divergence constraints treatment and schemes based on the PHM-based divergence cleaning.

Figures (16)

• Figure 1: Stencils for multidimensional Riemann solver.
• Figure 2: Stencil of second order reconstruction at the vertex $\left( i+\frac{1}{2}, j+\frac{1}{2} \right)$.
• Figure 3: Numerical results for \ref{['test:1d_brio']} using different schemes and grid resolutions.
• Figure 4: \ref{['test:1d_sheet']}: Comparison of the $B_y$ profile for explicit O2EXP-MultiD and IMEX scheme O2IMEX-MultiD using $400$ cells.
• Figure 5: \ref{['test:smooth_2d']}: Evolution of the divergence constraints errors of magnetic and electric fields for different schemes using $100 \times 100$ cells.

Theorems & Definitions (10)

• Theorem 3.1
• Remark 3.1
• Theorem 3.2: Evolution of the divergence of magnetic field
• proof
• Theorem 3.3: Evolution of the divergence of electric field
• proof
• Proposition 4.1: Evolution of the divergence constraint for the explicit scheme O2EXP-MultiD
• proof
• Proposition 4.2: Evolution of the divergence constraint for the explicit scheme O2IMEX-MultiD
• proof