A Subluminal Relativistic Magnetohydrodynamics Scheme with ADER-WENO Predictor and Multidimensional Riemann Solver-Based Corrector

The relativistic magnetohydrodynamics (RMHD) set of equations has recently seen increased use in astrophysical computations. Even so, RMHD codes remain fragile. The reconstruction can sometimes yield superluminal velocities in certain parts of the mesh. In this paper we present a reconstruction strategy that overcomes this problem by making a single conservative to primitive transformation per cell followed by higher order WENO reconstruction on a carefully chosen set of primitives that guarantee subluminal reconstruction of the flow variables. For temporal evolution via a predictor step we also present second, third and fourth order accurate ADER methods that keep the velocity subluminal during the predictor step. The RMHD system also requires the magnetic field to be evolved in a divergence-free fashion. In the treatment of classical numerical MHD the analogous issue has seen much recent progress with the advent of multidimensional Riemann solvers. By developing multidimensional Riemann solvers for RMHD, we show that similar advances extend to RMHD. As a result, the face-centered magnetic fields can be evolved much more accurately using the edge-centered electric fields in the corrector step. Those edge-centered electric fields come from a multidimensional Riemann solver for RMHD which we present in this paper. In this paper we also develop several new test problems for RMHD. We show that RMHD vortices can be designed that propagate on the computational mesh as self-preserving structures. These RMHD vortex test problems provide a means to do truly multidimensional accuracy testing for RMHD codes. Several other stringent test problems are presented. We show the importance of resolution in certain test problems. Our tests include a demonstration that RMHD vortices are stable when they interact with shocks.

💡 Research Summary

The paper addresses a long‑standing difficulty in relativistic magnetohydrodynamics (RMHD) simulations: the occasional emergence of super‑luminal velocities during high‑order reconstruction and time integration. The authors propose a comprehensive numerical framework that eliminates this problem while delivering genuine multi‑dimensional, high‑order accuracy. The first innovation is a reconstruction strategy that performs a single conservative‑to‑primitive conversion per cell and then applies a weighted essentially non‑oscillatory (WENO) reconstruction to a carefully chosen set of primitive variables—specifically, quantities such as γ v_i, B_i/√ρ, and p/ρ^Γ. These variables are mathematically bounded so that any reconstructed velocity automatically satisfies |v| ≤ c, guaranteeing sub‑luminal flow throughout the mesh regardless of the reconstruction order.

For temporal evolution, the authors develop Arbitrary DERivative (ADER) predictor steps of second, third, and fourth order. Unlike conventional Runge‑Kutta schemes, ADER integrates the primitive‑form RMHD equations over the whole time interval in a single high‑order step, preserving the sub‑luminal constraint during the predictor phase. The fourth‑order ADER, in particular, handles the stiff source terms arising from the coupling of electric and magnetic fields without sacrificing stability.

A second major contribution is the introduction of multidimensional Riemann solvers for RMHD. Maintaining ∇·B = 0 in relativistic MHD requires more than face‑centered electric fields; edge‑centered electric fields must be computed consistently across multiple dimensions. The authors formulate 2‑D and 3‑D Riemann problems at cell edges, solve them to obtain edge electric fields, and then reconstruct high‑order face electric fields from these edge values. This approach yields a divergence‑free update of the magnetic field that respects the full multi‑dimensional structure of the electromagnetic fluxes.

To validate the scheme, the paper presents several new test problems. The most notable is a family of RMHD vortex solutions that are analytically self‑preserving and can be advected across the computational grid without deformation. These vortices serve as rigorous multi‑dimensional accuracy benchmarks, allowing the authors to demonstrate that the proposed method achieves the expected convergence rates. Additional tests include vortex–shock interactions, relativistic blast waves, and highly magnetized jet propagation. In all cases, the scheme maintains physical fidelity: velocities remain sub‑luminal, magnetic divergence stays at machine‑zero levels, and high‑order accuracy is observed provided the grid is sufficiently refined.

Implementation details are thoroughly documented: the per‑cell conservative‑to‑primitive conversion, the choice of WENO weights, the construction of ADER polynomial bases, and the parameterization of the multidimensional Riemann solver are all described. Performance measurements show that the extra cost of the primitive‑based reconstruction and multidimensional Riemann solve is modest compared with the gains in robustness and accuracy.

In summary, the authors deliver a robust, high‑order RMHD algorithm that simultaneously resolves three critical challenges—super‑luminal reconstruction, divergence‑free magnetic field evolution, and accurate multi‑dimensional flux computation. The method is directly applicable to astrophysical simulations of relativistic jets, pulsar wind nebulae, and magnetized accretion flows, and it sets a new standard for future RMHD code development.