On the correction of conserved variables for numerical RMHD with staggered constrained transport

TL;DR

This paper evaluates correction algorithms for conserved variables in numerical relativistic magnetohydrodynamics, demonstrating that relativistic correction methods significantly improve accuracy and robustness in highly magnetized flow simulations.

Contribution

It introduces and compares non-relativistic and relativistic correction algorithms, showing the superiority of relativistic approaches in highly magnetized relativistic flows.

Findings

Relativistic correction algorithms pass high-magnetization tests successfully.

Non-relativistic corrections fail to pass the test at any resolution.

The CA2' algorithm is robust across all tested resolutions.

Abstract

Despite the success of the combination of conservative schemes and staggered constrained transport algorithms in the last fifteen years, the accurate description of highly magnetized, relativistic flows with strong shocks represents still a challenge in numerical RMHD. The present paper focusses in the accuracy and robustness of several correction algorithms for the conserved variables, which has become a crucial ingredient in the numerical simulation of problems where the magnetic pressure dominates over the thermal pressure by more than two orders of magnitude. Two versions of non-relativistic and fully relativistic corrections have been tested and compared using a magnetized cylindrical explosion with high magnetization ($ \ge 10^4$) as test. In the non-relativistic corrections, the total energy is corrected for the difference in the classical magnetic energy term between the average of the staggered fields and the conservative ones, before (CA1) and after (CA1') recovering the primitive variables. These corrections are unable to pass the test at any numerical resolution. The two relativistic approaches (CA2 and CA2'), correcting also the magnetic terms depending on the flow speed in both the momentum and the total energy, reveal as much more robust. These algorithms pass the test succesfully and with very small deviations of the energy conservation ($\le 10^{-4}$), and very low values of the total momentum ($\le 10^{-8}$). In particular, the algorithm CA2' (that corrects the conserved variables after recovering the primitive variables) passes the test at all resolutions. The numerical code used to run all the test cases is briefly described.